Research ArticleLong Time Behavior for a System of Differential Equations withNon-Lipschitzian Nonlinearities
Nasser-Eddine Tatar
Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia
Correspondence should be addressed to Nasser-Eddine Tatar tatarnkfupmedusa
Received 10 May 2014 Revised 7 September 2014 Accepted 8 September 2014 Published 14 September 2014
Academic Editor Ozgur Kisi
Copyright copy 2014 Nasser-Eddine TatarThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider a general system of nonlinear ordinary differential equations of first orderThe nonlinearities involve distributed delaysin addition to the states In turn the distributed delays involve nonlinear functions of the different variables and states An explicitbound for solutions is obtained under some rather reasonable conditions Several special cases of this systemmay be found in neuralnetwork theory As a direct application of our result it is shown how to obtain global existence and more importantly convergenceto zero at an exponential rate in a certain norm All these nonlinearities (including the activation functions) may be non-Lipschitzand unbounded
1 Introduction
Of concern is the following system
1199091015840
119894(119905) = minus119886
119894(119905) 119909119894(119905)
+
119898
sum
119895=1
119891119894119895(119905 119909119895(119905) int
119905
minusinfin
119870119894119895(119905 119904 119909
119895(119904)) 119889119904) + 119888
119894(119905)
(1)
with continuous data 119909119895(119905) = 119909
119894119895are nonlinear continuous functions This is a
general nonlinear version of several systems that arise inmany applications (see [1ndash9] and Section 4 below)
The literature is very rich of works on the asymptoticbehavior of solutions for special cases of system (1) (seefor instance [10ndash19]) Here the integral terms representsome kind of distributed delays but discrete delays may berecovered as well by considering delta Dirac distributionsDifferent sufficient conditions on the coefficients the func-tions and the kernels have been established ensuring con-vergence to equilibrium or (uniform global and asymptotic)stability In applications it is important to have global asymp-totic stability at a very rapid rate like the exponential rate
Roughly speaking it has been assumed that the coefficients119886119894(119905) must dominate the coefficients of some ldquobadrdquo similar
terms that appear in the estimations For the nonlinearities(activation functions) the first assumptions of boundednessmonotonicity and differentiability have been all weakenedto a Lipschitz condition According to [8 20] and otherreferences even this condition needs to be weakened furtherUnfortunately we can find only few papers on continuous butnot Lipschitz continuous activation functions Assumptionslike partially Lipschitz and linear growth 120572-inverse Holdercontinuous or inverse Lipschitz non-Lipschitz but boundedwere used (see [16 21 22])
For Holder continuous activation functions we referthe reader to [23] where exponential stability was provedunder some boundedness and monotonicity conditions onthe activation functions and the coefficients form a Lyapunovdiagonally stable matrix (see also [24 25] for other resultswithout these conditions)
There are however a good number of papers dealing withdiscontinuous activation functions under certain strongerconditions like 119872-Matrix the LMI condition (linear matrixinequality) and some extra conditions on the matricesand growth conditions on the activation functions (see[20 26ndash37]) Global asymptotic stability of periodic solutionshave been investigated for instance in [38 39]
Hindawi Publishing CorporationAdvances in Artificial Neural SystemsVolume 2014 Article ID 252674 7 pageshttpdxdoiorg1011552014252674
2 Advances in Artificial Neural Systems
Here we assume that the functions 119891119894119895and 119870
119894119895are (or
bounded by) continuous monotone nondecreasing functionsthat are not necessarily Lipschitz continuous and they maybe unbounded (like power type functions with powers biggerthan one) We prove that for sufficiently small initial datasolutions decay to zero exponentially
The local existence and global existence are standard seethe Gronwall-type Lemma 1 below and the estimation in ourtheorem However the uniqueness of the equilibrium is notan issue here (even in case of constant coefficients) as we areconcerned with convergence to zero rather than stability ofequilibrium
After the Preliminaries section where we present ourmain hypotheses and the main lemma used in our proofwe state and prove the convergence result in Section 3 Thesection is ended by some corollaries and important remarksIn the last section we give an application where this type ofsystems (or special cases of it) appears in real world problems
Remark 6 We have assumed that 120572119894119895and 120573
119894119895are greater than
one but the case when they are smaller than one may betreated similarlyWhen their sum is smaller than one we haveglobal existence without adding any extra condition
Remark 7 The decay rate obtained in Corollary 5 is to becompared with the one in the theorem (case (b)) It appearsthat the estimation in Corollary 5 holds for more generalinitial data (not as small as the ones in case (b)) Howeverthe decay rate is smaller than the one in (b) besides assumingthat int1199050119886(119904)119889119904 minus 119871119905 rarr infin as 119905 rarr infin
Remark 8 If we consider the following new functions thenthe monotonicity condition and the order imposed in thetheorem may be dropped
1206011(119905) = max
0le119904le119905
1198921(119904)
120601119896(119905) = max
0le119904le119905
119892119896(119904)
120601119896minus1
(119904)
120601119896minus1
(119905)
(29)
and 120595(119905) = 120601119896(119905)120601119896minus1
(119905)
4 Application
(Artificial)Neural networks are built in an attempt to performdifferent tasks just as the nervous system Typically a neuralnetwork consists of several layers (input layer hidden layersand output layer) Each layer contains one or more cells(neurons) with many connections between them The cellsin one layer receive inputs from the previous layer makesome transformations and send the results to the cells of thesubsequent layer
One may encounter neural networks in many fields suchas control patternmatching settlement of structures classifi-cation of soil supply chain management engineering designmarket segmentation product analysis market developmentforecasting signature verification bond rating recognitionof diseases robust pattern detection text mining price fore-cast botanical classification and scheduling optimization
Neural networks not only can perform many of the tasksa traditional computer can do but also excel in for instanceclassifying incomplete or noisy data predicting future eventsand generalizing
The system (1) is a general version of simpler systems thatappear in neural network theory [1ndash9] like
It is well established by now that (for constant coefficientsand constant 119888
119894(119905)) solutions converge in an exponential
manner to the equilibrium Notice that zero in our case isnot an equilibrium This equilibrium exists and is unique incase of Lipschitz continuity of the activation functions Inour case the system is much more general and the activationfunctions as well as the nonlinearities are not necessarily Lip-schitz continuous However in case of Lipschitz continuityand existence of a unique equilibrium we expect to haveexponential stability using the standard techniques at leastwhen we start away from zero
may be taken as power functions see alsoCorollary 5) our theorem gives sufficient conditions guaran-teeing the estimation
119909 (119905) le 120596119899(119905) exp [minusint
119905
0
119886 (119904) 119889119904] 0 le 119905 lt 1205730 (33)
Then Corollaries 3 and 4 provide practical situations wherewe have global existence and decay to zero at an exponentialrate
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful for the financial support and thefacilities provided by King Fahd University of Petroleum andMinerals through Grant no IN111052
References
[1] J Cao K Yuan and H-X Li ldquoGlobal asymptotical stabilityof recurrent neural networks with multiple discrete delays anddistributed delaysrdquo IEEE Transactions on Neural Networks vol17 no 6 pp 1646ndash1651 2006
[2] B Crespi ldquoStorage capacity of non-monotonic neuronsrdquoNeuralNetworks vol 12 no 10 pp 1377ndash1389 1999
6 Advances in Artificial Neural Systems
[3] G de Sandre M Forti P Nistri and A Premoli ldquoDynamicalanalysis of full-range cellular neural networks by exploiting dif-ferential variational inequalitiesrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 54 no 8 pp 1736ndash1749 2007
[4] C Feng and R Plamondon ldquoOn the stability analysis of delayedneural networks systemsrdquo Neural Networks vol 14 no 9 pp1181ndash1188 2001
[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982
[6] J J Hopfield and D W Tank ldquoComputing with neural circuitsa modelrdquo Science vol 233 no 4764 pp 625ndash633 1986
[7] J I Inoue ldquoRetrieval phase diagrams of non-monotonic Hop-field networksrdquo Journal of Physics A Mathematical and Generalvol 29 no 16 pp 4815ndash4826 1996
[8] B Kosko Neural Network and Fuzzy SystemmdashA DynamicalSystem Approach toMachine Intelligence Prentice-Hall of IndiaNew Delhi India 1991
[9] H-F Yanai and S-I Amari ldquoAuto-associative memory withtwo-stage dynamics of nonmonotonic neuronsrdquo IEEE Transac-tions on Neural Networks vol 7 no 4 pp 803ndash815 1996
[10] X Liu and N Jiang ldquoRobust stability analysis of generalizedneural networks with multiple discrete delays and multipledistributed delaysrdquo Neurocomputing vol 72 no 7ndash9 pp 1789ndash1796 2009
[11] SMohamadKGopalsamy andHAkca ldquoExponential stabilityof artificial neural networks with distributed delays and largeimpulsesrdquo Nonlinear Analysis Real World Applications vol 9no 3 pp 872ndash888 2008
[12] J Park ldquoOn global stability criterion for neural networks withdiscrete and distributed delaysrdquo Chaos Solitons and Fractalsvol 30 no 4 pp 897ndash902 2006
[13] J Park ldquoOn global stability criterion of neural networks withcontinuously distributed delaysrdquo Chaos Solitons and Fractalsvol 37 no 2 pp 444ndash449 2008
[14] Z Qiang M A Run-Nian and X Jin ldquoGlobal exponentialconvergence analysis of Hopfield neural networks with con-tinuously distributed delaysrdquo Communications in TheoreticalPhysics vol 39 no 3 pp 381ndash384 2003
[15] Y Wang W Xiong Q Zhou B Xiao and Y Yu ldquoGlobal expo-nential stability of cellular neural networks with continuouslydistributed delays and impulsesrdquo Physics Letters A vol 350 no1-2 pp 89ndash95 2006
[16] H Wu ldquoGlobal exponential stability of Hopfield neural net-works with delays and inverse Lipschitz neuron activationsrdquoNonlinear Analysis Real World Applications vol 10 no 4 pp2297ndash2306 2009
[17] Q Zhang X P Wei and J Xu ldquoGlobal exponential stability ofHopfield neural networkswith continuously distributed delaysrdquoPhysics Letters A vol 315 no 6 pp 431ndash436 2003
[19] J Zhou S Li and Z Yang ldquoGlobal exponential stability ofHopfield neural networks with distributed delaysrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1513ndash15202009
[20] R Gavalda and H T Siegelmann ldquoDiscontinuities in recurrentneural networksrdquo Neural Computation vol 11 no 3 pp 715ndash745 1999
[21] H Wu F Tao L Qin R Shi and L He ldquoRobust exponentialstability for interval neural networks with delays and non-Lipschitz activation functionsrdquoNonlinearDynamics vol 66 no4 pp 479ndash487 2011
[22] H Wu and X Xue ldquoStability analysis for neural networks withinverse Lipschitzian neuron activations and impulsesrdquo AppliedMathematical Modelling vol 32 no 11 pp 2347ndash2359 2008
[23] M Forti M Grazzini P Nistri and L Pancioni ldquoGeneralizedLyapunov approach for convergence of neural networks withdiscontinuous or non-LIPschitz activationsrdquo Physica D Nonlin-ear Phenomena vol 214 no 1 pp 88ndash99 2006
[25] N-E Tatar ldquoControl of systems with Holder continuousfunctions in the distributed delaysrdquo Carpathian Journal ofMathematics vol 30 no 1 pp 123ndash128 2014
[26] G Bao and Z Zeng ldquoAnalysis and design of associative mem-ories based on recurrent neural network with discontinuousactivation functionsrdquoNeurocomputing vol 77 no 1 pp 101ndash1072012
[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003
[28] YHuang H Zhang and ZWang ldquoDynamical stability analysisof multiple equilibrium points in time-varying delayed recur-rent neural networks with discontinuous activation functionsrdquoNeurocomputing vol 91 pp 21ndash28 2012
[29] L Li and L Huang ldquoDynamical behaviors of a class of recur-rent neural networks with discontinuous neuron activationsrdquoApplied Mathematical Modelling vol 33 no 12 pp 4326ndash43362009
[30] L Li and L Huang ldquoGlobal asymptotic stability of delayedneural networks with discontinuous neuron activationsrdquo Neu-rocomputing vol 72 no 16-18 pp 3726ndash3733 2009
[31] Y Li and H Wu ldquoGlobal stability analysis for periodic solutionin discontinuous neural networks with nonlinear growth acti-vationsrdquo Advances in Difference Equations vol 2009 Article ID798685 14 pages 2009
[32] X Liu and J Cao ldquoRobust state estimation for neural networkswith discontinuous activationsrdquo IEEE Transactions on SystemsMan and Cybernetics B Cybernetics vol 40 no 6 pp 1425ndash1437 2010
[33] J Liu X Liu and W-C Xie ldquoGlobal convergence of neuralnetworks with mixed time-varying delays and discontinuousneuron activationsrdquo Information Sciences vol 183 pp 92ndash1052012
[34] S Qin and X Xue ldquoGlobal exponential stability and global con-vergence in finite time of neural networks with discontinuousactivationsrdquoNeural Processing Letters vol 29 no 3 pp 189ndash2042009
[35] J Wang L Huang and Z Guo ldquoGlobal asymptotic stabilityof neural networks with discontinuous activationsrdquo NeuralNetworks vol 22 no 7 pp 931ndash937 2009
[36] Z Wang L Huang Y Zuo and L Zhang ldquoGlobal robuststability of time-delay systems with discontinuous activationfunctions under polytopic parameter uncertaintiesrdquo Bulletin ofthe KoreanMathematical Society vol 47 no 1 pp 89ndash102 2010
[37] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009
Advances in Artificial Neural Systems 7
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
Here we assume that the functions 119891119894119895and 119870
119894119895are (or
bounded by) continuous monotone nondecreasing functionsthat are not necessarily Lipschitz continuous and they maybe unbounded (like power type functions with powers biggerthan one) We prove that for sufficiently small initial datasolutions decay to zero exponentially
The local existence and global existence are standard seethe Gronwall-type Lemma 1 below and the estimation in ourtheorem However the uniqueness of the equilibrium is notan issue here (even in case of constant coefficients) as we areconcerned with convergence to zero rather than stability ofequilibrium
After the Preliminaries section where we present ourmain hypotheses and the main lemma used in our proofwe state and prove the convergence result in Section 3 Thesection is ended by some corollaries and important remarksIn the last section we give an application where this type ofsystems (or special cases of it) appears in real world problems
Remark 6 We have assumed that 120572119894119895and 120573
119894119895are greater than
one but the case when they are smaller than one may betreated similarlyWhen their sum is smaller than one we haveglobal existence without adding any extra condition
Remark 7 The decay rate obtained in Corollary 5 is to becompared with the one in the theorem (case (b)) It appearsthat the estimation in Corollary 5 holds for more generalinitial data (not as small as the ones in case (b)) Howeverthe decay rate is smaller than the one in (b) besides assumingthat int1199050119886(119904)119889119904 minus 119871119905 rarr infin as 119905 rarr infin
Remark 8 If we consider the following new functions thenthe monotonicity condition and the order imposed in thetheorem may be dropped
1206011(119905) = max
0le119904le119905
1198921(119904)
120601119896(119905) = max
0le119904le119905
119892119896(119904)
120601119896minus1
(119904)
120601119896minus1
(119905)
(29)
and 120595(119905) = 120601119896(119905)120601119896minus1
(119905)
4 Application
(Artificial)Neural networks are built in an attempt to performdifferent tasks just as the nervous system Typically a neuralnetwork consists of several layers (input layer hidden layersand output layer) Each layer contains one or more cells(neurons) with many connections between them The cellsin one layer receive inputs from the previous layer makesome transformations and send the results to the cells of thesubsequent layer
One may encounter neural networks in many fields suchas control patternmatching settlement of structures classifi-cation of soil supply chain management engineering designmarket segmentation product analysis market developmentforecasting signature verification bond rating recognitionof diseases robust pattern detection text mining price fore-cast botanical classification and scheduling optimization
Neural networks not only can perform many of the tasksa traditional computer can do but also excel in for instanceclassifying incomplete or noisy data predicting future eventsand generalizing
The system (1) is a general version of simpler systems thatappear in neural network theory [1ndash9] like
It is well established by now that (for constant coefficientsand constant 119888
119894(119905)) solutions converge in an exponential
manner to the equilibrium Notice that zero in our case isnot an equilibrium This equilibrium exists and is unique incase of Lipschitz continuity of the activation functions Inour case the system is much more general and the activationfunctions as well as the nonlinearities are not necessarily Lip-schitz continuous However in case of Lipschitz continuityand existence of a unique equilibrium we expect to haveexponential stability using the standard techniques at leastwhen we start away from zero
may be taken as power functions see alsoCorollary 5) our theorem gives sufficient conditions guaran-teeing the estimation
119909 (119905) le 120596119899(119905) exp [minusint
119905
0
119886 (119904) 119889119904] 0 le 119905 lt 1205730 (33)
Then Corollaries 3 and 4 provide practical situations wherewe have global existence and decay to zero at an exponentialrate
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful for the financial support and thefacilities provided by King Fahd University of Petroleum andMinerals through Grant no IN111052
References
[1] J Cao K Yuan and H-X Li ldquoGlobal asymptotical stabilityof recurrent neural networks with multiple discrete delays anddistributed delaysrdquo IEEE Transactions on Neural Networks vol17 no 6 pp 1646ndash1651 2006
[2] B Crespi ldquoStorage capacity of non-monotonic neuronsrdquoNeuralNetworks vol 12 no 10 pp 1377ndash1389 1999
6 Advances in Artificial Neural Systems
[3] G de Sandre M Forti P Nistri and A Premoli ldquoDynamicalanalysis of full-range cellular neural networks by exploiting dif-ferential variational inequalitiesrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 54 no 8 pp 1736ndash1749 2007
[4] C Feng and R Plamondon ldquoOn the stability analysis of delayedneural networks systemsrdquo Neural Networks vol 14 no 9 pp1181ndash1188 2001
[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982
[6] J J Hopfield and D W Tank ldquoComputing with neural circuitsa modelrdquo Science vol 233 no 4764 pp 625ndash633 1986
[7] J I Inoue ldquoRetrieval phase diagrams of non-monotonic Hop-field networksrdquo Journal of Physics A Mathematical and Generalvol 29 no 16 pp 4815ndash4826 1996
[8] B Kosko Neural Network and Fuzzy SystemmdashA DynamicalSystem Approach toMachine Intelligence Prentice-Hall of IndiaNew Delhi India 1991
[9] H-F Yanai and S-I Amari ldquoAuto-associative memory withtwo-stage dynamics of nonmonotonic neuronsrdquo IEEE Transac-tions on Neural Networks vol 7 no 4 pp 803ndash815 1996
[10] X Liu and N Jiang ldquoRobust stability analysis of generalizedneural networks with multiple discrete delays and multipledistributed delaysrdquo Neurocomputing vol 72 no 7ndash9 pp 1789ndash1796 2009
[11] SMohamadKGopalsamy andHAkca ldquoExponential stabilityof artificial neural networks with distributed delays and largeimpulsesrdquo Nonlinear Analysis Real World Applications vol 9no 3 pp 872ndash888 2008
[12] J Park ldquoOn global stability criterion for neural networks withdiscrete and distributed delaysrdquo Chaos Solitons and Fractalsvol 30 no 4 pp 897ndash902 2006
[13] J Park ldquoOn global stability criterion of neural networks withcontinuously distributed delaysrdquo Chaos Solitons and Fractalsvol 37 no 2 pp 444ndash449 2008
[14] Z Qiang M A Run-Nian and X Jin ldquoGlobal exponentialconvergence analysis of Hopfield neural networks with con-tinuously distributed delaysrdquo Communications in TheoreticalPhysics vol 39 no 3 pp 381ndash384 2003
[15] Y Wang W Xiong Q Zhou B Xiao and Y Yu ldquoGlobal expo-nential stability of cellular neural networks with continuouslydistributed delays and impulsesrdquo Physics Letters A vol 350 no1-2 pp 89ndash95 2006
[16] H Wu ldquoGlobal exponential stability of Hopfield neural net-works with delays and inverse Lipschitz neuron activationsrdquoNonlinear Analysis Real World Applications vol 10 no 4 pp2297ndash2306 2009
[17] Q Zhang X P Wei and J Xu ldquoGlobal exponential stability ofHopfield neural networkswith continuously distributed delaysrdquoPhysics Letters A vol 315 no 6 pp 431ndash436 2003
[19] J Zhou S Li and Z Yang ldquoGlobal exponential stability ofHopfield neural networks with distributed delaysrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1513ndash15202009
[20] R Gavalda and H T Siegelmann ldquoDiscontinuities in recurrentneural networksrdquo Neural Computation vol 11 no 3 pp 715ndash745 1999
[21] H Wu F Tao L Qin R Shi and L He ldquoRobust exponentialstability for interval neural networks with delays and non-Lipschitz activation functionsrdquoNonlinearDynamics vol 66 no4 pp 479ndash487 2011
[22] H Wu and X Xue ldquoStability analysis for neural networks withinverse Lipschitzian neuron activations and impulsesrdquo AppliedMathematical Modelling vol 32 no 11 pp 2347ndash2359 2008
[23] M Forti M Grazzini P Nistri and L Pancioni ldquoGeneralizedLyapunov approach for convergence of neural networks withdiscontinuous or non-LIPschitz activationsrdquo Physica D Nonlin-ear Phenomena vol 214 no 1 pp 88ndash99 2006
[25] N-E Tatar ldquoControl of systems with Holder continuousfunctions in the distributed delaysrdquo Carpathian Journal ofMathematics vol 30 no 1 pp 123ndash128 2014
[26] G Bao and Z Zeng ldquoAnalysis and design of associative mem-ories based on recurrent neural network with discontinuousactivation functionsrdquoNeurocomputing vol 77 no 1 pp 101ndash1072012
[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003
[28] YHuang H Zhang and ZWang ldquoDynamical stability analysisof multiple equilibrium points in time-varying delayed recur-rent neural networks with discontinuous activation functionsrdquoNeurocomputing vol 91 pp 21ndash28 2012
[29] L Li and L Huang ldquoDynamical behaviors of a class of recur-rent neural networks with discontinuous neuron activationsrdquoApplied Mathematical Modelling vol 33 no 12 pp 4326ndash43362009
[30] L Li and L Huang ldquoGlobal asymptotic stability of delayedneural networks with discontinuous neuron activationsrdquo Neu-rocomputing vol 72 no 16-18 pp 3726ndash3733 2009
[31] Y Li and H Wu ldquoGlobal stability analysis for periodic solutionin discontinuous neural networks with nonlinear growth acti-vationsrdquo Advances in Difference Equations vol 2009 Article ID798685 14 pages 2009
[32] X Liu and J Cao ldquoRobust state estimation for neural networkswith discontinuous activationsrdquo IEEE Transactions on SystemsMan and Cybernetics B Cybernetics vol 40 no 6 pp 1425ndash1437 2010
[33] J Liu X Liu and W-C Xie ldquoGlobal convergence of neuralnetworks with mixed time-varying delays and discontinuousneuron activationsrdquo Information Sciences vol 183 pp 92ndash1052012
[34] S Qin and X Xue ldquoGlobal exponential stability and global con-vergence in finite time of neural networks with discontinuousactivationsrdquoNeural Processing Letters vol 29 no 3 pp 189ndash2042009
[35] J Wang L Huang and Z Guo ldquoGlobal asymptotic stabilityof neural networks with discontinuous activationsrdquo NeuralNetworks vol 22 no 7 pp 931ndash937 2009
[36] Z Wang L Huang Y Zuo and L Zhang ldquoGlobal robuststability of time-delay systems with discontinuous activationfunctions under polytopic parameter uncertaintiesrdquo Bulletin ofthe KoreanMathematical Society vol 47 no 1 pp 89ndash102 2010
[37] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009
Advances in Artificial Neural Systems 7
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
Remark 6 We have assumed that 120572119894119895and 120573
119894119895are greater than
one but the case when they are smaller than one may betreated similarlyWhen their sum is smaller than one we haveglobal existence without adding any extra condition
Remark 7 The decay rate obtained in Corollary 5 is to becompared with the one in the theorem (case (b)) It appearsthat the estimation in Corollary 5 holds for more generalinitial data (not as small as the ones in case (b)) Howeverthe decay rate is smaller than the one in (b) besides assumingthat int1199050119886(119904)119889119904 minus 119871119905 rarr infin as 119905 rarr infin
Remark 8 If we consider the following new functions thenthe monotonicity condition and the order imposed in thetheorem may be dropped
1206011(119905) = max
0le119904le119905
1198921(119904)
120601119896(119905) = max
0le119904le119905
119892119896(119904)
120601119896minus1
(119904)
120601119896minus1
(119905)
(29)
and 120595(119905) = 120601119896(119905)120601119896minus1
(119905)
4 Application
(Artificial)Neural networks are built in an attempt to performdifferent tasks just as the nervous system Typically a neuralnetwork consists of several layers (input layer hidden layersand output layer) Each layer contains one or more cells(neurons) with many connections between them The cellsin one layer receive inputs from the previous layer makesome transformations and send the results to the cells of thesubsequent layer
One may encounter neural networks in many fields suchas control patternmatching settlement of structures classifi-cation of soil supply chain management engineering designmarket segmentation product analysis market developmentforecasting signature verification bond rating recognitionof diseases robust pattern detection text mining price fore-cast botanical classification and scheduling optimization
Neural networks not only can perform many of the tasksa traditional computer can do but also excel in for instanceclassifying incomplete or noisy data predicting future eventsand generalizing
The system (1) is a general version of simpler systems thatappear in neural network theory [1ndash9] like
It is well established by now that (for constant coefficientsand constant 119888
119894(119905)) solutions converge in an exponential
manner to the equilibrium Notice that zero in our case isnot an equilibrium This equilibrium exists and is unique incase of Lipschitz continuity of the activation functions Inour case the system is much more general and the activationfunctions as well as the nonlinearities are not necessarily Lip-schitz continuous However in case of Lipschitz continuityand existence of a unique equilibrium we expect to haveexponential stability using the standard techniques at leastwhen we start away from zero
may be taken as power functions see alsoCorollary 5) our theorem gives sufficient conditions guaran-teeing the estimation
119909 (119905) le 120596119899(119905) exp [minusint
119905
0
119886 (119904) 119889119904] 0 le 119905 lt 1205730 (33)
Then Corollaries 3 and 4 provide practical situations wherewe have global existence and decay to zero at an exponentialrate
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful for the financial support and thefacilities provided by King Fahd University of Petroleum andMinerals through Grant no IN111052
References
[1] J Cao K Yuan and H-X Li ldquoGlobal asymptotical stabilityof recurrent neural networks with multiple discrete delays anddistributed delaysrdquo IEEE Transactions on Neural Networks vol17 no 6 pp 1646ndash1651 2006
[2] B Crespi ldquoStorage capacity of non-monotonic neuronsrdquoNeuralNetworks vol 12 no 10 pp 1377ndash1389 1999
6 Advances in Artificial Neural Systems
[3] G de Sandre M Forti P Nistri and A Premoli ldquoDynamicalanalysis of full-range cellular neural networks by exploiting dif-ferential variational inequalitiesrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 54 no 8 pp 1736ndash1749 2007
[4] C Feng and R Plamondon ldquoOn the stability analysis of delayedneural networks systemsrdquo Neural Networks vol 14 no 9 pp1181ndash1188 2001
[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982
[6] J J Hopfield and D W Tank ldquoComputing with neural circuitsa modelrdquo Science vol 233 no 4764 pp 625ndash633 1986
[7] J I Inoue ldquoRetrieval phase diagrams of non-monotonic Hop-field networksrdquo Journal of Physics A Mathematical and Generalvol 29 no 16 pp 4815ndash4826 1996
[8] B Kosko Neural Network and Fuzzy SystemmdashA DynamicalSystem Approach toMachine Intelligence Prentice-Hall of IndiaNew Delhi India 1991
[9] H-F Yanai and S-I Amari ldquoAuto-associative memory withtwo-stage dynamics of nonmonotonic neuronsrdquo IEEE Transac-tions on Neural Networks vol 7 no 4 pp 803ndash815 1996
[10] X Liu and N Jiang ldquoRobust stability analysis of generalizedneural networks with multiple discrete delays and multipledistributed delaysrdquo Neurocomputing vol 72 no 7ndash9 pp 1789ndash1796 2009
[11] SMohamadKGopalsamy andHAkca ldquoExponential stabilityof artificial neural networks with distributed delays and largeimpulsesrdquo Nonlinear Analysis Real World Applications vol 9no 3 pp 872ndash888 2008
[12] J Park ldquoOn global stability criterion for neural networks withdiscrete and distributed delaysrdquo Chaos Solitons and Fractalsvol 30 no 4 pp 897ndash902 2006
[13] J Park ldquoOn global stability criterion of neural networks withcontinuously distributed delaysrdquo Chaos Solitons and Fractalsvol 37 no 2 pp 444ndash449 2008
[14] Z Qiang M A Run-Nian and X Jin ldquoGlobal exponentialconvergence analysis of Hopfield neural networks with con-tinuously distributed delaysrdquo Communications in TheoreticalPhysics vol 39 no 3 pp 381ndash384 2003
[15] Y Wang W Xiong Q Zhou B Xiao and Y Yu ldquoGlobal expo-nential stability of cellular neural networks with continuouslydistributed delays and impulsesrdquo Physics Letters A vol 350 no1-2 pp 89ndash95 2006
[16] H Wu ldquoGlobal exponential stability of Hopfield neural net-works with delays and inverse Lipschitz neuron activationsrdquoNonlinear Analysis Real World Applications vol 10 no 4 pp2297ndash2306 2009
[17] Q Zhang X P Wei and J Xu ldquoGlobal exponential stability ofHopfield neural networkswith continuously distributed delaysrdquoPhysics Letters A vol 315 no 6 pp 431ndash436 2003
[19] J Zhou S Li and Z Yang ldquoGlobal exponential stability ofHopfield neural networks with distributed delaysrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1513ndash15202009
[20] R Gavalda and H T Siegelmann ldquoDiscontinuities in recurrentneural networksrdquo Neural Computation vol 11 no 3 pp 715ndash745 1999
[21] H Wu F Tao L Qin R Shi and L He ldquoRobust exponentialstability for interval neural networks with delays and non-Lipschitz activation functionsrdquoNonlinearDynamics vol 66 no4 pp 479ndash487 2011
[22] H Wu and X Xue ldquoStability analysis for neural networks withinverse Lipschitzian neuron activations and impulsesrdquo AppliedMathematical Modelling vol 32 no 11 pp 2347ndash2359 2008
[23] M Forti M Grazzini P Nistri and L Pancioni ldquoGeneralizedLyapunov approach for convergence of neural networks withdiscontinuous or non-LIPschitz activationsrdquo Physica D Nonlin-ear Phenomena vol 214 no 1 pp 88ndash99 2006
[25] N-E Tatar ldquoControl of systems with Holder continuousfunctions in the distributed delaysrdquo Carpathian Journal ofMathematics vol 30 no 1 pp 123ndash128 2014
[26] G Bao and Z Zeng ldquoAnalysis and design of associative mem-ories based on recurrent neural network with discontinuousactivation functionsrdquoNeurocomputing vol 77 no 1 pp 101ndash1072012
[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003
[28] YHuang H Zhang and ZWang ldquoDynamical stability analysisof multiple equilibrium points in time-varying delayed recur-rent neural networks with discontinuous activation functionsrdquoNeurocomputing vol 91 pp 21ndash28 2012
[29] L Li and L Huang ldquoDynamical behaviors of a class of recur-rent neural networks with discontinuous neuron activationsrdquoApplied Mathematical Modelling vol 33 no 12 pp 4326ndash43362009
[30] L Li and L Huang ldquoGlobal asymptotic stability of delayedneural networks with discontinuous neuron activationsrdquo Neu-rocomputing vol 72 no 16-18 pp 3726ndash3733 2009
[31] Y Li and H Wu ldquoGlobal stability analysis for periodic solutionin discontinuous neural networks with nonlinear growth acti-vationsrdquo Advances in Difference Equations vol 2009 Article ID798685 14 pages 2009
[32] X Liu and J Cao ldquoRobust state estimation for neural networkswith discontinuous activationsrdquo IEEE Transactions on SystemsMan and Cybernetics B Cybernetics vol 40 no 6 pp 1425ndash1437 2010
[33] J Liu X Liu and W-C Xie ldquoGlobal convergence of neuralnetworks with mixed time-varying delays and discontinuousneuron activationsrdquo Information Sciences vol 183 pp 92ndash1052012
[34] S Qin and X Xue ldquoGlobal exponential stability and global con-vergence in finite time of neural networks with discontinuousactivationsrdquoNeural Processing Letters vol 29 no 3 pp 189ndash2042009
[35] J Wang L Huang and Z Guo ldquoGlobal asymptotic stabilityof neural networks with discontinuous activationsrdquo NeuralNetworks vol 22 no 7 pp 931ndash937 2009
[36] Z Wang L Huang Y Zuo and L Zhang ldquoGlobal robuststability of time-delay systems with discontinuous activationfunctions under polytopic parameter uncertaintiesrdquo Bulletin ofthe KoreanMathematical Society vol 47 no 1 pp 89ndash102 2010
[37] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009
Advances in Artificial Neural Systems 7
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
Remark 6 We have assumed that 120572119894119895and 120573
119894119895are greater than
one but the case when they are smaller than one may betreated similarlyWhen their sum is smaller than one we haveglobal existence without adding any extra condition
Remark 7 The decay rate obtained in Corollary 5 is to becompared with the one in the theorem (case (b)) It appearsthat the estimation in Corollary 5 holds for more generalinitial data (not as small as the ones in case (b)) Howeverthe decay rate is smaller than the one in (b) besides assumingthat int1199050119886(119904)119889119904 minus 119871119905 rarr infin as 119905 rarr infin
Remark 8 If we consider the following new functions thenthe monotonicity condition and the order imposed in thetheorem may be dropped
1206011(119905) = max
0le119904le119905
1198921(119904)
120601119896(119905) = max
0le119904le119905
119892119896(119904)
120601119896minus1
(119904)
120601119896minus1
(119905)
(29)
and 120595(119905) = 120601119896(119905)120601119896minus1
(119905)
4 Application
(Artificial)Neural networks are built in an attempt to performdifferent tasks just as the nervous system Typically a neuralnetwork consists of several layers (input layer hidden layersand output layer) Each layer contains one or more cells(neurons) with many connections between them The cellsin one layer receive inputs from the previous layer makesome transformations and send the results to the cells of thesubsequent layer
One may encounter neural networks in many fields suchas control patternmatching settlement of structures classifi-cation of soil supply chain management engineering designmarket segmentation product analysis market developmentforecasting signature verification bond rating recognitionof diseases robust pattern detection text mining price fore-cast botanical classification and scheduling optimization
Neural networks not only can perform many of the tasksa traditional computer can do but also excel in for instanceclassifying incomplete or noisy data predicting future eventsand generalizing
The system (1) is a general version of simpler systems thatappear in neural network theory [1ndash9] like
It is well established by now that (for constant coefficientsand constant 119888
119894(119905)) solutions converge in an exponential
manner to the equilibrium Notice that zero in our case isnot an equilibrium This equilibrium exists and is unique incase of Lipschitz continuity of the activation functions Inour case the system is much more general and the activationfunctions as well as the nonlinearities are not necessarily Lip-schitz continuous However in case of Lipschitz continuityand existence of a unique equilibrium we expect to haveexponential stability using the standard techniques at leastwhen we start away from zero
may be taken as power functions see alsoCorollary 5) our theorem gives sufficient conditions guaran-teeing the estimation
119909 (119905) le 120596119899(119905) exp [minusint
119905
0
119886 (119904) 119889119904] 0 le 119905 lt 1205730 (33)
Then Corollaries 3 and 4 provide practical situations wherewe have global existence and decay to zero at an exponentialrate
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful for the financial support and thefacilities provided by King Fahd University of Petroleum andMinerals through Grant no IN111052
References
[1] J Cao K Yuan and H-X Li ldquoGlobal asymptotical stabilityof recurrent neural networks with multiple discrete delays anddistributed delaysrdquo IEEE Transactions on Neural Networks vol17 no 6 pp 1646ndash1651 2006
[2] B Crespi ldquoStorage capacity of non-monotonic neuronsrdquoNeuralNetworks vol 12 no 10 pp 1377ndash1389 1999
6 Advances in Artificial Neural Systems
[3] G de Sandre M Forti P Nistri and A Premoli ldquoDynamicalanalysis of full-range cellular neural networks by exploiting dif-ferential variational inequalitiesrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 54 no 8 pp 1736ndash1749 2007
[4] C Feng and R Plamondon ldquoOn the stability analysis of delayedneural networks systemsrdquo Neural Networks vol 14 no 9 pp1181ndash1188 2001
[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982
[6] J J Hopfield and D W Tank ldquoComputing with neural circuitsa modelrdquo Science vol 233 no 4764 pp 625ndash633 1986
[7] J I Inoue ldquoRetrieval phase diagrams of non-monotonic Hop-field networksrdquo Journal of Physics A Mathematical and Generalvol 29 no 16 pp 4815ndash4826 1996
[8] B Kosko Neural Network and Fuzzy SystemmdashA DynamicalSystem Approach toMachine Intelligence Prentice-Hall of IndiaNew Delhi India 1991
[9] H-F Yanai and S-I Amari ldquoAuto-associative memory withtwo-stage dynamics of nonmonotonic neuronsrdquo IEEE Transac-tions on Neural Networks vol 7 no 4 pp 803ndash815 1996
[10] X Liu and N Jiang ldquoRobust stability analysis of generalizedneural networks with multiple discrete delays and multipledistributed delaysrdquo Neurocomputing vol 72 no 7ndash9 pp 1789ndash1796 2009
[11] SMohamadKGopalsamy andHAkca ldquoExponential stabilityof artificial neural networks with distributed delays and largeimpulsesrdquo Nonlinear Analysis Real World Applications vol 9no 3 pp 872ndash888 2008
[12] J Park ldquoOn global stability criterion for neural networks withdiscrete and distributed delaysrdquo Chaos Solitons and Fractalsvol 30 no 4 pp 897ndash902 2006
[13] J Park ldquoOn global stability criterion of neural networks withcontinuously distributed delaysrdquo Chaos Solitons and Fractalsvol 37 no 2 pp 444ndash449 2008
[14] Z Qiang M A Run-Nian and X Jin ldquoGlobal exponentialconvergence analysis of Hopfield neural networks with con-tinuously distributed delaysrdquo Communications in TheoreticalPhysics vol 39 no 3 pp 381ndash384 2003
[15] Y Wang W Xiong Q Zhou B Xiao and Y Yu ldquoGlobal expo-nential stability of cellular neural networks with continuouslydistributed delays and impulsesrdquo Physics Letters A vol 350 no1-2 pp 89ndash95 2006
[16] H Wu ldquoGlobal exponential stability of Hopfield neural net-works with delays and inverse Lipschitz neuron activationsrdquoNonlinear Analysis Real World Applications vol 10 no 4 pp2297ndash2306 2009
[17] Q Zhang X P Wei and J Xu ldquoGlobal exponential stability ofHopfield neural networkswith continuously distributed delaysrdquoPhysics Letters A vol 315 no 6 pp 431ndash436 2003
[19] J Zhou S Li and Z Yang ldquoGlobal exponential stability ofHopfield neural networks with distributed delaysrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1513ndash15202009
[20] R Gavalda and H T Siegelmann ldquoDiscontinuities in recurrentneural networksrdquo Neural Computation vol 11 no 3 pp 715ndash745 1999
[21] H Wu F Tao L Qin R Shi and L He ldquoRobust exponentialstability for interval neural networks with delays and non-Lipschitz activation functionsrdquoNonlinearDynamics vol 66 no4 pp 479ndash487 2011
[22] H Wu and X Xue ldquoStability analysis for neural networks withinverse Lipschitzian neuron activations and impulsesrdquo AppliedMathematical Modelling vol 32 no 11 pp 2347ndash2359 2008
[23] M Forti M Grazzini P Nistri and L Pancioni ldquoGeneralizedLyapunov approach for convergence of neural networks withdiscontinuous or non-LIPschitz activationsrdquo Physica D Nonlin-ear Phenomena vol 214 no 1 pp 88ndash99 2006
[25] N-E Tatar ldquoControl of systems with Holder continuousfunctions in the distributed delaysrdquo Carpathian Journal ofMathematics vol 30 no 1 pp 123ndash128 2014
[26] G Bao and Z Zeng ldquoAnalysis and design of associative mem-ories based on recurrent neural network with discontinuousactivation functionsrdquoNeurocomputing vol 77 no 1 pp 101ndash1072012
[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003
[28] YHuang H Zhang and ZWang ldquoDynamical stability analysisof multiple equilibrium points in time-varying delayed recur-rent neural networks with discontinuous activation functionsrdquoNeurocomputing vol 91 pp 21ndash28 2012
[29] L Li and L Huang ldquoDynamical behaviors of a class of recur-rent neural networks with discontinuous neuron activationsrdquoApplied Mathematical Modelling vol 33 no 12 pp 4326ndash43362009
[30] L Li and L Huang ldquoGlobal asymptotic stability of delayedneural networks with discontinuous neuron activationsrdquo Neu-rocomputing vol 72 no 16-18 pp 3726ndash3733 2009
[31] Y Li and H Wu ldquoGlobal stability analysis for periodic solutionin discontinuous neural networks with nonlinear growth acti-vationsrdquo Advances in Difference Equations vol 2009 Article ID798685 14 pages 2009
[32] X Liu and J Cao ldquoRobust state estimation for neural networkswith discontinuous activationsrdquo IEEE Transactions on SystemsMan and Cybernetics B Cybernetics vol 40 no 6 pp 1425ndash1437 2010
[33] J Liu X Liu and W-C Xie ldquoGlobal convergence of neuralnetworks with mixed time-varying delays and discontinuousneuron activationsrdquo Information Sciences vol 183 pp 92ndash1052012
[34] S Qin and X Xue ldquoGlobal exponential stability and global con-vergence in finite time of neural networks with discontinuousactivationsrdquoNeural Processing Letters vol 29 no 3 pp 189ndash2042009
[35] J Wang L Huang and Z Guo ldquoGlobal asymptotic stabilityof neural networks with discontinuous activationsrdquo NeuralNetworks vol 22 no 7 pp 931ndash937 2009
[36] Z Wang L Huang Y Zuo and L Zhang ldquoGlobal robuststability of time-delay systems with discontinuous activationfunctions under polytopic parameter uncertaintiesrdquo Bulletin ofthe KoreanMathematical Society vol 47 no 1 pp 89ndash102 2010
[37] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009
Advances in Artificial Neural Systems 7
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
Remark 6 We have assumed that 120572119894119895and 120573
119894119895are greater than
one but the case when they are smaller than one may betreated similarlyWhen their sum is smaller than one we haveglobal existence without adding any extra condition
Remark 7 The decay rate obtained in Corollary 5 is to becompared with the one in the theorem (case (b)) It appearsthat the estimation in Corollary 5 holds for more generalinitial data (not as small as the ones in case (b)) Howeverthe decay rate is smaller than the one in (b) besides assumingthat int1199050119886(119904)119889119904 minus 119871119905 rarr infin as 119905 rarr infin
Remark 8 If we consider the following new functions thenthe monotonicity condition and the order imposed in thetheorem may be dropped
1206011(119905) = max
0le119904le119905
1198921(119904)
120601119896(119905) = max
0le119904le119905
119892119896(119904)
120601119896minus1
(119904)
120601119896minus1
(119905)
(29)
and 120595(119905) = 120601119896(119905)120601119896minus1
(119905)
4 Application
(Artificial)Neural networks are built in an attempt to performdifferent tasks just as the nervous system Typically a neuralnetwork consists of several layers (input layer hidden layersand output layer) Each layer contains one or more cells(neurons) with many connections between them The cellsin one layer receive inputs from the previous layer makesome transformations and send the results to the cells of thesubsequent layer
One may encounter neural networks in many fields suchas control patternmatching settlement of structures classifi-cation of soil supply chain management engineering designmarket segmentation product analysis market developmentforecasting signature verification bond rating recognitionof diseases robust pattern detection text mining price fore-cast botanical classification and scheduling optimization
Neural networks not only can perform many of the tasksa traditional computer can do but also excel in for instanceclassifying incomplete or noisy data predicting future eventsand generalizing
The system (1) is a general version of simpler systems thatappear in neural network theory [1ndash9] like
It is well established by now that (for constant coefficientsand constant 119888
119894(119905)) solutions converge in an exponential
manner to the equilibrium Notice that zero in our case isnot an equilibrium This equilibrium exists and is unique incase of Lipschitz continuity of the activation functions Inour case the system is much more general and the activationfunctions as well as the nonlinearities are not necessarily Lip-schitz continuous However in case of Lipschitz continuityand existence of a unique equilibrium we expect to haveexponential stability using the standard techniques at leastwhen we start away from zero
may be taken as power functions see alsoCorollary 5) our theorem gives sufficient conditions guaran-teeing the estimation
119909 (119905) le 120596119899(119905) exp [minusint
119905
0
119886 (119904) 119889119904] 0 le 119905 lt 1205730 (33)
Then Corollaries 3 and 4 provide practical situations wherewe have global existence and decay to zero at an exponentialrate
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful for the financial support and thefacilities provided by King Fahd University of Petroleum andMinerals through Grant no IN111052
References
[1] J Cao K Yuan and H-X Li ldquoGlobal asymptotical stabilityof recurrent neural networks with multiple discrete delays anddistributed delaysrdquo IEEE Transactions on Neural Networks vol17 no 6 pp 1646ndash1651 2006
[2] B Crespi ldquoStorage capacity of non-monotonic neuronsrdquoNeuralNetworks vol 12 no 10 pp 1377ndash1389 1999
6 Advances in Artificial Neural Systems
[3] G de Sandre M Forti P Nistri and A Premoli ldquoDynamicalanalysis of full-range cellular neural networks by exploiting dif-ferential variational inequalitiesrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 54 no 8 pp 1736ndash1749 2007
[4] C Feng and R Plamondon ldquoOn the stability analysis of delayedneural networks systemsrdquo Neural Networks vol 14 no 9 pp1181ndash1188 2001
[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982
[6] J J Hopfield and D W Tank ldquoComputing with neural circuitsa modelrdquo Science vol 233 no 4764 pp 625ndash633 1986
[7] J I Inoue ldquoRetrieval phase diagrams of non-monotonic Hop-field networksrdquo Journal of Physics A Mathematical and Generalvol 29 no 16 pp 4815ndash4826 1996
[8] B Kosko Neural Network and Fuzzy SystemmdashA DynamicalSystem Approach toMachine Intelligence Prentice-Hall of IndiaNew Delhi India 1991
[9] H-F Yanai and S-I Amari ldquoAuto-associative memory withtwo-stage dynamics of nonmonotonic neuronsrdquo IEEE Transac-tions on Neural Networks vol 7 no 4 pp 803ndash815 1996
[10] X Liu and N Jiang ldquoRobust stability analysis of generalizedneural networks with multiple discrete delays and multipledistributed delaysrdquo Neurocomputing vol 72 no 7ndash9 pp 1789ndash1796 2009
[11] SMohamadKGopalsamy andHAkca ldquoExponential stabilityof artificial neural networks with distributed delays and largeimpulsesrdquo Nonlinear Analysis Real World Applications vol 9no 3 pp 872ndash888 2008
[12] J Park ldquoOn global stability criterion for neural networks withdiscrete and distributed delaysrdquo Chaos Solitons and Fractalsvol 30 no 4 pp 897ndash902 2006
[13] J Park ldquoOn global stability criterion of neural networks withcontinuously distributed delaysrdquo Chaos Solitons and Fractalsvol 37 no 2 pp 444ndash449 2008
[14] Z Qiang M A Run-Nian and X Jin ldquoGlobal exponentialconvergence analysis of Hopfield neural networks with con-tinuously distributed delaysrdquo Communications in TheoreticalPhysics vol 39 no 3 pp 381ndash384 2003
[15] Y Wang W Xiong Q Zhou B Xiao and Y Yu ldquoGlobal expo-nential stability of cellular neural networks with continuouslydistributed delays and impulsesrdquo Physics Letters A vol 350 no1-2 pp 89ndash95 2006
[16] H Wu ldquoGlobal exponential stability of Hopfield neural net-works with delays and inverse Lipschitz neuron activationsrdquoNonlinear Analysis Real World Applications vol 10 no 4 pp2297ndash2306 2009
[17] Q Zhang X P Wei and J Xu ldquoGlobal exponential stability ofHopfield neural networkswith continuously distributed delaysrdquoPhysics Letters A vol 315 no 6 pp 431ndash436 2003
[19] J Zhou S Li and Z Yang ldquoGlobal exponential stability ofHopfield neural networks with distributed delaysrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1513ndash15202009
[20] R Gavalda and H T Siegelmann ldquoDiscontinuities in recurrentneural networksrdquo Neural Computation vol 11 no 3 pp 715ndash745 1999
[21] H Wu F Tao L Qin R Shi and L He ldquoRobust exponentialstability for interval neural networks with delays and non-Lipschitz activation functionsrdquoNonlinearDynamics vol 66 no4 pp 479ndash487 2011
[22] H Wu and X Xue ldquoStability analysis for neural networks withinverse Lipschitzian neuron activations and impulsesrdquo AppliedMathematical Modelling vol 32 no 11 pp 2347ndash2359 2008
[23] M Forti M Grazzini P Nistri and L Pancioni ldquoGeneralizedLyapunov approach for convergence of neural networks withdiscontinuous or non-LIPschitz activationsrdquo Physica D Nonlin-ear Phenomena vol 214 no 1 pp 88ndash99 2006
[25] N-E Tatar ldquoControl of systems with Holder continuousfunctions in the distributed delaysrdquo Carpathian Journal ofMathematics vol 30 no 1 pp 123ndash128 2014
[26] G Bao and Z Zeng ldquoAnalysis and design of associative mem-ories based on recurrent neural network with discontinuousactivation functionsrdquoNeurocomputing vol 77 no 1 pp 101ndash1072012
[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003
[28] YHuang H Zhang and ZWang ldquoDynamical stability analysisof multiple equilibrium points in time-varying delayed recur-rent neural networks with discontinuous activation functionsrdquoNeurocomputing vol 91 pp 21ndash28 2012
[29] L Li and L Huang ldquoDynamical behaviors of a class of recur-rent neural networks with discontinuous neuron activationsrdquoApplied Mathematical Modelling vol 33 no 12 pp 4326ndash43362009
[30] L Li and L Huang ldquoGlobal asymptotic stability of delayedneural networks with discontinuous neuron activationsrdquo Neu-rocomputing vol 72 no 16-18 pp 3726ndash3733 2009
[31] Y Li and H Wu ldquoGlobal stability analysis for periodic solutionin discontinuous neural networks with nonlinear growth acti-vationsrdquo Advances in Difference Equations vol 2009 Article ID798685 14 pages 2009
[32] X Liu and J Cao ldquoRobust state estimation for neural networkswith discontinuous activationsrdquo IEEE Transactions on SystemsMan and Cybernetics B Cybernetics vol 40 no 6 pp 1425ndash1437 2010
[33] J Liu X Liu and W-C Xie ldquoGlobal convergence of neuralnetworks with mixed time-varying delays and discontinuousneuron activationsrdquo Information Sciences vol 183 pp 92ndash1052012
[34] S Qin and X Xue ldquoGlobal exponential stability and global con-vergence in finite time of neural networks with discontinuousactivationsrdquoNeural Processing Letters vol 29 no 3 pp 189ndash2042009
[35] J Wang L Huang and Z Guo ldquoGlobal asymptotic stabilityof neural networks with discontinuous activationsrdquo NeuralNetworks vol 22 no 7 pp 931ndash937 2009
[36] Z Wang L Huang Y Zuo and L Zhang ldquoGlobal robuststability of time-delay systems with discontinuous activationfunctions under polytopic parameter uncertaintiesrdquo Bulletin ofthe KoreanMathematical Society vol 47 no 1 pp 89ndash102 2010
[37] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009
Advances in Artificial Neural Systems 7
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
[3] G de Sandre M Forti P Nistri and A Premoli ldquoDynamicalanalysis of full-range cellular neural networks by exploiting dif-ferential variational inequalitiesrdquo IEEE Transactions on Circuitsand Systems I Regular Papers vol 54 no 8 pp 1736ndash1749 2007
[4] C Feng and R Plamondon ldquoOn the stability analysis of delayedneural networks systemsrdquo Neural Networks vol 14 no 9 pp1181ndash1188 2001
[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982
[6] J J Hopfield and D W Tank ldquoComputing with neural circuitsa modelrdquo Science vol 233 no 4764 pp 625ndash633 1986
[7] J I Inoue ldquoRetrieval phase diagrams of non-monotonic Hop-field networksrdquo Journal of Physics A Mathematical and Generalvol 29 no 16 pp 4815ndash4826 1996
[8] B Kosko Neural Network and Fuzzy SystemmdashA DynamicalSystem Approach toMachine Intelligence Prentice-Hall of IndiaNew Delhi India 1991
[9] H-F Yanai and S-I Amari ldquoAuto-associative memory withtwo-stage dynamics of nonmonotonic neuronsrdquo IEEE Transac-tions on Neural Networks vol 7 no 4 pp 803ndash815 1996
[10] X Liu and N Jiang ldquoRobust stability analysis of generalizedneural networks with multiple discrete delays and multipledistributed delaysrdquo Neurocomputing vol 72 no 7ndash9 pp 1789ndash1796 2009
[11] SMohamadKGopalsamy andHAkca ldquoExponential stabilityof artificial neural networks with distributed delays and largeimpulsesrdquo Nonlinear Analysis Real World Applications vol 9no 3 pp 872ndash888 2008
[12] J Park ldquoOn global stability criterion for neural networks withdiscrete and distributed delaysrdquo Chaos Solitons and Fractalsvol 30 no 4 pp 897ndash902 2006
[13] J Park ldquoOn global stability criterion of neural networks withcontinuously distributed delaysrdquo Chaos Solitons and Fractalsvol 37 no 2 pp 444ndash449 2008
[14] Z Qiang M A Run-Nian and X Jin ldquoGlobal exponentialconvergence analysis of Hopfield neural networks with con-tinuously distributed delaysrdquo Communications in TheoreticalPhysics vol 39 no 3 pp 381ndash384 2003
[15] Y Wang W Xiong Q Zhou B Xiao and Y Yu ldquoGlobal expo-nential stability of cellular neural networks with continuouslydistributed delays and impulsesrdquo Physics Letters A vol 350 no1-2 pp 89ndash95 2006
[16] H Wu ldquoGlobal exponential stability of Hopfield neural net-works with delays and inverse Lipschitz neuron activationsrdquoNonlinear Analysis Real World Applications vol 10 no 4 pp2297ndash2306 2009
[17] Q Zhang X P Wei and J Xu ldquoGlobal exponential stability ofHopfield neural networkswith continuously distributed delaysrdquoPhysics Letters A vol 315 no 6 pp 431ndash436 2003
[19] J Zhou S Li and Z Yang ldquoGlobal exponential stability ofHopfield neural networks with distributed delaysrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 33 no 3 pp 1513ndash15202009
[20] R Gavalda and H T Siegelmann ldquoDiscontinuities in recurrentneural networksrdquo Neural Computation vol 11 no 3 pp 715ndash745 1999
[21] H Wu F Tao L Qin R Shi and L He ldquoRobust exponentialstability for interval neural networks with delays and non-Lipschitz activation functionsrdquoNonlinearDynamics vol 66 no4 pp 479ndash487 2011
[22] H Wu and X Xue ldquoStability analysis for neural networks withinverse Lipschitzian neuron activations and impulsesrdquo AppliedMathematical Modelling vol 32 no 11 pp 2347ndash2359 2008
[23] M Forti M Grazzini P Nistri and L Pancioni ldquoGeneralizedLyapunov approach for convergence of neural networks withdiscontinuous or non-LIPschitz activationsrdquo Physica D Nonlin-ear Phenomena vol 214 no 1 pp 88ndash99 2006
[25] N-E Tatar ldquoControl of systems with Holder continuousfunctions in the distributed delaysrdquo Carpathian Journal ofMathematics vol 30 no 1 pp 123ndash128 2014
[26] G Bao and Z Zeng ldquoAnalysis and design of associative mem-ories based on recurrent neural network with discontinuousactivation functionsrdquoNeurocomputing vol 77 no 1 pp 101ndash1072012
[27] M Forti and P Nistri ldquoGlobal convergence of neural networkswith discontinuous neuron activationsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 50 no 11 pp 1421ndash1435 2003
[28] YHuang H Zhang and ZWang ldquoDynamical stability analysisof multiple equilibrium points in time-varying delayed recur-rent neural networks with discontinuous activation functionsrdquoNeurocomputing vol 91 pp 21ndash28 2012
[29] L Li and L Huang ldquoDynamical behaviors of a class of recur-rent neural networks with discontinuous neuron activationsrdquoApplied Mathematical Modelling vol 33 no 12 pp 4326ndash43362009
[30] L Li and L Huang ldquoGlobal asymptotic stability of delayedneural networks with discontinuous neuron activationsrdquo Neu-rocomputing vol 72 no 16-18 pp 3726ndash3733 2009
[31] Y Li and H Wu ldquoGlobal stability analysis for periodic solutionin discontinuous neural networks with nonlinear growth acti-vationsrdquo Advances in Difference Equations vol 2009 Article ID798685 14 pages 2009
[32] X Liu and J Cao ldquoRobust state estimation for neural networkswith discontinuous activationsrdquo IEEE Transactions on SystemsMan and Cybernetics B Cybernetics vol 40 no 6 pp 1425ndash1437 2010
[33] J Liu X Liu and W-C Xie ldquoGlobal convergence of neuralnetworks with mixed time-varying delays and discontinuousneuron activationsrdquo Information Sciences vol 183 pp 92ndash1052012
[34] S Qin and X Xue ldquoGlobal exponential stability and global con-vergence in finite time of neural networks with discontinuousactivationsrdquoNeural Processing Letters vol 29 no 3 pp 189ndash2042009
[35] J Wang L Huang and Z Guo ldquoGlobal asymptotic stabilityof neural networks with discontinuous activationsrdquo NeuralNetworks vol 22 no 7 pp 931ndash937 2009
[36] Z Wang L Huang Y Zuo and L Zhang ldquoGlobal robuststability of time-delay systems with discontinuous activationfunctions under polytopic parameter uncertaintiesrdquo Bulletin ofthe KoreanMathematical Society vol 47 no 1 pp 89ndash102 2010
[37] H Wu ldquoGlobal stability analysis of a general class of discontin-uous neural networks with linear growth activation functionsrdquoInformation Sciences vol 179 no 19 pp 3432ndash3441 2009
Advances in Artificial Neural Systems 7
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
[38] Z Cai and L Huang ldquoExistence and global asymptotic stabilityof periodic solution for discrete and distributed time-varyingdelayed neural networks with discontinuous activationsrdquo Neu-rocomputing vol 74 no 17 pp 3170ndash3179 2011
[39] D Papini and V Taddei ldquoGlobal exponential stability of theperiodic solution of a delayed neural network with discontinu-ous activationsrdquo Physics Letters A vol 343 no 1ndash3 pp 117ndash1282005
[40] M Pinto ldquoIntegral inequalities of Bihari-type and applicationsrdquoFunkcialaj Ekvacioj vol 33 no 3 pp 387ndash403 1990
[41] V Lakshmikhantam and S Leela Differential and IntegralInequalities Theory and Applications vol 55-I of Mathematicsin Sciences and Engineering Edited by Bellman R AcademicPress New York NY USA 1969
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