# Generalized Common Fixed Point Results via Greatest Lower Bound Property

Authors: Marwan Amin Kutbi; Jamshaid Ahmad; Akbar Azam; Ahmed Saleh Al-Rawashdeh
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The aim of this paper is to unify the concept of greatest lower bound (g.l.b) property and establish some generalized common fixed results. We support our results by a nontrivial example.


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## Body

## 1. Introduction and Preliminaries

The Banach fixed point theorem is used to establish the existence of a unique solution for a nonlinear integral equation [1]. Moreover this theorem plays an important role in several branches of mathematics. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool of mathematical analysis in many directions. Nadler [2] introduced the concept of multivalued contraction mappings and obtained the fixed points results for multivalued mappings. Huang and Zhang [3] introduced the notion of cone metric space which is a generalization of metric space. They extended Banach contraction principle to cone metric spaces. Since then, Arshad et al. [4], Azam and Arshad [5], Cho and Bae [6], and many others obtained fixed point theorems in cone metric spaces.Azam et al. in [7] introduced the notion of complex-valued metric space and obtained some common fixed points of a pair of mappings satisfying rational expressions contractive condition. Although complex-valued metric spaces form a special class of cone metric space, yet this idea is intended to define rational expressions which are not meaningful in cone metric spaces. Subsequently, Rouzkard and Imdad [8] and Abbas et al. [9, 10] established some common fixed point theorems satisfying certain rational expressions in complex-valued metric spaces which generalize, unify, and complement the results of Azam et al. [7]. Sintunavarat et al. [11, 12] obtained common fixed point results by replacing constant of contractive condition with control functions. Klin-eam and Suanoom [13] established a common fixed point result for two single valued mappings in complex-valued metric spaces. Abbas et al. [14] introduced complex-valued generalized metric space and obtained common fixed point results in this space. For more details in fixed point theory, we refer the reader to [15–23]. Very recently, Ahmad et al. [24] introduced the notion of greatest lower bound (g.l.b.) property of the multivalued mappings and obtained some common fixed point results in the context of complex-valued metric spaces. Then Azam et al. [25] extend the concept of greatest lower bound (g.l.b.) property and proved some new common fixed point theorems in the setting of complex-valued metric spaces. In this paper, we present some new common fixed results and generalized the results of [24, 25].LetC denote the set of complex numbers. Let z1,z2∈C. Define a partial order ≾ on C as follows:
(1)z1≾z2iffRe(z1)⩽Re(z2),Im⁡(z1)⩽Im⁡(z2).It follows that(2)z1≾z2,
if one of the following conditions is satisfied: (i)
Re(z1)=Re(z2), Im⁡(z1)<Im⁡(z2),(ii)
Re(z1)<Re(z2), Im⁡(z1)=Im⁡(z2),(iii)
Re(z1)<Re(z2), Im⁡(z1)<Im⁡(z2),(iv)
Re(z1)=Re(z2), Im⁡(z1)=Im⁡(z2).In particular, we will write z1⋨z2 if z1≠z2 and one of (i), (ii), and (iii) is satisfied and we will write z1≺z2 if only (iii) is satisfied. Note that
(3)0≾z1⋨z2⟹|z1|<|z2|,z1≾z2,z2≺z3⟹z1≺z3.Definition 1.
LetX be a nonempty set. Suppose that a mapping dc:X×X→C satisfies the following: (1)
0≾dc(ζ,η), for all ζ,η∈X and dc(ζ,η)=0 if and only if ζ=η;(2)
dc(ζ,η)=dc(η,ζ) for all ζ,η∈X;(3)
dc(ζ,η)≾dc(ζ,ς)+dc(ς,η), for all ζ,η,ς∈X.Thendc is called a complex-valued metric on X, and (X,dc)is called a complex-valued metric space. A point ζ∈X is called interior point of a set A⊆X whenever there exists 0≺r∈C such that
(4)B(ζ,r)={η∈X:dc(ζ,η)≺r}⊆A.A pointζ∈X is called a limit point of A whenever for every 0≺r∈C(5)B(ζ,r)∩(A∖{ζ})≠ϕ.A is called open set whenever each element of A is an interior point of A and a subset B⊆X is called closed set whenever each limit point of B belongs to B. The family
(6)F={B(ζ,r):ζ∈X,0≺r}
is a subbase for a Hausdorff topology τ on X. Let {ζn} be a sequence in X and ζ∈X. If for every c∈C with 0≺c there is n0∈N such that for all n>n0, dc(ζn,ζ)≺c, then {ζn} is said to be convergent, {ζn} converges to ζ, and ζ is the limit point of {ζn}. We denote this by lim⁡n→∞ζn=ζ, or ζn→ζ as n→∞. If for every c∈C with 0≺c there is n0∈N such that for all n>n0, dc(ζn,ζn+m)≺c, then {ζn} is called a Cauchy sequence in (X,dc). If every Cauchy sequence is convergent in (X,dc), then (X,dc) is called a complete complex-valued metric space. We require the following lemmas.Lemma 2 (see [7]).
Let(X,dc) be a complex-valued metric space and let {ζn} be a sequence in X. Then {ζn} converges to ζ if and only if  |dc(ζn,ζ)|→0 as n→∞.Lemma 3 (see [7]).
Let(X,dc) be a complex-valued metric space and let {ζn} be a sequence in X. Then {ζn} is a Cauchy sequence if and only if |dc(ζn,ζm)|→0 as n,m→∞.


## 2. Main Result

Let(X,dc) be a complex-valued metric space. In the sequel of [24], we denote nonempty, closed, and bounded subsets of X by CB(X), respectively.Throughout this paper, we denotes(z1)={z2∈C:z1⪯z2} for z1∈C and s(a,B)=∪b∈Bs(dc(a,b))=∪b∈B{z∈C:dc(a,b)⪯z} for a∈X and B∈CB(X).ForA,B∈CB(X) we denote
(7)s(A,B)=(∩a∈As(a,B))∩(∩b∈Bs(b,A)).Remark 4.
Let(X,dc) be a complex-valued metric space. If C=R, then (X,d) is a metric space. Moreover for A,B∈C(X), H(A,B)=inf⁡s(A,B) is the Hausdorff distance induced by dc.Let(X,dc) be a complex-valued metric space and CB(X) be a collection of nonempty closed subsets of X. Let S:X→CB(X) be a multivalued map. For ζ∈X and A∈CB(X), define
(8)Wζ(A)={dc(ζ,a):a∈A}.
Thus for ζ,η∈X,
(9)Wζ(Sη)={dc(ζ,u):u∈Sη}.Definition 5 (see [24]).
Let(X,dc) be a complex-valued metric space. A nonempty subset A of X is called bounded from below if there exists some z∈C, such that z⪯a for all a∈A.Definition 6 (see [24]).
Let(X,dc) be acomplex-valued metric space. A multivalued mapping F:X→2C is called bounded from below if for ζ∈X there exists zζ∈C such that
(10)zζ≾u,
for all u∈Fζ.Definition 7 (see [24]).
Let(X,dc) be a complex-valued metric space. The multivalued mapping S:X→CB(X) is said to have lower bound property (l.b. property) on (X,dc), if for any ζ∈X the multivalued mapping Fζ:X→2C defined by
(11)Fζ(Sη)=Wζ(Sη)
is bounded from below. That is, for ζ,η∈X there exists an element lζ(Sη)∈C such that
(12)lζ(Sη)≾u,
for all u∈Wζ(Sη), where lζ(Sη) is called lower bound of S associated with (ζ,η).Definition 8 (see [24]).
Let(X,dc) be a complex-valued metric space. The multivalued mapping S:X→CB(X) is said to have greatest lower bound property (g.l.b. property) on (X,dc), if greatest lower bound of Wζ(Sη) exists in C for all ζ,η∈X. We denote dc(ζ,Sη) by the g.l.b. of Wζ(Sη). That is
(13)dc(ζ,Sη)=inf⁡{dc(ζ,u):u∈Sη}.Theorem 9.
Let(X,dc) be a complete complex-valued metric space and let S,F:X→CB(X) be multivalued mappings with g.l.b. property such that
(14)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Sζ)dc(ζ,Fη)1+dc(ζ,η)+Ddc(ζ,Sζ)dc(ζ,Fη)1+dc(ζ,η)+Edc(η,Sζ)dc(η,Fη)1+dc(ζ,η)∈s(Sζ,Fη),
for all ζ,η∈X, where A, B, C, D, and E are nonnegative real numbers with A+B+C+2D+2E<1. Then S,F have a common fixed point.Proof.
Letζ0 be an arbitrary point in X and ζ1∈Sζ0. From (14), we have
(15)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)∈s(Sζ0,Fζ1).
This implies that
(16)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)∈(∩ζ∈Sζ0s(ζ,Fζ1)),Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)∈s(ζ,Fζ1),
for all ζ∈Sζ0. Since ζ1∈Sζ0, we have
(17)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)∈s(ζ1,Fζ1),Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)∈∪ζ∈Fζ1s(dc(ζ1,ζ)).
So there exists some ζ2∈Fζ1, such that
(18)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)∈s(dc(ζ1,ζ2)).
That is
(19)dc(ζ1,ζ2)⪯Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Ddc(ζ0,Sζ0)dc(ζ0,Fζ1)1+dc(ζ0,ζ1)+Edc(ζ1,Sζ0)dc(ζ1,Fζ1)1+dc(ζ0,ζ1).
By the greatest lower bound property (g.l.b. property) of S and F, we get
(20)dc(ζ1,ζ2)⪯Adc(ζ0,ζ1)+Bdc(ζ0,ζ1)dc(ζ1,ζ2)1+dc(ζ0,ζ1)+Cdc(ζ1,ζ1)dc(ζ0,ζ2)1+dc(ζ0,ζ1)+Ddc(ζ0,ζ1)dc(ζ0,ζ2)1+dc(ζ0,ζ1)+Edc(ζ1,ζ1)dc(ζ1,ζ2)1+dc(ζ0,ζ1)=Adc(ζ0,ζ1)+Bdc(ζ0,ζ1)dc(ζ1,ζ2)1+dc(ζ0,ζ1)+dcdc(ζ0,ζ1)dc(ζ0,ζ2)1+dc(ζ0,ζ1),
which implies that
(21)|dc(ζ1,ζ2)|≤A|dc(ζ0,ζ1)|+B|dc(ζ0,ζ1)||dc(ζ1,ζ2)||1+dc(ζ0,ζ1)|+D|dc(ζ0,ζ1)||dc(ζ0,ζ2)||1+dc(ζ0,ζ1)|=A|dc(ζ0,ζ1)|+B|dc(ζ1,ζ2)||dc(ζ0,ζ1)1+dc(ζ0,ζ1)|+D|dc(ζ0,ζ2)||dc(ζ0,ζ1)||1+dc(ζ0,ζ1)|,|dc(ζ1,ζ2)|≤A|dc(ζ0,ζ1)|+B|dc(ζ1,ζ2)|+D|dc(ζ0,ζ2)|.
Then
(22)|dc(ζ1,ζ2)|≤A+D(1-B-D)|dc(ζ0,ζ1)|.
Similarly, we get
(23)Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)∈s(Fζ1Sζ2).
This implies that
(24)Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)∈(∩ζ∈Sζ0s(ζ,Sζ2)),Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)∈s(ζ,Sζ2),
for all ζ∈Fζ1. Since ζ2∈Fζ1, we have
(25)Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)∈s(ζ2,Sζ2),Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)∈s(ζ2,Sζ2)=∪ζ∈Fζ1s(dc(ζ2,ζ)).
So there exists some ζ3∈Sζ2, such that
(26)Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)∈s(dc(ζ2,ζ3)).
That is,
(27)dc(ζ2,ζ3)⪯Adc(ζ1,ζ2)+Bdc(ζ1,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Cdc(ζ2,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2)+Ddc(ζ2,Fζ1)dc(ζ2,Sζ2)1+dc(ζ1,ζ2)+Edc(ζ1,Fζ1)dc(ζ1,Sζ2)1+dc(ζ1,ζ2).
By the greatest lower bound property (g.l.b. property) of S and F, we get
(28)dc(ζ2,ζ3)⪯Adc(ζ1,ζ2)+Bdc(ζ1,ζ2)dc(ζ2,ζ3)1+dc(ζ1,ζ2)+Cdc(ζ2,ζ2)dc(ζ1,ζ3)1+dc(ζ1,ζ2)+Ddc(ζ2,ζ2)dc(ζ2,ζ3)1+dc(ζ1,ζ2)+Edc(ζ1,ζ2)dc(ζ1,ζ3)1+dc(ζ1,ζ2),
which implies that
(29)|dc(ζ2,ζ3)|≤A|dc(ζ1,ζ2)|+B|dc(ζ2,ζ3)||dc(ζ1,ζ2)1+dc(ζ1,ζ2)|+E|dc(ζ1,ζ3)||dc(ζ1,ζ2)1+dc(ζ1,ζ2)|≤A|dc(ζ1,ζ2)|+B|dc(ζ2,ζ3)|+E|dc(ζ1,ζ3)|.
Then
(30)|dc(ζ2,ζ3)|≤A+E(1-B-E)|dc(ζ1,ζ2)|.
Putting λ=max⁡{((A+D)/(1-B-D)),((A+E)/(1-B-E))}, we obtain a sequence {ζn} in ζ such that for n=0,1,2,…,ζ2n+1∈Sζ2n and ζ2n+2∈Fζ2n+1(31)|dc(ζn,ζn+1)|≤λ|dc(ζn-1,ζn)|≤λ2|dc(ζn-2,ζn-1)|⋯≤λn|dc(ζ0,ζ1)|.
Now for m>n, we get
(32)|dc(ζn,ζm)|≤|dc(ζn,ζn+1)|+|dc(ζn+1,ζn+2)|+⋯+|dc(ζm-1,ζm)|≤[λn+λn+1+⋯+λm-1]|dc(ζ0,ζ1)|≤[λn1-λ]|dc(ζ0,ζ1)|,
and so
(33)|dc(ζn,ζm)|≤λn1-λ|dc(ζ0,ζ1)|⟶0asm,n⟶∞.
This implies that {ζn} is a Cauchy sequence in ζ. As X is complete space, there exists u∈X such that ζn→u as n→∞. We now show that u∈Fu and u∈Su. From (14), we have
(34)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)∈s(Sζ2k,Fu).
This implies that
(35)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)∈(∩ζ∈Sζ2ks(ζ,Fu)),Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)∈s(ζ,Fu),
for all ζ∈Sζ2k. Since ζ2k+1∈Sζ2k, we have
(36)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)∈s(ζ2k+1,Fu).
By definition
(37)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)∈s(ζ2k+1,Fu)=∪u′∈Fus(dc(ζ2k+1,u′)).
There exists some uk∈Fu such that
(38)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)∈s(dc(ζ2k+1,uk));
that is,
(39)dc(ζ2k+1,uk)⪯Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)1+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Ddc(ζ2k,Sζ2k)dc(ζ2k,Fu)1+dc(ζ2k,u)+Edc(u,Sζ2k)dc(u,Fu)1+dc(ζ2k,u).
By the greatest lower bound property (g.l.b. property) of S and F, we have
(40)dc(ζ2k+1,uk)⪯Adc(ζ2k,u)+Bdc(ζ2k,ζ2k+1)dc(u,uk)1+dc(ζ2k,u)+Cdc(u,ζ2k+1)dc(ζ2k,uk)1+dc(ζ2k,u)+Ddc(ζ2k,ζ2k+1)dc(ζ2k,uk)1+dc(ζ2k,u)+Edc(u,ζ2k+1)dc(u,uk)1+dc(ζ2k,u).
Since
(41)dc(u,uk)⪯dc(u,ζ2k+1)+dc(ζ2k+1,uk),
we get
(42)dc(u,uk)⪯dc(u,ζ2k+1)+Adc(ζ2k,u)+Bdc(ζ2k,ζ2k+1)dc(u,uk)1+dc(ζ2k,u)+Cdc(u,ζ2k+1)dc(ζ2k,uk)1+dc(ζ2k,u)+Ddc(ζ2k,ζ2k+1)dc(ζ2k,uk)1+dc(ζ2k,u)+Edc(u,ζ2k+1)dc(u,uk)1+dc(ζ2k,u),
which implies that
(43)|dc(u,uk)|≤|dc(u,ζ2k+1)|+A|dc(u,ζ2k+1)|+B|dc(ζ2k,ζ2k+1)||dc(u,uk)||1+dc(ζ2k,u)|+C|dc(u,ζ2k+1)||dc(ζ2k,uk)||1+dc(ζ2k,u)|+D|dc(ζ2k,ζ2k+1)||dc(ζ2k,uk)||1+dc(ζ2k,u)|+E|dc(u,ζ2k+1)||dc(u,uk)||1+dc(ζ2k,u)|.
Taking the limit as k→∞, we get |dc(u,uk)|→0 as k→∞. By lemma 1 [7], we have uk→u as k→∞. Since Fu is closed, u∈Fu. Similarly, it follows that u∈Su. Hence S and F have a common fixed point and our theorem follows.Consequently, we have the following corollaries.By settingS=F in Theorem 9, we get the following corollary.Corollary 10.
Let(X,dc) be a complete complex-valued metric space and let F:X→CB(X) be multivalued mapping with g.l.b. property such that
(44)Adc(ζ,η)+Bdc(ζ,Fζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Fζ)dc(ζ,Fη)1+dc(ζ,η)+Ddc(ζ,Fζ)dc(ζ,Fη)1+dc(ζ,η)+Edc(η,Fζ)dc(η,Fη)1+dc(ζ,η)∈s(Fζ,Fη),
for all ζ,η∈X, where A, B, C, D, and E are nonnegative real numbers with A+B+C+2D+2E<1. Then F has a fixed point.By choosingE=0 in Theorem 9, we get the following corollary.Corollary 11.
Let(X,dc) be a complete complex-valued metric space and let S,F:X→CB(X) be multivalued mappings with g.l.b. property such that
(45)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Sζ)dc(ζ,Fη)1+dc(ζ,η)+Ddc(ζ,Sζ)dc(ζ,Fη)1+dc(ζ,η)∈s(Sζ,Fη),
for all ζ,η∈X, where A, B, C, and D are nonnegative real numbers with A+B+C+2D<1. Then S,F have a common fixed point.By settingS=F in Corollary 11, we get the following corollary.Corollary 12.
Let(X,dc) be a complete complex-valued metric space and let F:X→CB(X) be multivalued mapping with g.l.b. property such that
(46)Adc(ζ,η)+Bdc(ζ,Fζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Fζ)dc(ζ,Fη)1+dc(ζ,η)+Ddc(ζ,Fζ)dc(ζ,Fη)1+dc(ζ,η)∈s(Fζ,Fη),
for all ζ,η∈X, where A, B, C, and D are nonnegative real numbers with A+B+C+2D<1. Then F has a fixed point.By choosingD=0 in Theorem 9, we get the following corollary.Corollary 13.
Let(X,dc) be a complete complex-valued metric space and let S,F:X→CB(X) be multivalued mappings with g.l.b. property such that
(47)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Sζ)dc(ζ,Fη)1+dc(ζ,η)+Edc(η,Sζ)dc(η,Fη)1+dc(ζ,η)∈s(Sζ,Fη),
for all ζ,η∈X, where A, B, C, and E are nonnegative real numbers with A+B+C+2E<1. Then S,F have a common fixed point.By settingS=F in Corollary 13, we get the following corollary.Corollary 14.
Let(X,dc) be a complete complex-valued metric space and let F:X→CB(X) be multivalued mapping with g.l.b. property such that
(48)Adc(ζ,η)+Bdc(ζ,Fζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Fζ)dc(ζ,Fη)1+dc(ζ,η)+Edc(η,Fζ)dc(η,Fη)1+dc(ζ,η)∈s(Fζ,Fη),
for all ζ,η∈X, where A, B, C, and E are nonnegative real numbers with A+B+C+2E<1. Then F has a fixed point.By choosingD=E=0 in Theorem 9, we get the following corollary.Corollary 15 (see [24]).
Let(X,dc) be a complete complex-valued metric space and let S,F:X→CB(X) be multivalued mappings with g.l.b. property such that
(49)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Sζ)dc(ζ,Fη)1+dc(ζ,η)∈s(Sζ,Fη),
for all ζ,η∈X, where A, B, and C are nonnegative real numbers with A+B+C<1. Then S,F have a common fixed point.By settingS=F in Corollary 15, we get the following corollary.Corollary 16 (see [24]).
Let(X,dc) be a complete complex-valued metric space and let F:X→CB(X) be multivalued mapping with g.l.b. property such that
(50)Adc(ζ,η)+Bdc(ζ,Fζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Fζ)dc(ζ,Fη)1+dc(ζ,η)∈s(Fζ,Fη),
for all ζ,η∈X, where A, B, and C are nonnegative real numbers with A+B+C<1. Then F has a fixed point.By choosingC=D=E=0 in Theorem 9, we get the following corollary.Corollary 17 (see [24]).
Let(X,dc) be a complete complex-valued metric space and let S,F:X→CB(X) be multivalued mappings with g.l.b. property such that
(51)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)1+dc(ζ,η)∈s(Sζ,Fη),
for all ζ,η∈X, where A,B are nonnegative real numbers with A+B<1. Then S,F have a common fixed point.By settingS=F in Corollary 17, we get the following corollary.Corollary 18.
Let(X,dc) be a complete complex-valued metric space and let F:X→CB(X) be multivalued mapping with g.l.b. property such that
(52)Adc(ζ,η)+Bdc(ζ,Fζ)dc(η,Fη)1+dc(ζ,η),
for all ζ,η∈X, where A,B are nonnegative real numbers with A+B<1. Then F has a fixed point.By Remark4, we get the following corollaries.Corollary 19.
Let(X,d) be a complete metric space and let S,T:X→CB(X) be multivalued mappings such that
(53)H(Sx,Ty)≤Ad(x,y)+Bd(x,Sx)d(y,Ty)1+d(x,y)+Cd(y,Sx)d(x,Ty)1+d(x,y)+Dd(x,Sx)d(x,Ty)1+d(x,y)+Ed(y,Sx)d(y,Ty)1+d(x,y),
for all x,y∈X and A, B, C, D, and E are nonnegative real numbers with A+B+C+2D+2E<1. Then S,T have a common fixed point.Corollary 20.
Let(X,d) be a complete metric space and let T:X→CB(X) be multivalued mappings such that
(54)H(Tx,Ty)≤Ad(x,y)+Bd(x,Tx)d(y,Ty)1+d(x,y)+Cd(y,Tx)d(x,Ty)1+d(x,y)+Dd(x,Tx)d(x,Ty)1+d(x,y)+Ed(y,Tx)d(y,Ty)1+d(x,y),
for all x,y∈X and A, B, C, D, and E are nonnegative real numbers with A+B+C+2D+2E<1. Then T has a fixed point.Remark 21.
By equatingA, B, C, D, and E to 0 in all possible combinations, one can derive a host of corollaries which include Banach fixed point theorem for multivalued mappings in complete metric space.Theorem 22.
Let(X,dc) be a complete complex-valued metric space and let S,F:X→CB(X) be multivalued mappings with g.l.b. property such that
(55)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)dc(ζ,Fη)+dc(η,Sζ)+dc(ζ,η)+Cdc(η,Sζ)dc(ζ,Fη)dc(ζ,Fη)+dc(η,Sζ)+dc(ζ,η)∈s(Sζ,Fη),
for all ζ,η∈X, where A, B, and C are nonnegative real numbers with A+B+C<1. Then S,F have a common fixed point.Proof.
Letζ0 be an arbitrary point in X and ζ1∈Sζ0. From (55), we have
(56)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)∈s(Sζ0,Fζ1).
This implies that
(57)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)∈(∩ζ∈Sζ0s(ζ,Fζ1)),Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)∈s(ζ,Fζ1),
for all ζ∈Sζ0. Since ζ1∈Sζ0, we have
(58)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)∈s(ζ1,Fζ1),Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)∈∪ζ∈Fζ1s(dc(ζ1,ζ)).
So there exists some ζ2∈Fζ1, such that
(59)Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)∈s(dc(ζ1,ζ2)).
That is,
(60)dc(ζ1,ζ2)⪯Adc(ζ0,ζ1)+Bdc(ζ0,Sζ0)dc(ζ1,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1)+Cdc(ζ1,Sζ0)dc(ζ0,Fζ1)dc(ζ0,Fζ1)+dc(ζ1,Sζ0)+dc(ζ0,ζ1).
By using the greatest lower bound property (g.l.b. property) of S and F, we get
(61)dc(ζ1,ζ2)⪯Adc(ζ0,ζ1)+Bdc(ζ0,ζ1)dc(ζ1,ζ2)dc(ζ0,ζ2)+dc(ζ0,ζ1),
which implies that
(62)|dc(ζ1,ζ2)|≤A|dc(ζ0,ζ1)|+B|dc(ζ0,ζ1)||dc(ζ1,ζ2)||dc(ζ0,ζ2)+dc(ζ0,ζ1)|.
As earlier, by the triangular inequality
(63)|dc(ζ1,ζ2)|≤|dc(ζ1,ζ0)+dc(ζ0,ζ2)|,
we get
(64)|dc(ζ1,ζ2)|≤(A+B)|dc(ζ0,ζ1)|.
Similarly we can prove that
(65)|dc(ζ2,ζ3)|≤(A+B)|dc(ζ1,ζ2)|.
By putting A+B=l<1 and continuing in a similar way to the proof of Theorem 9, we obtain that {ζn} is a Cauchy sequence in X and ζn→u as n→∞. We now show that u∈Fu and u∈Su. From (55), we have
(66)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)∈s(Sζ2k,Fu).
This implies that
(67)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)∈(∩ζ∈Sζ2ks(ζ,Fu)),(68)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)∈s(ζ,Fu),
for all ζ∈Sζ2k. Since ζ2k+1∈Sζ2k, we have
(69)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)∈s(ζ2k+1,Fu).
By definition
(70)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)∈s(ζ2k+1,Fu)=∪u′∈Fus(dc(ζ2k+1,u′)).
There exists some uk∈Fu such that
(71)Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)∈s(dc(ζ2k+1,uk));
that is,
(72)dc(ζ2k+1,uk)⪯Adc(ζ2k,u)+Bdc(ζ2k,Sζ2k)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,Sζ2k)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u).
By the greatest lower bound property (g.l.b. property) of S and F, we have
(73)dc(ζ2k+1,uk)⪯Adc(ζ2k,u)+Bdc(ζ2k,ζ2k+1)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,ζ2k+1)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u).
Since
(74)dc(u,uk)⪯dc(u,ζ2k+1)+dc(ζ2k+1,uk),
we get
(75)dc(u,uk)⪯dc(u,ζ2k+1)+Adc(ζ2k,u)+Bdc(ζ2k,ζ2k+1)dc(u,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)+Cdc(u,ζ2k+1)dc(ζ2k,Fu)dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u),
which implies that
(76)|dc(u,uk)|≤|dc(u,ζ2k+1)|+A|dc(ζ2k,u)|+B|dc(ζ2k,ζ2k+1)||dc(u,Fu)||dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)|+C|dc(u,ζ2k+1)||dc(ζ2k,Fu)||dc(ζ2k,Fu)+dc(u,Sζ2k)+dc(ζ2k,u)|.
Taking the limit as k→∞, we get |dc(u,uk)|→0 as k→∞. By lemma 1 [7], we have uk→u as k→∞. Since Fu is closed, so u∈Fu. Similarly, it follows that u∈Su. Hence S and F have a common fixed point and our theorem follows.Corollary 23.
Let(X,d) be a complete complex-valued metric space and let T:X→CB(X) be multivalued mapping with g.l.b. property such that
(77)Ad(x,y)+Bd(x,Tx)d(y,Ty)d(x,Ty)+d(y,Tx)+d(x,y)+Cd(y,Tx)d(x,Ty)d(x,Ty)+d(y,Tx)+d(x,y)∈s(Tx,Ty),
for all x,y∈X, where A, B, and C are nonnegative real numbers with A+B+C<1. Then T has fixed point.Now, let us consider the following example.Example 24.
LetX=[0,1]; define dc:X×X→C by
(78)dc(ζ,η)=|ζ-η|eiθ,θ=tan-1|ηζ|.
Then (X,dc) is a complex-valued metric space. Let S,F:X→CB(X) be the mappings defined by
(79)Sζ=[0,16ζ],Fζ=[0,112ζ].
The contractive condition of main theorem is trivial for the case when ζ=η=0. Suppose without any loss of generality that all ζ,η are nonzero and ζ<η. Then
(80)dc(ζ,η)=|η-ζ|eiθ,dc(ζ,Sζ)=|ζ-ζ6|eiθ,dc(η,Fη)=|η-η12|eiθ,dc(η,Sζ)=|η-ζ6|eiθ,dc(ζ,Fη)=|ζ-η12|eiθ,s(Sζ,Fη)=s(|ζ6-η12|eiθ).It is clear that, for any value ofB, C, D, E, and A=1/6, we have
(81)|ζ6-η12|≤16|η-ζ|.
Thus
(82)Adc(ζ,η)+Bdc(ζ,Sζ)dc(η,Fη)1+dc(ζ,η)+Cdc(η,Sζ)dc(ζ,Fη)1+dc(ζ,η)+Ddc(ζ,Sζ)dc(ζ,Fη)1+dc(ζ,η)+Edc(η,Sζ)dc(η,Fη)1+dc(ζ,η)∈s(Sζ,Fη).
Hence all the conditions of our main Theorem 9 are satisfied and 0 is a common fixed point of S and F.



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*Source: 265865-2014-07-10.xml*