# Fixed Point Results forα-ψλ-Contractions on Gauge Spaces and Applications

Authors: Mohamed Jleli; Erdal Karapınar; Bessem Samet
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We extend the concept ofα-ψ-contractive mappings introduced recently by Samet et al. (2012) to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory plays an important role in nonlinear analysis. This is because many practical problems in applied science, economics, physics, and engineering can be reformulated as a problem of finding fixed points of nonlinear mappings. The Banach contraction principle [1] is one of the fundamental results in fixed point theory. It guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to approximate those fixed points.Theorem 1 (see [1]).
Let(X,d) be a complete metric space. Let T:X→X be a contraction self-mapping on X; that is, there exists a constant k∈(0,1) such that
(1)d(Tx,Ty)≤kd(x,y),
for all (x,y)∈X×X. Then the mapTadmits a unique fixed point. Moreover, for any x0∈X, the sequence {Tnx0} converges to this fixed point.During the last few decades, several extensions of this famous principle have been established. In 1961, Edelstein [2] established the following result.Theorem 2 (see [2]).
Let(X,d) be complete and ε-chainable for some ε>0. Let T:X→X be such that
(2)x,y∈X,d(x,y)<ε⟹d(Tx,Ty)≤kd(x,y),
where k∈(0,1) is a constant. ThenThas a unique fixed point.Kirk et al. [3] introduced the concept of cyclic mappings and proved the following fixed point theorem.Theorem 3 (see [3]).
LetA and B be two nonempty closed subsets of a complete metric space (X,d). Let T:X→X be a self-mapping such that
(3)x,y∈X,(x,y)∈A×B⟹d(Tx,Ty)≤kd(x,y),
where k∈(0,1) is a constant. Suppose also that T(A)⊆B and T(B)⊂A. Then T has a unique fixed point in A∩B.Ran and Reurings [4] extended the Banach contraction principle to a metric space endowed with a partial order. They established the following result.Theorem 4 (see [4]).
Let(X,d) be a complete metric space endowed with a partial order ⪯. Let T:X→X be a continuous mapping such that
(4)x,y∈X,x⪯y⟹d(Tx,Ty)≤kd(x,y),
where k∈(0,1) is a constant. Suppose also that there exists x0∈X such that x0⪯Tx0. Then T has a fixed point.Many extensions of the previous result exist in the literature; for more details, we refer the reader to [5–11] and the references therein.Observe that all the contractive conditions (2), (3), and (4) can be written as
(5)d(Tx,Ty)≤kd(x,y),∀(x,y)∈H,
where H is a subset of X×X. In Theorem 2, we have
(6)H≔{(x,y)∈X×X:d(x,y)<ε}.
In Theorem 3, we have
(7)H≔(A×B)∪(B×A).
In Theorem 4, we have
(8)H≔{(x,y)∈X×X:(x⪯y)∧(y⪯x)}.
The contractive condition (5) is said to be a partial contraction, that is, a contraction satisfied only on a subset of H⊆X×X.Very recently, Samet et al. [12] observed that a partial contraction can be considered as a total contraction, that is, a contraction satisfied for every pair (x,y)∈X×X. More precisely, if we define the function α:X×X→[0,∞) by
(9)α(x,y)≔{1,if(x,y)∈H,0,otherwise,
we show that (5) is equivalent to
(10)α(x,y)d(Tx,Ty)≤kd(x,y),∀(x,y)∈X×X.
In [12], the authors considered a more general inequality; that is,
(11)α(x,y)d(Tx,Ty)≤ψ(d(x,y)),∀(x,y)∈X×X,
where ψ:[0,∞)→[0,∞) is a function satisfying some conditions. The above inequality is called an α-ψ-contraction. In [12], some fixed point results were established under this contractive condition. For other works in this direction, we refer the reader to [13–16].The aim of this work is to extend, generalize, and improve the obtained results in [12]. More precisely, the concept of α-ψ-contractive mappings is extended to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.Through this paper,𝔼 will denote a gauge space endowed with a separating gauge structure 𝒟={dλ}λ∈Λ, where Λ is a directed set.A sequence{xn}⊂𝔼 is said to be convergent if there exists an x∈𝔼 such that for every ε>0 and λ∈Λ, there is an N∈ℕ with dλ(xn,x)<ε, for all n≥N.A sequence{xn}⊂𝔼 is said to be Cauchy, if for every ε>0 and λ∈Λ, there is an N∈ℕ with dλ(xn,xn+p)<ε, for all n≥N and p∈ℕ.A gauge space is called complete if any Cauchy sequence is convergent.A subset of𝔼is said to be closed if it contains the limit of any convergent sequence of its elements.For more details on guage spaces, we refer the reader to Dugundji [17].We denote byΨ the set of functions ψ:[0,∞)→[0,∞) satisfying the following conditions:(C1)
ψ is nondecreasing;(C2)
∑n=1∞‍ψn(t)<∞, for all t>0, where ψn is the nth iterate of ψ;(C3)
ψ(a)+ψ(b)≤ψ(a+b), for all a,b≥0.It is easy to show that under the conditions (C1) and (C2), we have ψ(t)<t, for all t>0. Moreover, under the conditions (C1) and (C3), we have
(12)ψn(a)+ψn(b)≤ψn(a+b),
for all n∈ℕ and a,b≥0.Example 5.
Letψ:[0,∞)→[0,∞) be the function defined by
(13)ψ(t)≔{t24,if  ort∈[0,1),t2,ift≥1.
Then ψ∈Ψ.Definition 6.
Letα:𝔼×𝔼→[0,∞) be a given function. Let N∈ℕ and x,y∈𝔼. We say that (xi)i=0N⊂𝔼 is an N-α-path fromxtoyif
(14)x0=x,xN=y,α(xi-1,xi)≥1,∀i=1,…,N.We denote(15)x[N,α]≔{y∈𝔼:thereisanN-α-pathfromxtoy}.
Let N∈ℕ and x∈𝔼. For λ∈Λ and y∈x[N,α], and let
(16)pλN(x,y)≔inf{∑i=1N‍dλ(xi-1,xi):(xi)i=0N⊂X0000000isanN-α-pathfromxtoy∑i=1N‍}.


## 2. Fixed Point Results forα-ψλ-Contractions

Definition 7.
LetT:𝔼→𝔼 be a given self-mapping. Let α:𝔼×𝔼→[0,∞) be a given function, and let {ψλ}λ∈Λ⊂Ψ. We say that T is an α-ψλ-contraction if
(17)α(x,y)dλ(Tx,Ty)≤ψλ(dλ(x,y)),
for all λ∈Λ and x,y∈𝔼.Definition 8.
LetT:𝔼→𝔼 be a given self-mapping. Let α:𝔼×𝔼→[0,∞) be a given function. We say that T is α-admissible if
(18)α(x,y)≥1⟹α(Tx,Ty)≥1,∀x,y∈𝔼.The following lemma will be useful to establish our fixed point results.Lemma 9.
LetT:𝔼→𝔼 be a self-mapping. Suppose that there exist α:𝔼×𝔼→[0,∞) and {ψλ}λ∈Λ⊂Ψ such that the following conditions hold:(i)
Tis anα-ψλ-contraction;(ii)
Tisα-admissible.Letε>0andN∈ℕ. Then, for everyλ∈Λ,x∈X, andy∈x[N,α], one has(I)
Ty∈Tx[N,α]with(19)pλN(Tx,Ty)≤ψλ(pλN(x,y)+ε);(II)
for allk∈ℕ∪{0}, one hasTky∈Tkx[N,α]with(20)pλN(Tkx,Tky)≤ψλk(pλN(x,y)+ε).Proof.
Letλ∈Λ, x∈X, and y∈x[N,α]. Let {xi}i=0N be an N-α-path from x toysuch that
(21)∑i=1N‍dλ(xi-1,xi)≤pλN(x,y)+ε.
Since α(x,x1)≥1, we have
(22)dλ(Tx,Tx1)≤α(x,x1)dλ(Tx,Tx1)≤ψλ(dλ(x,x1)).
Since α(x1,x2)≥1, we have
(23)dλ(Tx1,Tx2)≤α(x1,x2)dλ(Tx1,Tx2)≤ψλ(dλ(x1,x2)).
Recursively from i=3 to N, since α(xi-1,xi)≥1, we have
(24)dλ(Txi-1,Txi)≤α(xi-1,xi)dλ(Txi-1,Txi)≤ψλ(dλ(xi-1,xi)).
On the other hand, sinceTisα-admissible, {Txi}i=0N is an N-α-path from Tx to Ty. So, we have
(25)pλN(Tx,Ty)≤∑i=1N‍dλ(Txi-1,Txi)≤∑i=1N‍ψλ(dλ(xi-1,xi))≤ψλ(∑i=1N‍dλ(xi-1,xi))≤ψλ(pλN(x,y)+ε).
Thus, we proved (I).
Again, sinceα(Tx,Tx1)≥1, we have
(26)dλ(T2x,T2x1)≤α(Tx,Tx1)dλ(T2x,T2x1)≤ψλ(dλ(Tx,Tx1))≤ψλ2(dλ(x,x1)).
Since α(Tx1,Tx2)≥1, we have
(27)dλ(T2x1,T2x2)≤α(Tx1,Tx2)dλ(T2x1,T2x2)≤ψλ(dλ(Tx1,Tx2))≤ψλ2(dλ(x1,x2)).
Recursively from i=3 to N, since α(Txi-1,Txi)≥1, we have
(28)dλ(T2xi-1,T2xi)≤α(Txi-1,Txi)dλ(T2xi-1,T2xi)≤ψλ(dλ(Txi-1,Txi))≤ψλ2(dλ(xi-1,xi)).
On the other hand, since T is α-admissible, {T2xi}i=0N is an N-α-path from T2x to T2y. So, we have
(29)pλN(T2x,T2y)≤∑i=1N‍dλ(T2xi-1,T2xi)≤∑i=1N‍ψλ2(dλ(xi-1,xi))≤ψλ2(∑i=1N‍dλ(xi-1,xi))≤ψλ2(pλN(x,y)+ε).
Continuing this process, by induction, we get (II).Definition 10.
LetT:𝔼→𝔼 be a self-mapping, and let α:𝔼×𝔼→[0,∞) be a given function. For N∈ℕ, we say that a sequence {xn}⊂𝔼 is an N-α-Picard trajectory from x0 if xn=Txn-1∈xn-1[N,α] for all n∈ℕ. We denote by 𝒯N(T,α,x0), the set of all N-α-Picard trajectories fromx0.Definition 11.
LetT:𝔼→𝔼 be a self-mapping, and let α:𝔼×𝔼→[0,∞) be a given function. For N∈ℕ, we say that T is N-α-Picard continuous from x0∈𝔼 if the limit of any convergent sequence {xn}∈𝒯N(T,α,x0) is a fixed point of T.We have the following fixed point result.Theorem 12.
LetT:𝔼→𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψλ}λ∈Λ⊂Ψ and α:𝔼×𝔼→[0,∞) be a given function. Suppose that the following conditions hold:(i)
Tis anα-ψλ-contraction;(ii)
Tisα-admissible;(iii)
there existN∈ℕandx0∈𝔼such thatTx0∈x0[N,α];(iv)
TisN-α-Picard continuous fromx0.ThenThas a fixed point.Proof.
Letλ∈Λ and ε>0. From condition (iii) and Lemma 9, we have T2x0∈Tx0[N,α] and
(30)dλ(Tx0,T2x0)≤pλN(Tx0,T2x0)≤ψλ(pλN(x0,Tx0)+ε).
Again, from Lemma 9, we have T3x0∈T2x0[N,α] and
(31)dλ(T2x0,T3x0)≤pλN(T2x0,T3x0)≤ψλ2(pλN(x0,Tx0)+ε).
Continuing this process, by induction, for n≥2, we have Tn+1x0∈Tnx0[N,α] and
(32)dλ(Tnx0,Tn+1x0)≤pλN(Tnx0,Tn+1x0)≤ψλn(pλN(x0,Tx0)+ε).
Thus, {Tnx0}∈𝒯N(T,α,x0) and, for m≥1,
(33)dλ(Tnx0,Tn+mx0)≤∑i=0m-1‍dλ(Tn+ix0,Tn+i+1x0)≤∑i=0m-1‍ψλn+i(pλN(x0,Tx0)+ε)≤∑j≥n‍ψλj(pλN(x0,Tx0)+ε).
From condition (C2), we have
(34)∑j≥n‍ψλj(pλN(x0,Tx0)+ε)⟶0asn⟶∞,
which implies that {Tnx0} is a Cauchy sequence in the complete gauge space 𝔼. Since T is N-α-Picard continuous from x0, the limit of {Tnx0} is a fixed point of T.Corollary 13.
LetT:𝔼→𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψλ}λ∈Λ⊂Ψ and α:𝔼×𝔼→[0,∞) be a given function. Suppose that the following conditions hold:(i)
Tis anα-ψλ-contraction;(ii)
Tisα-admissible;(iii)
there existN∈ℕandx0∈𝔼such thatTx0∈x0[N,α];(iv)
Tis continuous.ThenThas a fixed point.Proof.
Let{xn}∈𝒯N(T,α,x0) be such that xn→x∈𝔼. Since T is continuous, we have xn+1=Txn→Tx. Since 𝔼 is endowed with a separating gauge structure, we have x=Tx. The conclusion follows from Theorem 12.Corollary 14.
LetT:𝔼→𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψλ}λ∈Λ⊂Ψ and α:𝔼×𝔼→[0,∞) be a given function. Suppose that the following conditions hold:(i)
T is an α-ψλ-contraction;(ii)
T is α-admissible;(iii)
there existN∈ℕandx0∈𝔼such thatTx0∈x0[N,α];(iv)
for every{xn}∈𝒯N(T,α,x0)such thatxn→x∈𝔼, there exist a subsequence{xn(k)}of{xn}  andk0∈ℕ  such thatα(xn(k),x)≥1fork≥k0.ThenThas a fixed point.Proof.
Let{xn}∈𝒯N(T,α,x0) be such that xn→x∈𝔼. From condition (iv), there exist a subsequence {xn(k)} of {xn} and k0∈ℕ such that α(xn(k),x)≥1 for k≥k0. Since T is an α-ψλ-contraction, for all λ∈Λ and k≥k0, we have
(35)dλ(Txn(k),Tx)≤α(xn(k),x)dλ(Txn(k),Tx)≤ψλ(dλ(xn(k),x)).
Letting k→∞ in the above inequality, we obtain that
(36)xn(k)+1=Txn(k)⟶Tx,ask⟶∞.
Since 𝔼 is endowed with a separating gauge structure, we have x=Tx. The conclusion follows from Theorem 12.ForT:𝔼→𝔼, we denote by Fix(T) the set of fixed points of T; that is,
(37)Fix(T)≔{x∈𝔼:x=Tx}.
The next result gives us a sufficient condition that ensures the uniqueness of the fixed point.Theorem 15.
Suppose that all the conditions of Theorem12 are satisfied. Moreover, suppose that (v)
for every(x,y)∈Fix(T)×Fix(T)withx≠y, there existsN(x,y)∈ℕsuch thaty∈x[N(x,y),α].Then T has a unique fixed point.Proof.
From Theorem12, The mapping T has at least one fixed point. Suppose that u,v∈𝔼 are two fixed points of T with u≠v. From the condition (v), there exists N(u,v)∈ℕ such that v∈u[N(u,v),α]. Let λ∈Λ and ε>0. From Lemma 9, we have
(38)dλ(u,v)≤pλN(u,v)(u,v)≤ψλn(pλN(u,v)(u,v)+ε),
for all n∈ℕ. Letting n→∞ in the above inequality, we obtain that dλ(u,v)=0 for all λ∈Λ, which is a contradiction with u≠v (since we have a separating gauge structure). We deduce that u=v.The following result follows immediately from Theorems12 and 15 with N=1 and α(x,y)=1 for every x,y∈𝔼.Corollary 16.
LetT:𝔼→𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψλ}λ∈Λ⊂Ψ. Suppose that for all λ∈Λ, for all x,y∈𝔼, one has
(39)dλ(Tx,Ty)≤ψλ(dλ(x,y)).
Then T has a unique fixed point.Corollary 17.
LetT:𝔼→𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψλ}λ∈Λ⊂Ψ and α:𝔼×𝔼→[0,∞) be a given function. Suppose that the following conditions hold:(i)
Tis anα-ψλ-contraction;(ii)
Tisα-admissible;(iii)
there existsx0∈𝔼such thatα(x0,Tx0)≥1;(iv)
Tis continuous(or) for any sequence {xn}⊂𝔼 such that xn=Txn-1, xn→x∈𝔼 and α(xn-1,Txn-1)≥1 for n∈ℕ, there exist a subsequence {xn(k)} of {xn} and k0∈ℕ such that α(xn(k),x)≥1 for k≥k0.
ThenT has a fixed point. Moreover, if (v)
for every(x,y)∈Fix(T)×Fix(T)withx≠y, there exists z∈𝔼such thatα(x,z)≥1andα(z,y)≥1,one has uniqueness of the fixed point.Proof.
The existence follows from Theorem12 with N=1. The uniqueness follows from Theorem 15 with N(x,y)=2.Corollary 18.
Let⪯ be a partial order on the complete gauge space 𝔼. Let T:𝔼→𝔼 be a self-mapping and {ψλ}λ∈Λ⊂Ψ. Suppose that the following conditions hold:(i)
for allλ∈Λ, for all x,y∈𝔼 such that x and y are comparable, one has
(40)dλ(Tx,Ty)≤ψλ(dλ(x,y));(ii)
x,y∈𝔼,x and y are comparable ⇒Tx and Ty are comparable;(iii)
there existsx0∈𝔼 such that x0 and Tx0 are comparable;(iv)
T is continuous,(or) for any sequence {xn}⊂𝔼 such that xn=Txn-1, xn→x∈𝔼, xn-1 and xn=Txn-1 are comparable for n∈ℕ, there exist a subsequence {xn(k)} of {xn} and k0∈ℕ such that xn(k) and x are comparable for k≥k0.
ThenT has a fixed point. Moreover, if (v)
for every(x,y)∈Fix(T)×Fix(T) with x≠y, there exists z∈𝔼 such that x and z are comparable, z and y are comparable,one has uniqueness of the fixed point.Proof.
It follows from Corollary17 with
(41)α(x,y)≔{1,if(x⪯y)∨(y⪯x),0,otherwise.


## 3. Applications

In this section, we are interested in the study of the existence of solutions to the nonlinear integral equation on the real axis(42)x(t)=h(t)+∫0t‍F(t,s,x(s))ds,t≥0,
where h∈C([0,∞),ℰ) and F∈C([0,∞)×[0,∞)×ℰ,ℰ). Here, ℰ is a Banach space with respect to a given norm ∥·∥ℰ.Let𝔼:=C([0,∞),ℰ) and the family of pseudonorms {∥·∥n}n∈ℕ defined by
(43)∥x∥n≔maxt∈[0,n]∥x(t)∥ℰe-τt,τ>0.
For every n∈ℕ, define now
(44)dn(x,y)≔∥x-y∥n,forx,y∈𝔼.
Then 𝒟≔{dn}n∈ℕ is a separating gauge structure on𝔼.We have the following existence result.Theorem 19.
Suppose that the following conditions hold:(i)
there exist a nonempty setΓ⊆ℰ×ℰ and a constant k<τ such that
(45)∥F(t,s,u)-F(t,s,v)∥ℰ≤k∥u-v∥ℰ,for all t,s≥0, (u,v)∈Γ;(ii)
there exists a nonempty setΓ𝔼⊆𝔼×𝔼 such that
(46)(x,y)∈Γ𝔼⟺(x(t),y(t))∈Γ,∀t≥0;(iii)
for all(x,y)∈Γ𝔼, one has
(47)(h(t)+∫0t‍F(t,s,x(s))ds,h(t)+∫0t‍F(t,s,y(s))ds)∈Γ,for all t≥0;(iv)
there existsx0∈𝔼 such that
(48)(x0(t),h(t)+∫0t‍F(t,s,x0(s))ds)∈Γ,for all t≥0;(v)
if{xp}⊂𝔼 is a sequence such that (xp,xp+1)∈Γ𝔼  for p∈ℕ and xp→x∈𝔼 (with respect to 𝒟), then there exist a subsequence {xp(k)} of {xp} and k0∈ℕ such that
(49)(h(t)+∫0t‍F(t,s,xp(k)(s))ds,x(t))∈Γ,for all k≥k0, t≥0.
Then (42) has at least one solution in 𝔼.Proof.
Consider the mappingT:𝔼→𝔼 defined by
(50)Tx(t)≔h(t)+∫0t‍F(t,s,x(s))ds,t≥0,
for all x∈𝔼. We have to prove that T has at least a fixed point.
Define the functionα:𝔼×𝔼→[0,∞) by
(51)α(x,y)≔{1,if(x,y)∈Γ𝔼,0,otherwise.
We claim that for all n∈ℕ, for all x,y∈𝔼,
(52)α(x,y)dn(Tx,Ty)≤ψn(dn(x,y)),
where ψn(t)=(k/τ)t for all n∈ℕ, for all t≥0. Clearly, since k<τ, {ψn}n∈ℕ⊂Ψ. If α(x,y)=0, (52) holds immediately. So, suppose that α(x,y)≠0; that is, (x,y)∈Γ𝔼. Let n∈ℕ. From conditions (i) and (ii), for all t∈[0,n], we have
(53)∥Tx(t)-Ty(t)∥ℰ≤∫0t∥F(t,s,x(s))-F(t,s,y(s))∥ℰds≤∫0tkeτs(∥x(s)-y(s)∥ℰe-τs)ds≤kτdn(x,y)eτt,
which implies that
(54)α(x,y)dn(Tx,Ty)=dn(Tx,Ty)≤kτdn(x,y).
Thus, we proved (52).
We will prove thatTisα-admissible. Let (x,y)∈𝔼×𝔼 such that α(x,y)≥1; that is, (x,y)∈Γ𝔼. From condition (iii), we have (Tx(t),Ty(t))∈Γ for all t≥0, which implies from condition (ii) that (Tx,Ty)∈ΓE; that is, α(Tx,Ty)≥1. So, T is α-admissible.
From conditions (iv) and (ii), we have(x0,Tx0)∈Γ𝔼, which is equivalent to say that α(x0,Tx0)≥1.
Finally, condition (v) implies that for every{xp}∈𝒯1(T,α,x0) such that xp→x∈𝔼, there exist a subsequence {xp(k)} of {xp} and k0∈ℕ such that α(xp(k),x)≥1 for k≥k0.
Now, All the hypotheses of Corollary14 are satisfied; we deduce that T has at least a fixed point, which is a solution to (52).Theorem 20.
In addition to the assumptions of Theorem19, suppose that (vi)
for all(x,y)∈𝔼×𝔼, there exists z∈𝔼 such that (x,z)∈Γ𝔼 and (z,y)∈Γ𝔼.Then (42) has one and only one solution in 𝔼.Proof.
It follows immediately from Theorem15.



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