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Fixed points of asymptotically regular multivalued mappings

Article  in  Journal of the Australian Mathematical Society · November 1992

DOI: 10.1017/S1446788700036491

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/. Austral. Math. Soc. (Series A) 53 (1992), 313-326

FIXED POINTS OF ASYMPTOTICALLY REGULARMULTIVALUED MAPPINGS

ISMAT BEG and AKBAR AZAM

(Received 10 May 1990; revised 24 March 1991)

Communicated by E. N. Dancer

Abstract

Some results on fixed point of asymptotically regular multivalued mapping are obtained in metricspaces. The structure of common fixed points and coincidence points of a pair of compatiblemultivalued mappings is also discussed. Our work generalizes known results of Aubin and Siegel,Dube, Dube and Singh, Hardy and Rogers, Hu, Iseki, Jungck, Kaneko, Nadler, Ray and Shiau,Tan and Wong.

1991 Mathematics subject classification (Amer. Math. Soc): 54 H 25, 47 H 10.Keywords and phrases: metric space, multivalued mapping, fixed point, coincidence point.

1. Introduction

Let T be a single valued self mapping on a metric space X. A sequence {xn}in X is said to be asymptotically T-regular if d(xn , Txn) —> 0. The presenceof a sequence {xn} for which d{xn , Txn) -> 0 is related to some propertyof T (see [3], [7], [8], [24], [25], and [27]) and hence is exploited to obtainfixed points of T. The aim of the present paper is to bring out the thrustof a similar assumption for multivalued mappings. The weakly dissipativemultivalued mappings recently introduced by Aubin and Siegel in [3] satisfysuch an assumption. As stated in Aubin and Siegel [3], such fixed pointtheorems have application to control theory, system theory and optimizationproblems. Moreover, such a sequence (for multivalued mappings) has beenused by Itoh and Takahashi [12] and Rhoades, Singh and Kulshrestha [26].

© 1992 Australian Mathematical Society 0263-6115/92 $A2.00 + 0.00

313

314 Ismat Beg and Akbar Azam [2]

Rhoades [24], [25] had compared these contractive conditions. Most of thecontractive conditions used imply the asymptotic regularity of the mappingsunder consideration, so the study of such mappings play an important rolein fixed point theory.

In Section 2, some notation, definitions and facts, used in subsequentsections are listed. In Section 3 we prve the existence of a common fixedpoint of two multivalued mappings satisfying a contractive type condition ina metric space. In Section 4 a class of multivalued mappings is introducedwhich is larger (even in the case of single valued mappings) than those thatWong [32] refers to as Kannan mappings. The fixed point theorems thereinare proved under less restrictive hypotheses and for wider classes than theresults of Shiau, Tan and Wong [30]. In Section 5 we extend the idea ofJungck [13] to multivalued mappings and obtain a coincidence theorem fora pair of compatible multivalued mappings. The structure of common fixedpoints of these mappings is also studied.

2. Preliminaries

Let (X, d) be a metric space and let CB(X) denote the family of allnonempty bounded closed subsets of X. For A, B e CB(X), let H(A, B)denote the distance between A and B in Hausdorff metric, that is

f infE. „ if £ \ „ ^ 0 ,H(A,B) = \ A'B ,e

A'Br

where N(e, A) = {x e X: d(x, A) < e} andEA,B = {e>O:ACN(e,B),BC N(e,A)}.

Let T: X —> CB(X) be a mapping and {xn} a sequence in X. Then {xn}is said to be asymptotically T-regular if d(xn , Txn) -> 0. Let f: X ^ Xbe a mapping such that TX c / X . Then {xn} is called asymptotically T-regular with respect to f if d{fxn, 7"xn) —» 0. (cf. [27]). A point x is saidto be a ybcerf point of a single valued mapping / (multivalued mapping T)provided x = fx (x € Tx). The point x is called a coincidence point of/ and r if fx € Tx . We shall require the following well-known facts (cf.[21]).

LEMMA 2.1. If A, B e CB(X) with H(A, B) <e, then for each a e A,there exists an element b € B such that d(a, b) < e.

LEMMA 2.2. Let {An} be a sequence in CB(X) and limn^oc H(An, A) =0 for Ae CB(X). If xne An and limn^oo d{xn, x) = 0, then x e A .

[3] Fixed points of asymptotically regular multivalued mappings 315

3. Common fixed point of multivalued generalized contractions

Wong [31] extended the result of Hardy and Rogers [9] by showing that twoself mappings S and T on a complete metric space satisfying a contractivetype condition have a common fixed point. In this section we extend thisresult of Wong to the case when S and T are multivalued and satisfy amore general contractive type condition.

THEOREM 3.1. Let X be a complete metric space, S: X -* CB(X) andT: X —> CB{X). If there exists a constant a, 0 < a < 1, such that for eachx,yeX,

(1) H(Tx,Sy)

< a max{d(x, y), d(x, Sx), d(y, Ty), (d(x, Ty) + d(y, Sx))/2}

then there exists a common fixed point of S and T.

PROOF. Assume that /? = y/a.Let x0 be an arbitrary but fixed element of X and choose x{ e Sx0.

Then

H(Sx0 ,Txx)<p max{d(x0, x , ) , d(x0, Sx0), d{xx ,Txx),

Lemma 2.1 implies that there exists a point x2 e Txx such that

d(xl ,x2)<$ max{^(x0, x , ) , d(x0, Sx0), d{xl, Txx),

{d{xo,TXl) + d(Xl,Sxo

< 0 max{d{xQ, x , ) , rf(x,, x2), (rf(x0, x,) + d{xx, x2))/2}.

If d{xx, x2) > d{xQ, x , ) , then d(xx, x2) < ftd{x{, x2) , a contradiction.Thus d(x{, x2) < fid{x0, x , ) . Now

H(Txx, Sx2) < 0 max{fi?(x,, x2), d(xl, Txx), d(x2, Sx2),

Again using Lemma 2.1, we obtain a point x3 € Sx2 such that

d(x2, x3) < pmax{d(xx ,x2), d(xx ,Txx), d(x2 ,Sx2),

(d(xx,Sx2) + d(x2,Txx))/2}

< pd(xx, x2).

316 Ismat Beg and Akbar Azam [4]

By induction we produce a sequence {xn} of points of X, such that, forA:>0,

X2k+\ £ Sx2k , X2k+2 e Tx2k+\

and

d(xn,xn+l)<pd(xn_,,xn)

<P"d(xo,Xl).

Furthermore, for m> n ,

d(xn , x m ) < d(xn, x n + x ) + d(xn+l, x n + 2 ) + ••• + d{xm_x, x m )

<Vr + fi*+1 + ~. + ftm-1}d{x0,xl).

It follows that {xn} is a Cauchy sequence and there exists a point t e Xsuch that xn —> t. It further implies that x2k+ { —> t, and x2k+2 —> ?. Thuswe have,

d(t, St) < d(t, x2k+2) + ̂ (x 2 t + 2 , St)

<d(t,x2k+2) + H(Tx2k+1,St)

< d(t, x2k+2) + p max{d(x2k+l, t), d(t, St),d(X2k+l > *2*+2)' ( r f ( ? ' X2k+l) + d(X2k+l ' 5 0 ) / 2 } -

Letting k -» oo, we have rf(?, 5/) < fid{t, St). Hence t e St. Similarly,

d(t, Tt) < d(t, x2k+l) + H(Sx2k, Tt) < pd{t, Tt).

Therefore t e Tt.

COROLLARY 3.2. Let X be a complete metric space and T: X —> CB(X).If there exists a constant a, 0 < a < 1, such that for each x, y € X,

H(Tx, Ty)<amax{d(x,y),d(x, Tx),d(y, Ty),

(d(x,Ty) + d(y,Tx))/2}

then there exists a sequence {xn} which is asymptotically T-regular and con-verges to a fixed point of T.

REMARK 3.3. Theorem 3.1 improved the results of Kaneko [20], whichconsidered the mapping r of a reflexive Banach space X into the family ofweakly compact subsets of X. The proximinality of the set Tx is a conse-quence of his assumption and it is used in his proof. No such assumption isrequired in Theorem 3.1.

REMARK 3.4. In [22], Ray proved a fixed point theorem for a multivaluedmapping T: X -• CB{X) satisfying

H(Tx, Ty) < ad(x, y) + b(d(x, Tx) + d(y, Ty))

+ c(d(x,Ty) + d(y,Tx)),

[5] Fixed points of asymptotically regular multivalued mappings 317

where a, b and c are non-negative real numbers and 0<a + 2b + c< 1.Theorem 3.1 (even in the particular case for S = T) is not a special caseof the theorem of Ray [22] since T is not assumed to have closed graph. Italso illustrates that the compactness of Tx is not necessary for the theoremof Aubin and Siegel [3].

Several other results may also be seen to follow as immediate corollariesto Theorem 3.1. Included among these are Dube [5, Theorem 1], Dube andSingh [6, Theorem 1], Iseki [11], Nadler [21, Theorem 5], Hardy and Rogers[9] and Wong [31].

4. Fixed point of Kannan type multivalued mappings

In this section we consider the mapping T: X -* CB{X) satisfying thecondition

H(Tx, Ty)<a{{d{x, Tx))d(x, Tx) + a2(d(y, Ty))d(y, Ty),

where al;: E —> [0, 1) (/ = 1, 2). Such a mapping T is not a special case ofthe mapping considered in Section 1. In 1968 Kannan [17] had establisheda fixed point theorem for a single valued mapping T defined on a completemetric space X satisfying

d(Tx, Ty) < a(d(x, Tx) + d(y, Ty)),

where 0 < a < \ and x, y e X. Within the context of a complete metricspace the assumption 0 < a < j is crucial even to the existence part of thisresult, but within a more restrictive yet quite natural setting, an elaboratefixed point theory exists for the case a = j . Mappings of this wider classwere studied by Kannan in [18]. In recent years, Beg and Azam [4], ShiauTan and Wong [29] and Wong [32] have also studied such mappings.

THEOREM 4.1. Let X be a complete metric space and T: X -> CB(X) amapping satisfying

(2) H(Tx, Ty)<a,{d{x, Tx))d{x, Tx) + a2(d(y, Ty))d(y, Ty),

for all x,y e X, where a(: E —> [0, 1) (/ = 1,2). If there exists anasymptotically T-regular sequence {xn} in X, then T has a fixed point x*in X. Moreover Txn -> Tx*.

PROOF. By hypothesis, we have

H(Txn , TxJ < ax{d{xn , Txn))d(xn , Txn) + a2(d{xm , Txm))d(xm , TxJ.

318 Ismat Beg and Akbar Azam [6]

Thus {Txn} is a Cauchy sequence. Since (CB(X), H) is complete (see[2]), there exists a K* e CB(X), such that H{Txn,K*)^0. Let x* € K*.Then

d(x*, Tx*)<H(K*, Tx*)

= lim H{Txn , Tx*)n—>oo "

< \imo(al(d(xn , Txn))d{xn , Txn) + a2(d(x*, Tx*))d(x*, Tx*))

<a2(d(x*, Tx*))d{x*, Tx*).

It further implies that

(1 - a2(d{x*, Tx*)))d(x*, Tx*) < 0.

Therefore d(x*, Tx*) = 0. Thus x* e Tx*. Now,

H(K* ,Tx*)= lim H(Txn , Tx*)

<a2(d(x*, Tx*))d{x*, Tx*)

<d(x*, Tx*) = 0.

It follows thatTx* = K* = lim Txn.

THEOREM 4.2. Let X be a complete metric space and T: X —> CB(X) amapping satisfying (2). If there exists an asymptotically T-regular sequence{xn} in X and Txn is compact for each n, then each cluster point of {xn}is a fixed point of T.

PROOF. Let yn € Txn be such that d(xn, yn) = d{xn , Txn). Obviously,a cluster point of {xn} is a cluster point of {yn}. If y* is such a clusterpoint of {xn} and {yn} , then with x* (as in Theorem 4.1),

d(yn,Tx*)<H(Txn,Tx*)

< ax{d{xn , Txn))d{xn , Txn) + a2(d(x*, Tx*))d(x*, Tx*)<a{{d{xn,Txn))d(xn,Txn).

Therefore y* e Tx*. Now

d(y*, Ty*) < H(Tx*, Ty*)

<ai(d(x*,Tx*))d(x*,Tx*) + a2(d(y*,Ty*))d(y*, Ty*).

It follows that {1 - a2(d(y*, Ty*))}d(y*, Ty*) < 0. Hence y* e Ty*.

[7] Fixed points of asymptotically regular multivalued mappings 319

THEOREM 4.3. Let X be a complete metric space and T: X —> CB(X) amapping satisfying (2) with ax(d(x, Tx)) + a2(d(y, Ty)) < 1. Ifinf{d(x, Tx): x € X) = 0 then T has a fixed point.

PROOF. It is sufficient to show that there exists an asymptotically ^-regularsequence {xn} in X.

Let x0 be an arbitrary but fixed element of X. Consider the sequence{xn}, xn e TxnX. The inequality (2) implies that

d{xn,Txn)<H{Txn_x,Txn)

+ a2(d(xn,Txn))d(xn,Txn)

<d{xn_x,Txn_x).

It follows that the sequence {d{xn , Txn)} is decreasing. Therefore

d{xn, Txn) -> inf{d{xn , Txn): n e N} and d(xn , Txn) - 0.

Hence {xn} is asymptotically T-regular.

Theorems 4.1, 4.2 and 4.3 generalize results of Shiau, Tan and Wong [30].Here we desire to emphasize not only that our T belongs to a wider classof mappings but also that the hypothesis of compactness of Tx (in [30,Theorem 1]) is dropped.

5. Coincidence point of compatible multivalued mappings

Jungck [14] introduced a contraction condition for single valued compat-ible mappings on a metric space. He also pointed out in [15] and [16] thepotential of compatible mappings for generalized fixed point theorems. Sub-sequently a variety of extensions, generalizations and applications of thisfollowed; for example, see [1], [28] and [29]. This section is a continuationof these investigations for multivalued compatible mappings.

DEFINITION. Let X be a metric space. Mappings T: X —> CB(X),f:X—>X are compatible if, whenever there is a sequence {xn} c X satis-fying lim fxn e lim Txn (provided lim fxn exists in X and lim Txn existsin CB(X)), then lim H(fTxn, Tfxn) = 0.

If T is a single valued self mapping on X, this definition of compatibilitybecomes that of Jungck [14]. Let X = R, with Euclidean metric, Tx =

320 Ismat Beg and Akbar Azam [8]

[x2/4, x2/2], fx = x2/&. Then / and T are compatible but they do notcommute.

Let (p: (0, oo) —> [0, 1) be a function having the following property (cf.[10], [23]):(P) for t > 0, there exists d{t) > 0, s(t) < 1, such that

0 < r - t < d(t) implies <p{r) < s{t).

The following Theorem is a generalization of Hu [10, Theorem 2], Jungck[13], Kaneko [19] and Nadler [21, Theorem 5].

THEOREM 5.1. Let T be a mapping from a complete metric space X intoCB(X). Let f: X -* X be a continuous mapping such that TX C fX. Iff and T are compatible and for all x, y € X,

(3) H(Tx, Ty) < <p(d(fx, fy))d(fx, fy),

then there exists a sequence {xn} which is asymptotically T-regular withrespect to f, and fxn converges to a coincidence point of f and T.

PROOF. Let x0 be an arbitrary, but fixed element of X. We shall constructtwo sequences {xn} and {yn} of points of X as follow. Let y0 = fxQ andXj G X be such that y{ = fxl e Tx0 . Then inequality (3) implies that

H(Tx0, Txx) < <p(d(fx0, fxx))d{fx0, fx,).

Using Lemma 2.1 and the fact that TX c fX, we may choose x2e X suchthat y2 = fx2 e Txx and

d{yx, y2) = d(fxx, fx2)

< <p(d(fx0, fxx))d(fx0, fxx)<d(fxo,fxx).

By induction we produce two sequences of points of X such that yn = fxn eTxn_x, n > 0. Furthermore,

d(yn+l, yn+2) = d(fxn+l, fxn+2)< ?(d(fxn , fxn+l))d(fxn , fxn+l)<d(fxn,fxn+l) = d(yn,yn+l).

It follows that the sequence {d(yn , yn+1)} is decreasing and converges to itsgreatest lower bound which we denote by t. Now t > 0; in fact t = 0.Otherwise by property (P) of <p , there exists S{t) > 0, s(t) < 1, such that,

0 < r - t < S(t) implies (p{r) < s{t).

[9] Fixed points of asymptotically regular multivalued mappings 321

For this S(t) > 0, there exists a natural number N such that,

0 < d(yn , yn+i)) - t < 5{t), whenever n > N.

Hence<P(d{yn, yn+x)) < s{t), whenever n > N.

Let K = max{ip(d(yQ,yl)),(p(d(yl,y2)), ... ,9>(d{yN_l,yN)),s(t)}.T h e n , for n = 1 , 2 , 3 , . . . ,

d(yn, yn+l) < vidiy^ , yn))d{yn_,, yn)

<Kd{yn_x,yn)

</(:V(yo,y1) ->0 as n - ^ o o ,

which contradicts the assumption that t > 0. Consequently,

which implies that d(fxn, Txn) —» 0. Hence the sequence {xn} is asymp-totically T-regular with respect to / .

Assume that {fxn} is not a Cauchy sequence. Then there exists a positivenumber t* and subsequences {«(*)} > {w(/)} of the natural numbers with«(/) < m(i) and such that d{yn(i),ym{i)) > t \ d{yn(i),ym(i)_x) < t* for/ = 1, 2, 3 . . . . Then

t* < d{yn(i), ym(l))

Letting /' -» oo and using the fact that d{y,i),ym(i)_l) < t*, we obtainl i m , . ^ r f O ^ , ym{i)) = t*. For this t* > 0, there exists 8{t*) > 0, s{t*) <1, such that

0 < r - t* < S(t*) implies (p{r) < s(t*).

For this S(t*) > 0, there exists a natural number NQ such that,

/ > No implies 0 < d(yn{i), ym{i)) - t* < S(t*).

Hence <p{d{yn(i), ym(i))) < s{t") for i>N0. Thus

{i), ym{i)))d(yn{i), ym(i)

Letting / -> oo, we get t* < s(t*)t* < t*, a contradiction. Hence {/*„} is aCauchy sequence. By completeness of the space, there exists an element p e

322 Ismat Beg and Akbar Azam [10]

X such that d(yn, p) —* 0. Continuity of / implies that d(fyn , fp) —• 0.Hence

H(Tyn , Tp) < <p(d(fyn , fp))d{fyn , fp)

Inequality (3) and the fact that {/*„} is a Cauchy sequence imply that thereexists A e CB{X) such that Txn -> A . Furthermore,

d(p,A)<limoH(Txn_l,Txn) = 0.

Nowd(fyn+l,Tyn)<H(fTxn,Tfxn).

Letting n —» oo, we obtain d(fp, Tp) = 0. Hence fpeTp.

EXAMPLE 5.2. Let X = [0, oo) with the Euclidean metric Tx = [0, x]and fx = 104JC. Then / and T do not satisfy the condition of the theoremsin [10], [13] and [21]. Considering the function <p(x) = c, where 10~4 <c < 1, it is easily seen that all the hypotheses of Theorem 5.1 are valid. Thus/ and T have a coincidence point.

COROLLARY 5.3. If, in addition to the hypotheses of Theorem 5.1 the map-ping f satisfies, for all x, y € X,(4)d(fx ,fy)<y max{d(x, y), d(x, fx), d(y, fy), (d(x, fy)+d(y, fx))/2}

where 0 < y < 1, then there exists a common fixed point of f and T.

PROOF. Let /? = y/y. As in the proof of Theorem 5.1 there is a coinci-dence point p of f and T. Define the iterative sequence {tn} as follows:t0 = p and tn = ftn_l = ftQ, n = 0, 1 ,2. . . . Now inequality (4) impliesthat

d(tn,tn+l) = d(ftn_l,ftn)

(fn_1, tn), d(tn , tn+l), d{tn_x, tn+l)/2}

tn_x, tn)

<pnd(t0,tx).

It further implies that {tn} is a Cauchy sequence. By the completeness ofX, we have fnt0 -> x* e X.

Now consider a constant sequence {un} c X as follows: un = t0. Then

lim fu = ft0 € TtQ = lim Tu .

[11] Fixed points of asymptotically regular multivalued mappings 323

Thus by the compatibility of / and T,

H(fTt0, Tft0) = \w^H(fTun , Tfun) = 0.

Hence f2t0 = fft0 6 fTt0 = TftQ. Choose another constant sequence,vn = ft0 . Then

andH(fTft0, Tf2t0) = \imoH(fTvn , TfvH) = 0.

Thus ft0 = ff2t0 € fTftQ = Tf2t0. Consequently, we have f"+lt0 eTftQ. Using (3), we get l i m ^ ^ Txn = Tx*. Hence by Lemma 2.2, weobtain, x* € Tx*. Moreover,

fx* = f lim ft0 = lim /" + ' r 0 = x*.

Hence x* is a common fixed point of / and T.

In Theorem 5.1 our hypothesis that / is continuous implies that T iscontinuous. And we use the continuity of / and T in our proof. In thenext theorem we show that if fX is complete then the continuity and com-patibility of / and T are not required.

THEOREM 5.4. Let T be a mapping of a metric space X into CB{X).Let f: X -* X be a mapping such that TX c fX, fX is complete and thecondition (3) is satisfied. Then

(i) there exists a sequence {xn} which is asymptotically T-regular withrespect to f and

(ii) / and T have a coincidence point.

PROOF. Examining the proof of Theorem 5.1, we see that the only changeis that the completeness of fX allows us to obtain z e X such that fxn -*p — fz. Then

d(fz, Tz) < d(fz , fxn+l) + d(fxn+i, Tz)<d(fz,fxn+l) + H(Txn,Tz)< d(fz, fxn+l) + <p(d(fxn , fz))d(fxn , fz)<d(fz,fxn+l) + d(fxn,fz).

Letting n -> oo, we obtain

d(fz, Tz) < d(fz ,p) + d(p, fz) = 0.Hence fz eTz.

324 Ismat Beg and Akbar Azam [ 12]

COROLLARY 5.5. Suppose that, in addition to the hypotheses of Theorem5.4, / satisfies (4) and f and T are compatible. Then {fxn} converges toa coincidence point (say p) of f and T, and {fp} converges to a commonfixed point of f and T.

PROOF. By Theorem 5.4, there exists z e X such that fz e Tz. As inCorollary 5.3, compatibility of / and T implies that ffz — ftz — Tfz.

Since fxn —* fz (see Theorem 5.4), fxn converges to a coincidence pointof / a n d T.

Now, inequality (4) implies that {f'z} is a Cauchy sequence. Let f"z —>x*. Since (as in Corollary 5.3) fn+lz € Tfz, we have

d{x*, Tx*) < d(x*, f+lz) + H(Tf"z, Tx*)

< d(x*, f+lz) + <p(d(fz, x*))d(fz, x*)<d(x*,f+lz) + d(fz,x*).

Letting n -+ oo, we obtain d(x*, Tx*) = 0, i.e., x* e Tx*. Moreover,

d(x*,fx*)<d(x*,f+lz) + d(f+lz,fx*)<d(x*,f+lz)

+ ymax{d(fz,x*),d(fz,f+lz),d(x*,fx*),,fx*) + d(f+lz,x*))/2}.

Letting « -» oo, we have d(x*, fx*) < yd{x*, fx*). Hence x* = fx*.

We show that the assumption of TX C fX (Theorem 5.4) and compati-bility of / and T (Corollary 5.5) cannot be dropped.

EXAMPLE 5.6. Let X - R with the Euclidean metric Tx = [0, |JC|/3] ,fx = (x + 3)/2 and <p(x) = 2/3 . Then all the hypotheses of Theorem 5.4are satisfied and / ( -2) e T(-2). Moreover / and T are not compatible,but the other assumptions of Corollary 5.5 are satisfied, f{-2) —> 3 and 3is not a common fixed point of / and T.

Acknowledgment

Thanks are due to Professor B. E. Rhoades for providing us with a preprintof [28].

[13] Fixed points of asymptotically regular multivalued mappings 325

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Quaid-i-Azam University F. G. Post-graduate CollegeIslamabad IslamabadPakistan Pakistan

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