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Authors: Erdal Karapinar; Stefan Czerwik; Hassen Aydi
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We present a fixed point theorem for generalized(α,ψ)-Meir-Keeler type contractions in the setting of generalized b-metric spaces. The presented results improve, generalize, and unify many existing famous results in the corresponding literature.


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## Body

## 1. Introduction and Preliminaries

The idea of ab-metric has been introduced in the papers [1, 2]. Very recently, this idea was extended in [3] to a generalized b-metric space in the following manner.Definition 1.
LetX be a nonempty set and s≥1 be a fixed constant. A function d:X×X→[0,∞] is called a generalized b-metric space (in brief, gbms) if and only if for x,y,z∈X the conditions are satisfied:(d1)
d(x,y)=0 if and only if x=y.(d2)
d(x,y)=d(y,x).(d3)
d(x,y)≤s[d(x,z)+d(z,y)].A triple(X,d,s) is called a generalized b-metric space.On the other hand, Meir and Keeler [4] have proved the following very general result on the existence of fixed points of Meir-Keeler contraction mappings in metric spaces.Theorem 2 (see [4]).
Let(X,d) be a complete metric space and f:X→X satisfy the following condition: (d)
Givenε>0, there exists δ>0 such that (1)ε≤dx,y<ε+δimpliesdfx,fy<ε.
f has a unique fixed point ξ. Moreover, for any x∈X, (2)limn→∞⁡fnx=ξ,where fnx denotes the nth iteration of f at a point x.This result has been generalized and extended in many directions; see [5–15]. Using some auxiliary functions, the main purpose of this paper is to extend and generalize this result on generalized b-metric spaces.For the sake of explicitness, we recall some notations. The symbolsN,R denote the natural and real numbers, respectively. Furthermore, N0≔{0}∪N and R0+≔[0,∞).Berinde [16] characterized comparison functions to define the contraction mappings in the setting of b-metric spaces.Definition 3.
Lets≥1 be a real number. A function ϕ:[0,∞)→[0,∞) is called a (c)-comparison function if (1)
ϕ is increasing;(2)
there existk0∈N, a∈(0,1), and a convergent nonnegative series ∑k=1∞vk such that sk+1ϕk+1(t)≤askϕk(t)+vk, for k≥k0 and any t≥0.DenoteΨ as the set of (c)-comparison functions. We will need the following essential properties in our further discussion.Lemma 4 (see [16–18]).
For a(c)-comparison function φ:[0,+∞)→[0,+∞), the following statements hold: (1)
The series∑k=0∞‍skφk(t) converges for any t∈[0,+∞).(2)
The functionbs:[0,+∞)→[0,+∞) defined by bs(t)=∑k=0∞‍skφk(t), t∈[0,∞), is increasing and continuous at 0.(3)
Each iterateφk of φ for k≥1 is also a (c)-comparison function.(4)
φ is continuous at 0.(5)
φ(t)<t for any t>0.Inspired by Popescu [19], we introduce the concept of generalized α-orbital admissible mappings.Definition 5.
LetT:X→X be a mapping and α:X×X→[0,∞] be a function. We say that T is a generalized α-orbital admissible if (3)αx,Tx≥1⟹αTx,T2x≥1,αx,Tx<∞⟹αTx,Tx2<∞.Notice that eachα-orbital admissible mapping [19] is generalized α-orbital admissible.Based on the concept of generalizedα-orbital admissibility, we are the first who establish a fixed point result for a Meir-Keeler type contraction in the setting of generalized b-metric spaces.


## 2. Main Results

We start with this definition.Definition 6.
For an arbitrary constants≥1, let T be a self-mapping defined on a generalized b-metric space (X,d,s). Then T is called an (α,ψ)-Meir-Keeler contractive mapping if there exist two auxiliary mappings α:X×X→[0,∞] and ψ∈Ψ such that(4)ε≤ψdx,y<ε+δimpliesαx,ydTx,Ty<ε,∀x,y∈X.Remark 7.
Forx≠y and d(x,y)<∞ with α(x,y)<∞, from (4) we derive that(5)αx,ydTx,Ty<ψdx,y.Our main result is as follows.Theorem 8.
Lets≥1 be a fixed constant and (X,d,s) be a complete generalized b-metric space. Suppose that a self-mapping T:X→X is an (α,ψ)-Meir-Keeler type contraction. Assume also that (i)
T is generalized α-orbital admissible;(ii)
there existsx∈X such that 1≤α(x,Tx)<∞;(iii)
T is continuous.
Then for suchx, one of the following statements holds: (A)
For everyn∈N0, (6)dTnx,Tn+1x=∞(7)orαTnx,Tn+1x=∞.(B)
There existsk∈N0 such that d(Tkx,Tk+1x)<∞ and α(Tkx,Tk+1x)<∞. In this case, there exists u∈X such that Tu=u.Proof.
On account of assumption (ii), there existsx∈X such that α(x,Tx)≥1. We suppose that case (A) is not satisfied. Consequently, we have to examine case (B). Consequently, there exists k∈N0 such that d(Tkx,Tk+1x)<∞ and α(Tkx,Tk+1x)<∞. If Tkx=Tk+1x, the proof is completed. Assume that d(Tkx,Tk+1)>0. By property of ψ and Remark 7, we have(8)αTkx,Tk+1xdTk+1x,Tk+2x<ψdTkx,Tk+1x<dTkx,Tk+1x<∞.Since T is a generalized α-orbital admissible mapping, by (ii), we derive that (9)1≤αx,Tx<∞⟹1≤αTx,T2x<∞. Recursively, we obtain that(10)1≤αTk+nx,Tk+n+1x<∞∀n∈N0.Applying (10) in (8), we get(11)dTk+1x,Tk+2x<ψdTkx,Tk+1x<∞.Thus(12)dTk+nx,Tk+n+1x<∞∀n∈N0.Again, on account of (10) and (12) in (8), by induction, one gets(13)dTk+nx,Tk+n+1x<ψndTkx,Tk+1x.Consequently, for n,v∈N0, by (13) we have(14)dTk+nx,Tk+n+vx≤sdTk+nx,Tk+n+1x+⋯+sv-1dTk+n+v-2x,Tk+n+v-1x+svdTk+n+v-1x,Tk+n+vx<sψndTkx,Tk+1x+⋯+sv-1ψn+v-2dTkx,Tk+1x+svψn+v-1dTkx,Tk+1x≤s∑m=0∞smψn+mdTkx,Tk+1x≤s∑m=0∞smψmdTkx,Tk+1x.Finally,(15)dTk+nx,Tk+n+vx≤s∑m=0∞smψmdTkx,Tk+1x,for all n,v∈N0. By (15) and the fact that ψ∈Ψ, it follows that {Tnx} is a Cauchy sequence of elements of X.
SinceX is complete, there exists u∈X with (16)limn→∞⁡dTnx,u=0.Since T is continuous, we get (17)u=limn→∞⁡Tn+1x=Tlimn→∞⁡Tnx=Tu, and u is a fixed point of T, which ends the proof.Definition 9.
Lets≥1 be a fixed constant. We say that a generalized b-metric space (X,d,s) is regular if {xn} is a sequence in X such that 1≤α(xn,xn+1)<∞ for all n and xn→x∈X as n→∞; then there exists a subsequence {xn(k)} of {xn} such that 1≤α(xn(k),x)<∞ and 0<d(xn(k),x)<∞ for all k.Theorem 10.
Lets≥1 be a fixed constant and (X,d,s) be a complete generalized b-metric space. Suppose that a self-mapping T:X→X is an (α,ψ)-Meir-Keeler type contraction. Assume also that (i)
T is a generalized α-orbital admissible mapping;(ii)
there existsx∈X such that 1≤α(x,Tx)<∞;(iii)
(X,d,s) is regular.
Then for suchx, one of the following statements holds: (A)
For everyn∈N0, (18)dTnx,Tn+1x=∞(19)orαTnx,Tn+1x=∞.(B)
There existsk∈N0 such that d(Tkx,Tk+1x)<∞ and α(Tkx,Tk+1x)<∞. In this case, there exists u∈X such that Tu=u.Proof.
In case (B), following the proof of Theorem 8, we know that the sequence {Tn(x)} converges to some u∈X. By Definition 9 and condition (iii), there exists a subsequence {xn(k)} of {xn} such that 1≤α(Tn(k)x,u)<∞ and 0<d(Tn(k)x,u)<∞ for all k. Applying (5) for all k, we get that(20)dTnk+1x,Tu=dTTnkx,Tu≤αTnkx,udTTnkx,Tu<ψdTnkx,u.Letting k→∞ in the above equality, we get d(u,Tu)=0; that is, u=Tu.For the uniqueness of a fixed point of an(α,ψ)-Meir-Keeler type contraction mapping T in Y≔{t∈X:d(Tkx,t)<∞}, we shall consider the following condition:(U)
For allx,y∈Fix(T), we have 1≤α(x,y)<∞, where Fix(T) denotes the set of fixed points of T.Theorem 11.
By adding condition(U) to the hypotheses of Theorem 8 (resp., Theorem 10), T has at most one fixed point in Y≔{t∈X:d(Tkx,t)<∞}.Proof.
LetT be an (α,ψ)-Meir-Keeler type contraction. Owing to Theorem 8 (resp., Theorem 10), T has a fixed point u∈X.
Now, we shall show thatT has at most one fixed point in Y. We argue by contradiction. For this, assume that there exist two distinct fixed points u1 and u2 of T, where u1,u2∈Y; that is,(21)dTkx,ui<∞∀i=1,2. We deduce(22)du1,u2≤sdu1,Tkx+sdTkx,u2<∞. By condition (U), 1≤α(u1,u2)<∞ and since 0<d(u1,u2)<∞, in view of (5), one writes(23)du1,u2=dTu1,Tu2≤αu1,u2dTu1,Tu2<ψdu1,u2<du1,u2, which is a contradiction, so u1=u2. This completes the proof.


## 3. Consequences

### 3.1. Meir-Keeler Contraction Mappings in gbms

In this section, we present our main result. By lettingα(x,y)=1 and ϕ(t)=t/2s, we get the following result.Theorem 12.
Let(X,d,s) be a generalized complete b-metric space and T:X→X satisfy the following: given ε>0, there exists δ>0 such that(24)ε≤dx,y<ε+δimpliesdTx,Ty<ε2s,x,y∈X.Letx∈X. Then one of the following alternatives holds:(A)
For everyn∈N0 (N0 being the set of all nonnegative integers), (25)dTnx,Tn+1x=∞.(B)
There existsk∈N0 such that d(Tkx,Tk+1x)<∞.In case(B), we assert the following:(i)
The sequence{Tmx} is Cauchy in X.(ii)
There exists a pointu∈X such that Tu=u and limn→∞⁡d(Tnx,u)=0.(iii)
u is the unique fixed point of T in B≔{t∈X:d(Tkx,t}<∞}.(iv)
For everyt∈B, (26)limn→∞⁡dTnt,u=0.Remark 13.
Unfortunately, if(X,d) is a metric space, we do not get the result of Meir-Keeler [4].


## 3.1. Meir-Keeler Contraction Mappings in gbms

In this section, we present our main result. By lettingα(x,y)=1 and ϕ(t)=t/2s, we get the following result.Theorem 12.
Let(X,d,s) be a generalized complete b-metric space and T:X→X satisfy the following: given ε>0, there exists δ>0 such that(24)ε≤dx,y<ε+δimpliesdTx,Ty<ε2s,x,y∈X.Letx∈X. Then one of the following alternatives holds:(A)
For everyn∈N0 (N0 being the set of all nonnegative integers), (25)dTnx,Tn+1x=∞.(B)
There existsk∈N0 such that d(Tkx,Tk+1x)<∞.In case(B), we assert the following:(i)
The sequence{Tmx} is Cauchy in X.(ii)
There exists a pointu∈X such that Tu=u and limn→∞⁡d(Tnx,u)=0.(iii)
u is the unique fixed point of T in B≔{t∈X:d(Tkx,t}<∞}.(iv)
For everyt∈B, (26)limn→∞⁡dTnt,u=0.Remark 13.
Unfortunately, if(X,d) is a metric space, we do not get the result of Meir-Keeler [4].



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*Source: 3264620-2018-01-28.xml*