Research ArticleFixed Point Results Satisfying Rational TypeContraction in πΊ-Metric Spaces
Branislav Z. PopoviT,1 Muhammad Shoaib,2 and Muhammad Sarwar2
1Faculty of Science, University of Kragujevac, Radoja Domanovica 12, 34000 Kragujevac, Serbia2Department of Mathematics, University of Malakand, Chakdara, Lower Dir 18800, Pakistan
Correspondence should be addressed to Branislav Z. Popovic; [email protected]
A unique fixed point theorem for three self-maps under rational type contractive condition is established. In addition, a uniquefixed point result for six continuous self-mappings through rational type expression is also discussed.
1. Introduction
Fixed point theory is one of the core subjects of nonlinearanalysis. This theory is not constrained to mathematics; it isalso applicable to other disciplines. It is closely linked withsocial and medical science, military applications, graph the-ory [1], game theory, economics [2], statistics, and medicine.This theory is divided into three categories: topological fixedpoint theory, metric fixed point theory, and discrete fixedpoint theory.
In metric fixed point theory, the first result proved byBanach [3] is known as Banach contraction principle. Manyresearchers extended this principle for the study of fixedpoints and common fixed points using different types ofcontraction such as weak contraction [4, 5], integral typecontraction [6], rational type contraction [7], and T-HardyRogers type contraction [8]. For more details, see [9β11] andso forth.
Dass and Gupta [12] gave the extension of Banachβscontraction mapping principle by using a contractive con-dition of rational type. Jaggi [7] proved some unique fixedpoint results through contractive condition of rational typein metric spaces. Harjani et al. [13] studied the results ofJaggi in the setting of partially ordered metric spaces. Usinggeneralized weak contractions Luong and Thuan [14] gener-alized the results of [13] through rational type expressions
in the context of partially ordered metric spaces. Chandokand Karapinar [15] generalized the results of Harjani andestablished common fixed point results for weak contractiveconditions satisfying rational type expressions in partiallyordered metric spaces. Mustafa et al. [16] discussed fixedpoint results by almost generalized contraction via rationaltype expression which generalizes, extends, and unifies theresults of Jaggi [7], Harjani et al. [13], and Luong and Thuan[14], respectively. Fixed point theorems for contractive typeconditions satisfying rational inequalities in metric spaceshave been developed in a number of works; see [17β20] andso forth.
Mustafa and Sims [21] generalized the notion of metricspace as an appropriate notion of generalized metric spacecalled πΊ-metric space. They have investigated convergencein πΊ-metric spaces, introduced completeness of πΊ-metricspaces, and proved a Banach contraction mapping theoremand some other fixed point theorems involving contractivetype mappings in πΊ-metric spaces using different contractiveconditions. Later, various authors have proved some commonfixed point theorems in these spaces (see [8, 22β24]).
Sanodia et al. [25] used rational type contraction andinvestigated a unique fixed point theorem for single mappingin πΊ-metric spaces. Gandhi and Bajpai [26] generalized theresult of Sanodia et al. and proved unique common fixedpoint results for three mappings in πΊ-metric space satisfying
Hindawi Publishing CorporationJournal of Function SpacesVolume 2016, Article ID 9536765, 7 pageshttp://dx.doi.org/10.1155/2016/9536765
2 Journal of Function Spaces
rational type contractive condition. Recently, Shrivastava etal. [27] established some unique fixed point theorem for somenew rational type contraction.
The aim of this paper is to establish two common fixedpoint theorems satisfying rational type contraction. In thefirst result, we discuss the existence and uniqueness ofcommon fixed point for three self-maps in the context ofπΊ-metric space, while in the second one we studied theuniqueness of common fixed point for six continuous self-mappings in the setting of πΊ-metric through rational typeexpression.
2. Preliminaries
We recall some definitions that will be used in our discussion.
Definition 1 (see [21]). Letπ be a nonempty set and letπΊ : πΓ
πΓπ β R+ be a function satisfying the following conditions:
(1) πΊ(π₯, π¦, π§) = 0 implies that π₯ = π¦ = π§ for all π₯, π¦, π§ β
π.
(2) πΊ(π₯, π₯, π¦) β€ πΊ(π₯, π¦, π§) for all π₯, π¦, π§ β π.
Sanodia et al. [25] proved the following fixed point theoremin the setting of πΊ-metric space.
Theorem4. Let (π, πΊ) be aπΊ-completeπΊ-metric space and letπ : π β π be a self-map satisfying the condition
πΊ (ππ₯, ππ¦, ππ§) β€ π΄
β
max {πΊ2(π₯, ππ₯, ππ¦) , πΊ
2(π¦, ππ¦, ππ§) , πΊ
2(π§, ππ§, ππ₯)}
πΊ (π₯, π¦, π§)
(2)
for all π₯, π¦, π§ β π with 0 β€ π΄ < 1. Then, π has a uniquecommon fixed point in π.
Theorem 5. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π, π : π β π be two self-maps such that π(π) β π(π)
satisfying the following condition:
πΊ (ππ₯, ππ¦, ππ§) β€ π΄
β
max {πΊ2(ππ₯, ππ₯, ππ¦) , πΊ
2(ππ¦, ππ¦, ππ§) , πΊ
2
(ππ§, ππ§, ππ₯)}
πΊ (ππ₯, ππ¦, ππ§)
(3)
for all π₯, π¦, π§ β πwith 0 β€ π΄ < 1.Then, π andπ have a uniquecommon fixed point in π.
Gandhi and Bajpai [26] proved unique common fixedpoint results satisfying the following rational type contractivecondition.
Theorem6. Let (π, πΊ) be aπΊ-completeπΊ-metric space and letπ, π, β : π β π be three self-mappings satisfying the condition
πΊ (ππ₯, ππ¦, βπ§) β€ π΄
β
max {πΊ2(π₯, ππ₯, ππ¦) , πΊ
2(π¦, ππ¦, βπ§) , πΊ
2(π§, βπ§, ππ₯)}
πΊ (π₯, π¦, π§)
(4)
for all π₯, π¦, π§ β π with 0 β€ π΄ < 1. Then, π, π, and β have aunique common fixed point in π.
Currently, Shrivastava et al. [27] studied the followingresult.
Theorem7. Let (π, πΊ) be aπΊ-completeπΊ-metric space and letπ : π β π be a self-map satisfying the condition
for all π₯, π¦, π§ β π with 0 β€ π΄ + π΅ < 1/2. Then, π has a uniquecommon fixed point in π and π is πΊ-continuous at π’.
3. Main Results
Our first new result is the following.
Theorem8. Let (π, πΊ) be aπΊ-completeπΊ-metric space and letπ, π, π : π β π be three self-mappings satisfying the followingcondition:
π΅+πΆ < 1,πΊ(π₯, ππ₯, ππ¦)+πΊ(π₯, π¦, π§)+πΊ(π₯, ππ¦, π π§) = 0.Then, π,π, and π have a common fixed point. Further, ifπΊ(π₯, ππ₯, ππ¦)+
πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, π π§) = 0 implies πΊ(ππ₯, ππ¦, π π§) = 0, thenπ, π, and π have a unique common fixed point in π.
Proof. Let π₯0be arbitrary in π; we define a sequence π₯
πby
the following rules:
π₯3π+1
= ππ₯3π
,
π₯3π+2
= ππ₯3π+1
,
π₯3π+3
= π π₯3π+2
,
βπ β X.
(7)
Now, we have to show that π₯πis a πΊ-Cauchy sequence in π.
Consider πΊ(π₯, ππ₯, ππ¦) + πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, π π§) = 0; from(6), we have
πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) = πΊ (ππ₯3π
, ππ₯3π+1
, π π₯3π+2
) β€ π΄
β [πΊ (π₯3π
, ππ₯3π
, ππ₯3π+1
) πΊ (π₯3π+1
, ππ₯3π+1
, π π₯3π+2
)
+ [πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)]2
+ πΊ (π₯3π
, ππ₯3π
, ππ₯3π+1
)
β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)] (πΊ (π₯3π
, ππ₯3π
, ππ₯3π+1
)
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)
+ πΊ (π₯3π+1
, ππ₯3π+1
, π π₯3π+2
))β1
+ π΅
β (πΊ (π₯3π+1
, ππ₯3π+1
, π π₯3π+2
) [1
+ πΊ (π₯3π
, ππ₯3π
, ππ₯3π+1
)] (1
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
))β1
) + πΆ β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)
= π΄ β [πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
)
+ [πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)]2
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)
β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)] (πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) + πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
))β1
+ π΅ β (πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) [1
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)] (1
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
))β1
) + πΆ β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)
= π΄ β [πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) (πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
)
+ πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) + πΊ (π₯3π
, π₯3π+1
, π₯3π+2
))]
β (πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) + πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)
+ πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
))β1
+ π΅
β (πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) [1 + πΊ (π₯3π
, π₯3π+1
, π₯3π+2
)]
β (1 + πΊ (π₯3π
, π₯3π+1
, π₯3π+2
))β1
) + πΆ β πΊ (π₯3π
, π₯3π+1
,
π₯3π+2
) = π΄ β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) + π΅ β πΊ (π₯3π+1
,
π₯3π+2
, π₯3π+3
) + πΆ β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) = (π΄ + πΆ)
β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) + π΅ β πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) ,
(8)
which implies that
πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) β€ β β πΊ (π₯3π
, π₯3π+1
, π₯3π+2
) , (9)
where β = (π΄ + πΆ)/(1 β π΅).Similarly,
πΊ (π₯3π+3
, π₯3π+4
, π₯3π+5
) β€ β β πΊ (π₯3π+2
, π₯3π+3
, π₯3π+4
) . (10)
Therefore, for all π, we have
πΊ (π₯π+1
, π₯π+2
, π₯π+3
) β€ β β πΊ (π₯π, π₯π+1
, π₯π+2
) β€ β β β
β€ βπ+1
β πΊ (π₯0, π₯1, π₯2) .
(11)
Now, for all π, π, π, with π > π > π, using rectangularinequality, the second axiom of the πΊ-metric, and (11), wehave
where πΊ(π₯π, π₯π, π₯π) β 0 as π,π, π β β.
This shows that π₯πis a πΊ-Cauchy sequence. But (π, πΊ) is
πΊ-complete πΊ-metric space so there exists π€ in π such thatπ₯πβ π€ as π tends to infinity.Now, we assume that π π€ = π€. Using condition (6), we
have
πΊ (ππ€, π₯3π+2
, π₯3π+3
) = πΊ (ππ€, ππ₯3π+1
, π π₯3π+2
) β€ π΄
β [πΊ (π€, ππ€, ππ₯3π+1
) πΊ (π₯3π+1
, ππ₯3π+1
, π π₯3π+2
)
+ [πΊ (π€, π₯3π+1
, π₯3π+2
)]2
+ πΊ (π€, ππ€, ππ₯3π+1
)
β πΊ (π€, π₯3π+1
, π₯3π+2
)] (πΊ (π€, ππ€, ππ₯3π+1
)
+ πΊ (π€, π₯3π+1
, π₯3π+2
)
+ πΊ (π₯3π+1
, ππ₯3π+1
, π π₯3π+2
))β1
+ π΅
4 Journal of Function Spaces
β (πΊ (π₯3π+1
, ππ₯3π+1
, π π₯3π+2
)
β [1 + πΊ (π€, ππ€, ππ₯3π+1
)]
β (1 + πΊ (π€, π₯3π+1
, π₯3π+2
))β1
) + πΆ
β πΊ (π€, π₯3π+1
, π₯3π+2
) = π΄ β [πΊ (π€, ππ€, π₯3π+2
)
β πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) + [πΊ (π€, π₯3π+1
, π₯3π+2
)]2
+ πΊ (π€, ππ€, π₯3π+2
) πΊ (π€, π₯3π+1
, π₯3π+2
)]
β (πΊ (π€, π π€, π₯3π+2
) + πΊ (π€, π₯3π+1
, π₯3π+2
)
+ πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
))β1
+ π΅
β (πΊ (π₯3π+1
, π₯3π+2
, π₯3π+3
) [1 + πΊ (π€, ππ€, π₯3π+2
)]
β (1 + πΊ (π€, π₯3π+1
, π₯3π+2
))β1
) + πΆ
β πΊ (π€, π₯3π+1
, π₯3π+2
) .
(13)
As π₯πis πΊ-Cauchy sequence and converges to π€, therefore,
by taking limit π β β, we get πΊ(ππ€,π€, π€) β€ 0 which is heldonly if πΊ(ππ€,π€, π€) = 0 implies that ππ€ = π€. Similarly, it canbe shown that ππ€ = π€ and π π€ = π€. Hence, π€ is a commonfixed point of π, π and π .
Uniqueness. Suppose that π, π, and π have two common fixedpoints π§ andπ€ such that π§ = π€. Since conditionπΊ(π₯, ππ₯, ππ¦)+
πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, π π§) = 0 implies πΊ(ππ₯, ππ¦, π π§) = 0, wehave thatπΊ(π§, ππ§, ππ€)+πΊ(π§, π€, π€)+πΊ(π§, ππ€, π π€) = 0 impliesπΊ(ππ§, ππ€, π π€) = 0. Therefore, one can get the following:
which is a contradiction. Therefore, the common fixed pointis unique.
Corollary 9. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π, π, π : π β π be three self-mappings satisfying thecondition
for all π₯, π¦, π§ β π with π₯ = π¦ = π§ = π₯ π΄ β₯ 0 with 0 β€ π΄ <
1, πΊ(π₯, ππ₯, ππ¦) + πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, π π§) = 0. Then, π, π,and π have a common fixed point. Further, if πΊ(π₯, ππ₯, ππ¦) +
πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, π π§) = 0 implies πΊ(ππ₯, ππ¦, π π§) = 0, thenπ, T, and π have a unique common fixed point in π.
Proof. The proof follows by taking π΅ = πΆ = 0 in Theorem 8.
Corollary 10. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π, π, π : π β π be three self-mappings satisfying thecondition
for all π₯, π¦, π§ β π with π₯ = π¦ = π§ = π₯ π΅, πΆ β₯ 0 with 0 β€
π΅+πΆ < 1,πΊ(π₯, ππ₯, ππ¦)+πΊ(π₯, π¦, π§)+πΊ(π₯, ππ¦, π π§) = 0.Then π,π, and π have a common fixed point. Further, ifπΊ(π₯, ππ₯, ππ¦)+
πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, π π§) = 0 implies πΊ(ππ₯, ππ¦, π π§) = 0, thenπ, π, and π have a unique common fixed point in π.
Proof. The proof follows by taking π΄ = 0 in Theorem 8.
Corollary 11. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π, π : π β π be two self-mappings satisfying the condition
π΅ +πΆ < 1, πΊ(π₯, ππ₯, ππ¦) + πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, ππ§) = 0. Then,π and π have a common fixed point. Further, if πΊ(π₯, ππ₯, ππ¦) +
πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, ππ§) = 0 implies πΊ(ππ₯, ππ¦, ππ§) = 0, thenπ and π have a unique common fixed point in π.
Proof. The proof follows by taking π = π in Theorem 8.
By setting π = π = π inTheorem 8, we have the followingcorollary.
Corollary 12. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π : π β π be a self-mapping satisfying the condition
for all π₯, π¦, π§ β π with π₯ = π¦ = π§ = π₯ π΄, π΅, πΆ β₯ 0 with0 β€ π΄+π΅+πΆ < 1,πΊ(π₯, ππ₯, ππ¦)+πΊ(π₯, π¦, π§)+πΊ(π₯, ππ¦, ππ§) = 0.Then, π has a unique fixed point. Further, if πΊ(π₯, ππ₯, ππ¦) +
πΊ(π₯, π¦, π§) + πΊ(π₯, ππ¦, ππ§) = 0 implies πΊ(ππ₯, ππ¦, ππ§) = 0, thenπ has a unique common fixed point in π.
Journal of Function Spaces 5
The second main result in this section is the following.
Theorem 13. Let (π, πΊ) be a πΊ-complete πΊ-metric space. Letπ , π, π, πΌ, π½, π : π β π be six continuous self-maps and let{π, πΌ}, {π, π½}, and {π , π} be weakly commuting pairs of self-mapping such that π(π) β πΌ(π), π(π) β π½(π), and π (π) β
which is a Cauchy sequence in πΊ-complete πΊ-metric spaceand therefore converges to a limit point π€. But all subse-quences of a convergent sequence converge; so, we have
limπββ
π π₯3π
= limπββ
ππ₯3π+1
= π€,
limπββ
ππ₯3π
= limπββ
π½π₯3π+1
= π€,
limπββ
ππ₯3πβ1
= limπββ
πΌπ₯3π
= π€.
(24)
Since {π, πΌ} are weakly commuting mappings, thus we have
πΊ (ππΌπ₯3π
, πΌππ₯3π
, πΌππ₯3π
) β€ πΊ (πΌπ₯3π
, ππ₯3π
, ππ₯3π
) . (25)
Taking limit π β β and noting that π and πΌ are continuousmappings, we have
which gives the notion that ππ€ = πΌπ€. Analogously, we canget ππ€ = π½π€ and π π€ = ππ€. We claim that π π€ = ππ€ andππ€ = ππ€ and then from condition (3)
Similarly, using similar arguments to those given above, weobtain a contradiction for π π€ = ππ€ and ππ€ = ππ€ or forπ π€ = ππ€ and ππ€ = ππ€. Hence, in all the cases, we concludethat π π€ = ππ€ = ππ€. We prove that any fixed point of π is afixed point of π, π, π, πΌ, and π½. Assume thatπ€ β π is such thatπ π€ = π€. Now, we prove that π€ = ππ€ = ππ€. If it is not thecase, then, for π€ = ππ€ and π€ = ππ€, we get
πΊ(ππ§, π½π§, πΌπ€) = 0, which can be written asπΊ(ππ§, ππ§, π π€) = 0
or πΊ(ππ§, π½π§, πΌπ€) = 0.Therefore, one can get the following:
πΊ (π§, π§, π€) = 0
or πΊ (π§, π§, π€) = 0 implies that π§ = π€.
(30)
Theorem 13 produces the following corollaries.
Corollary 14. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π , π, π, πΌ, π½, π : π β π be three self-maps and let {π, πΌ},{π, π½}, and {π , π} be weakly commuting pairs of self-mappingsuch that π(π) β πΌ(π), π(π) β π½(π), and π (π) β π(π),satisfying
for all π₯, π¦, π§ in π with π₯ = π¦ = π§ = π₯ with 0 β€ π΅ < 1. Then,π , π, π, πΌ, π½, and π have a unique common fixed point in π.
Proof. It follows by taking π΄ = 0 in Theorem 13.
Corollary 15. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π , π, π, πΌ, π½, π : π β π be three self-maps and let {π, πΌ},{π, π½}, and {π , π} be weakly commuting pairs of self-mappingsuch that π(π) β πΌ(π), π(π) β π½(π), and π (π) β π(π),satisfying
for all π₯, π¦, π§ in π with π₯ = π¦ = π§ = π₯ π΄ β₯ 0 with 0 β€
π΄ < 1, πΊ(π π₯, ππ₯, πΌπ₯) + πΊ(ππ₯, π½π¦, πΌπ§) + πΊ(π π₯, ππ₯, πΌπ₯) = 0.Then, π , π, π, πΌ, π½, and π have a common fixed point. Further,if πΊ(π π₯, ππ₯, πΌπ₯) + πΊ(ππ₯, π½π¦, πΌπ§) + πΊ(π π₯, ππ₯, πΌπ₯) = 0 impliesπΊ(ππ₯, ππ¦, π π§)+πΊ(ππ₯, π½π¦, πΌπ§) = 0, thenπ , π, π, πΌ, π½, andπ havea unique common fixed point in π.
Proof. It follows by taking π΅ = 0 in Theorem 13.
Corollary 16. Let (π, πΊ) be a πΊ-complete πΊ-metric space andlet π, π , πΌ, π½ : π β π be three self-maps and let {π, πΌ}, {π, π½},and {π , πΌ} be weakly commuting pairs of self-mapping such thatπ(π) β πΌ(π), π(π) β π½(π), and π (π) β πΌ(π), satisfying
0. Then, π, π , πΌ, and π½ have a common fixed point. Further,if πΊ(π π₯, ππ₯, πΌπ₯) + πΊ(πΌπ₯, π½π¦, πΌπ§) + πΊ(π π₯, ππ₯, πΌπ₯) = 0 impliesπΊ(ππ₯, ππ¦, π π§) + πΊ(πΌπ₯, π½π¦, πΌπ§) = 0, then π, π , πΌ, and π½ have aunique common fixed point in π.
Proof. The proof follows by setting π = π and πΌ = π inTheorem 13.
Journal of Function Spaces 7
Competing Interests
The authors declare that they have no competing interests.
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