# Endpoints of Generalized Contractions inF-Metric Spaces with Application to Integral Equations

Authors: Nabil Mlaiki; Jamshaid Ahmad; Abdullah Eqal Al-Mazrooei
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

The purpose of this article is to introduce locallyα-ζ-multivalued contraction and rational Ćirić type α-ζ-multivalued contraction in the context of F-metric spaces and prove some endpoint results. We provide a nontrivial example to show the authenticity of our main result. Our results generalize some well-known results of literature. We also present some endpoint results in both graphic F-metric spaces and ordered F-metric spaces. As an application of our main result, we investigate the solution of an integral equation.


---

## Body

## 1. Introduction

In 2010, Amini-Harandi [1] showed that a multivalued mapping has a unique endpoint if and only if this multivalued mapping has the approximate endpoint property. Hussain et al. [2] established some approximate endpoints of the multivalued almost I-contractions in complete metric spaces. Later on, Moradi and Khojasteh [3] proved a result for generalized weak contractive multifunctions.On the other hand, Samet et al. [4] introduced the notion of α-admissibility and α-ζ-contraction in 2012. Asl et al. [5] extended this notion of α-admissibility to α∗-admissibility and proved some results for multivalued mappings. In 2015, Mohammadi and Rezapour [6] improved the α-admissibility concept and obtained endpoint of α-ζ-multivalued contraction. Later on, Choudhury et al. [7] used the notion of α-admissibility and proved end point results of multivalued mappings without continuity. Very recently, Isik et al. [8] proved endpoint results for α-ζ-contraction in the newly introduced space of Jleli and Samet [9] which is named as F-metric space (F-MS). In this artilce, we give locally α-ζ-multivalued contraction and rational Ćirić type α-ζ-multivalued contraction in the framework of F-metric space and generalized the main result of Isik et al. [8].


## 2. Preliminaries

LetM=∅ and T:M⟶2M (nonempty subsets of M) be a multivalued mapping. A point σ∈M is professed to be an endpoint (fixed point) of T if Tσ=σσ∈Tσ. Now, let M,d be a metric space, then T is said to satisfy the approximate fixed point property if
(1)infσ∈Msupÿ∈Tσdσ,ÿ=0.LetCBM represents the set of all nonempty, closed, and bounded subsets of M. The Hausdorff metric H is defined on CBM as follows:
(2)HA,B=maxsupσ∈Adσ,B,supÿ∈Bdÿ,A.In 2012, Samet et al. [4] used the following set Ψ of nondecreasing functions ζ:0,∞⟶0,∞ satisfying
(3)∑n=1∞ζnt<∞,forallt>0,and introduced α-ζ-contraction. Clearly, ζt<t for all t>0 ([30]).Samet et al. [4] also initiated the concept of α-admissibility of a single valued mapping in this way.Definition 1 (see [4]).
Letα:M×M⟶0,∞ and let T:M⟶M, then T is said to be α-admissible if ∀σ,ÿ∈M, ασ,ÿ≥1 implies αTσ,Tÿ≥1.They gave the following property ofM that is M is α-regular, if for each sequence σn in M with ασn,σn+1≥1, and σn⟶σ, then ασn,σ≥1, ∀n.In 2013, Asl et al. [5] extended this concept to multivalued mapping and gave the notion of α∗-admissibility as follows.Definition 2 (see [5]).
Letα:M×M⟶0,∞ and let T:M⟶CBM, then T is said to be α∗-admissible if for all σ,ÿ∈M, ασ,ÿ≥1 implies α∗Tσ,Tÿ≥1, where α∗A,B=infαa,b:a∈A,b∈B, for all A,B∈CBM.In 2015, Mohammadi and Rezapour [6] extended the above notion in this way.Definition 3 (see [6]).
Letα:M×M⟶0,∞ and T:M⟶2M, then T is α-admissible provided that for all σ∈M and ÿ∈Tσ with ασ,ÿ≥1, then αÿ,z≥1, for all z∈Tÿ.They proved endpoint results forα-ζ-multivalued contraction by using the following property.A multivalued mappingT:M⟶CBM is said to satisfy the property (BS), if for all σ∈M, there exists ÿ∈Tσ such that HTσ,Tÿ=supb∈Tÿdÿ,b. Isik et al. [8] used the property (SBS) of Mohammadi and Rezapour [6] to prove their results, that is, for each sequence σn with
(4)dσn,Tσn≤dσn,σn+1+ζdσn,σn+1,for all n and σn⟶σ, then dσn,Tσn≤dσn,σ+ζdσn,σ, for all n≥N.For more details in this direction, we refer the readers (see [10–14]).Recently, Jleli and Samet [9] introduced an interesting generalization of metric space which is called F-metric space (F-MS) as follows.LetF be the class of f:ℝ+⟶ℝ such that f (σ1)<f (σ2), for σn⊆ℝ+, limn⟶∞σn=0⇔limn⟶∞fσn=−∞.Definition 4 (see [9]).
LetM=∅, and let dF:M×M⟶0,+∞. Suppose that there exists f∈F and α∈0,+∞ such that
D1dFσ,ÿ=0⇐⇒σ=ÿ, for all σ,ÿ∈M×M
D2dFσ,ÿ=dFÿ,σ, for all σ,ÿ∈M×M
D3 for every σ,ÿ∈M×M, for every N∈ℕ, N≥2 and for every σii=1N⊂M with u1,uN=σ,ÿ, we have
(5)dFσ,ÿ>0⇒fdFσ,ÿ≤f∑i=1N−1dFui,ui+1+α
Then,M,dF is called an F-MS.Theorem 5 (see[9]).
LetM,dF be an F-MS and let T:M⟶M. Suppose that these assertions hold:
(i)
M,dF is F-complete(ii)
there existsk∈0,1 such that(6)dFTσ,Tÿ≤kdFσ,ÿThen, there existsσ∗∈M such that Tσ∗=σ∗ which is unique.Hussain and Kanwal [15] utilized an F-metric space and generalized the above result by considering the notion of α-ζ-contraction to prove a fixed point theorem. Many researchers (see [16–18]) worked in this newly generalized space.Very recently, Isik et al. [8] introduced the notion of Hausdorff metric HF(.,.) on CBM influence by F-metric dF as follows:
(7)HFA,B=maxsupσ∈AdFσ,B,supÿ∈BdFÿ,A,for all A,B∈CBM, where dFσ,B=infÿ∈BdFσ,ÿ and obtained endpoint results for α-ζ-multivalued contraction in this way.Theorem 6.
LetM,dF be an F-MS and T:M⟶CBM be an α-admissible mapping which satisfies the property (BS). Suppose there exists α:M×M⟶0,+∞ and ζ∈Ψ such that
(8)ασ,ÿ≥1⇒HFTσ,Tÿ≤ζdFσ,ÿ.
Also, suppose that these assertions hold:(i)
M,dF is F-complete(ii)
ασ0,σ1≥1foranσ0∈Mandσ1∈Tσ0(iii)
M is α-regular
Then,T has an endpoint.


## 3. Main Results

Definition 7.
LetM,dF be an F-MS. A mapping T:M⟶2M is called a locally α-ζ-multivalued contraction if there exists ζ∈Ψ and α:M×M⟶0,+∞ such that
(9)ασ,ÿ≥1⇒HFTσ,Tÿ≤ζdFσ,ÿ,for σ,ÿ∈Bσ0,r¯.Now, we state our main result regarding the existence of the endpoint of anα-ζ-multivalued contraction on the closed ball Bσ0,r¯ which is very advantageous in the perception that it needs the contractiveness of the multivalued mapping T:M⟶CBM only on the closed ball instead of the whole space.Theorem 8.
LetM,dF be an F-MS and T:M⟶CBM be an α-admissible, locally α-ζ-multivalued contraction such that T satisfies the property (BS) and for σ0∈M, there exists σ1∈Tσ0 such that
(10)ζidσ0,σ1<r,for all n=0,1,2,⋯ and r>0. Also, suppose that the following assertions hold:
(i)
M,dF is F-complete(ii)
ασ0,σ1≥1 for an σ0∈M and σ1∈Tσ0(iii)
M is α-regular
Then,T has an endpoint.Proof.
Chooseσ0∈M and σ1∈Tσ0 such that ασ0,σ1≥1. It follows directly from (10); we have
(11)dσ0,σ1<r,which implies that
(12)σ1∈Bσ0,r¯.It follows from (10) that
(13)ασ0,σ1≥1⇒HTσ0,Tσ1≤ζdσ0,σ1.SinceT satisfies the property (BS), so ∃σ2∈Tσ1 such that HFTσ1,Tσ2=supb∈Tσ2dFσ2,b. Now, from (13), we have
(14)dσ1,σ2≤supb∈Tσ1dFσ1,b=HFTσ0,Tσ1≤ζdσ0,σ1<r.This implies that
(15)σ2∈Bσ0,r¯.SinceT is α-admissible, ασ1,σ2≥1, so t follows from (9) that
(16)ασ1,σ2≥1⇒HTσ1,Tσ2≤ζdσ1,σ2.Continuing this process, we obtain a sequenceσn in Bσ0,r¯ such that σn+1∈Tσn, ασn,σn+1≥1 and HFTσn,Tσn+1=supb∈Tσn+1dFσn+1,b, for all n. If σn=σn+1 for some n∈ℕ, then we get that HFσn+1,Tσn+1=supb∈Tσn+1dFσn+1,b=HFTσn,Tσn+1=0. It implies that σn+1 is an endpoint. Hence, we suppose that σn≠σn+1, for all n∈ℕ.Now, sinceασn−1,σn≥1, so
(17)dFσn,σn+1≤supb∈TσndFσn,b=HFTσn−1,Tσn≤ζdFσn−1,σn≤ζ2dFσn−2,σn−1≤⋯≤ζndFσ0,σ1,for alln≥0. Assume that f,α∈F×0,+∞ be such that (D3) is satisfied and fix ε>0. By (F2),∃δ>0 such that
(18)0<t<δ⇒ft<fε−α.Suppose thatN∈ℕ be such that 0<∑i≥Nζi−1dFσ1,σ2<δ. Hence, by (17), (18) and (F1), we have
(19)f∑i=nm−1dFσi,σi+1≤f∑i=nm−1ζi−1dFσ1,σ2≤f∑i≥Nζi−1dFσ1,σ2<fε−α,for m>n≥N. Using (D3) and (19), we obtain that dFσn,σm>0 where m>n≥N which implies that
(20)fdFσn,σm≤f∑i=nm−1dFσi,σi+1+α<fε,which implies by (F1) that dFσn,σm<ε, for all m>n≥N. This proves that σn is F-Cauchy. Because of F-completeness of M, there exists σå∈Bσ0,r¯ such that σn⟶σå. We shall prove that σ∗ is an endpoint of T. We assume on the contrary that Tσ∗≠σ∗. Then HFσ∗,Tσ∗>0. Since M is locally α-regular, so ασn,σ∗≥1, for all n∈ℕ. Then, by (9) and (F1), we have
(21)fHFσn,Tσn=fHFTσn−1,Tσn≤fHFTσn−1,Tσ∗+HFTσn,Tσ∗+α≤fζdFσn−1,σ∗+ζdFσn,σ∗+α⟶−∞,as n⟶∞. Thus,
(22)limn⟶∞HFσn,Tσn=0.On the other side,(23)fHFσ∗,Tσ∗≤fHFσ∗,σn+HFσn,Tσn+HFTσn,Tσ∗+α≤fdσ∗,σn+HFσn,Tσn+ζdFσn,σ∗⟶−∞,as n⟶∞, that is a contradiction. Hence, σ∗=Tσ∗.Definition 9.
LetM,dF be an F-MS. A mapping T:M⟶2M is called a rational Ćirić type α-ζ-multivalued contraction if there exists two functions α:M×M⟶0,+∞ and ζ∈Ψ such that
(24)ασ,ÿHFTσ,Tÿ≤ζRFσ,ÿ,for σ,ÿ∈M×M, where
(25)RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ1+dFσ,ÿ.Theorem 10.
Suppose thatM,dF be an F-MS and T:M⟶CBM be an α-admissible and rational Ćirić type α-ζ-multivalued contraction such that T satisfies the property (BS). Also, suppose that these conditions hold:
(i)
M,dF is F-complete(ii)
ασ0,σ1≥1 for an σ0∈M and σ1∈Tσ0;(iii)
T is continuous
Then,T has an endpoint.Proof.
Chooseσ0∈M and σ1∈Tσ0 such that ασ0,σ1≥1. Since T satisfies the property (BS), there exists σ2∈Tσ1 such that
(26)HFTσ1,Tσ2=supb∈Tσ2dFσ2,b.SinceT is α-admissible, ασ1,σ2≥1. Continuing this process, we obtain a sequence σn such that σn+1∈Tσn, ασn,σn+1≥1 and
(27)HFTσn,Tσn+1=supb∈Tσn+1dFσn+1,b,for all n. If σn=σn+1 for some n∈ℕ, then we get that
(28)HFσn+1,Tσn+1=supb∈Tσn+1dFσn+1,b=HFTσn,Tσn+1=0.It implies thatσn+1 is an endpoint. Hence, we suppose that σn≠σn+1, for all n∈ℕ.Note that(29)dFσn,σn+1≤supb∈TσndFσn,b=HFTσn−1,Tσn≤ασn−1,σnHFTσn−1,Tσn≤ζRFσn−1,σn=ζmaxdFσn−1,σn,dFσn−1,Tσn−1dFσn,Tσn1+dFσn−1,σn≤ζmaxdFσn−1,σn,dFσn−1,σndFσn,σn+11+dFσn−1,σn≤ζmaxdFσn−1,σn,dFσn,σn+1,for all n≥2. If maxdFσn−1,σn,dFσn,σn+1=dFσn,σn+1, then
(30)dFσn,σn+1≤ζdFσn,σn+1<dFσn,σn+1,which is a contradiction. So, we have
(31)maxdFσn−1,σn,dFσn,σn+1=dFσn−1,σn,which implies
(32)dFσn,σn+1≤ζdFσn−1,σn.Continuing in this way, we obtain that(33)dFσn,σn+1≤ζdFσn−1,σn≤ζ2dFσn−2,σn−1≤ζ3dFσn−3,σn−2≤⋯≤ζndFσ0,σ1,for all n≥2 which yields that
(34)∑i=nm−1dFσi,σi+1≤∑i=nm−1ζidFσ0,σ1,for m>n≥2. Suppose that ε>0 be arbitrary. Next, let f,α∈F×0,+∞ be such that (dF3) is satisfied. By (F2), there exists δ>0 such that
(35)0<t<δ⇒ft<fε−α.Suppose thatN∈ℕ be such that ∑i≥NζidFσ1,σ2<δ. Hence, by (24), (35) and (F1), we have
(36)f∑i=nm−1dFσi,σi+1≤f∑i=nm−1ζidFσ0,σ1≤f∑i≥NζidFσ0,σ1<fε−α,for m>n≥N. Using (D3) and (36), we obtain that dFσn,σm>0 where m>n≥N which implies that
(37)fdFσn,σm≤f∑i=nm−1dFσi,σi+1+α<fε,which implies by (F1) that dFσn,σm<ε, for all m>n≥N. This proves that σn is F-Cauchy. As M is F-complete, so ∃σå∈M such that σn⟶σå. We shall prove that σ∗ is an endpoint of T. We assume on contrary that Tσ∗≠σ∗. Then, HFσ∗,Tσ∗>0. Now, we have
(38)fHFσn,Tσn=fHFTσn−1,Tσn≤fHFTσn−1,Tσ∗+HFTσn,Tσ∗+α.Note that we used the property (BS) in the above inequality. Taking the limit in both sides of the above inequality and using continuity assumption of T, we get limn⟶∞fHFσn,Tσn=−∞ which implies that limn⟶∞HFσn,Tσn=0. Hence,
(39)fHFσ∗,Tσ∗≤fHFσ∗,σn+HFσn,Tσn+HFTσn,Tσ∗+α≤fdσ∗,σn+HFσn,Tσn+HFTσn,Tσ∗⟶−∞,as n⟶∞, that is a contradiction. Hence, σ∗=Tσ∗.Example 1.
Consider the setM=1,2,3. Suppose that the mapping dF:M×M⟶0,+∞ be given by
(40)dF1,1=dF2,2=dF3,3=0,dF1,2=dF2,1=12,dF2,3=dF3,2=23,dF1,3=dF3,1=43.So,M,dF is an F-metric on M with ft=lnt and α=ln7/6. Now, define T:M⟶CBM by T1=T2=1 and T3=1,2. Taking ζt=3/4t, we have
(41)HFT1,T2=0,HFT1,T3=dF1,2=12≤3443=34RF1,3,where
(42)RF1,3=maxdF1,3,dF1,T1dF3,T31+dF1,3=maxdF1,3,dF1,1dF3,1,21+dF1,3,HFT2,T3=dF1,2=12≤3423=34RF2,3,where
(43)RF2,3=maxdF2,3,dF2,T2dF3,T31+dF1,3=maxdF1,3,dF2,1dF3,1,21+dF2,3.Therefore,(44)ασ,ÿHFTσ,Tÿ≤ζRFσ,ÿ,where
(45)RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ1+dFσ,ÿ,for all σ,ÿ∈M. Taking ασ,ÿ=1 for all σ,ÿ∈M, T satisfies all of the conditions of Theorem 10 and so T has an endpoint. Here, T1=1.


## 4. Endpoint Theorem in GraphicF-Metric Spaces

In the present section, we will discuss the existence of endpoints on anF-MS equiped with a graph G, i.e, (F-GMS).Jachymski [19] has obtained an extension of Banach’s contraction principle in metric space equiped with a graph G. Afterwars, Dinevari and Frigon [20] proved his results for multivalued mappings. Let M,dF be an F-MS. A set σ,σ:σ∈M is said to be a diagonal of M×M, and represented by Γ. Let G be a graph such that the set VG=M, that is, the set of its vertices and the set EG of its edges consists of all loops, i.e., Γ⊆EG.Definition 11.
[21] Let M=∅ equiped with a graph G and T:M⟶2M. The mapping T is said to preserves edges weakly if, for all σ∈M and ÿ∈Tσ with σ,ÿ∈EG, we get ÿ,z∈EG,∀z∈Tÿ.We give the following definition from [21] which is required in our proof.Definition 12.
LetM,dF be an F-GMS.
TheF-GMS M is called EG-complete if every Cauchy sequence σn in M with σn,σn+1∈EG, for all n∈ℕ converges in M.Definition 13.
A mappingT:M⟶CBM is called aEG-continuous mapping if, for any σ∈M and any sequence σn with limn⟶∞dFσn,σ=0 and σn,σn+1∈EG for all n∈ℕ, we have
(46)limn⟶∞HFTσn,Tσ=0.Definition 14.
A multivalued mappingT:M⟶CBM is called a rational Ćirić type EG,ζ-contraction multivalued mapping if there exist a function ζ∈Ψ such that
(47)σ,ÿ∈M,σ,ÿ∈EG⇒HFTσ,Tÿ≤ζRFσ,ÿ,where RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ/1+dFσ,ÿ.Theorem 15.
Suppose thatM,dF be an F-GMS and T:M⟶CBM be a rational Ćirić type EG,ζ-multivalued contraction. Suppose that the following conditions hold:
S1M,dF is an EG-complete F-GMS
S2T preserves edges weakly
S3 there exist σ0 and σ1∈Tσ0 such that σ0,σ1∈EG
S4T is an EG-continuous multivalued mapping
Then,T has an endpoint point in M.Proof.
This result can be obtain from Theorem10 if we define a mapping α:M×M⟶0,∞ by ασ,ÿ=1, if σ,ÿ∈EG and ασ,ÿ=0, otherwise.


## 5. Endpoint Theorem in OrderedF-Metric Spaces

In 2004, Ran and Reurings [22] gave the idea of ordered metric space (OMS) by combing classical metric space M,d and partial order ° on M. Fixed point results in OMS have many applications in integral and differential equations and other fields of mathematical analysis (see [23, 24]). In this section, we will consider (F-OMS), i.e., M,dF° where M,dF is an F-MS and ° is a partial order on M and we will derive some new results from Theorems 8 and 10. Remember that T:M⟶M is nondecreasing if ∀σ,ÿ∈M,σ°ÿ⇒Tσ°Tÿ.Here, we state the following notion motivated from [25].Definition 16.
LetM=∅ with partial order °onM and T:M⟶2M. Then, T is said to be weakly increasing if, f or all σ∈M and ÿ∈Tσ with σ°ÿ, we get that ÿ°z, for all z∈Tÿ.Definition 17.
LetM,dF° be an F-OMS.
TheF-OMS M is called °-complete if every Cauchy sequence σn in M with σn°σn+1, for all n∈ℕ converges in M.Definition 18.
A mappingT:M⟶CBM is said to be a°-continuous mapping if, for any σ∈M and any sequence σn with limn⟶∞dFσn,σ=0 and σn°σn+1, for all n∈ℕ, we get
(48)limn⟶∞HFTσn,Tσ=0.Motivated from [8], we define the notion of an ordered rational C′iric′ type ζ-multivalued contraction in an F-OMS.Definition 19.
A multivaluedT:M⟶CBM is called an ordered rational C′iric′ type ζ-multivalued contraction if there exists ζ∈Ψ such that
(49)σ,ÿ∈M,σ°ÿ⇒HFTσ,Tÿ≤ζRFσ,ÿ,where RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ/1+dFσ,ÿ.Theorem 20.
LetM,dF, be an F-OMS ° and T:M⟶CBM be an ordered rational Ćirić type ζ-multivalued contraction. Assume that these hold:
S1M,dF is an °-complete F-OMS
S2T is weakly increasing
S3 there exist σ0 and σ1∈Tσ0 such that σ0°σ1
S4T is an °-continuous multivalued mapping
Then,T has an endpoint point in M.Proof.
This result can be obtained from Theorem10 if we define a mapping α:M×M⟶0,∞ by ασ,ÿ=1, if σ°ÿ, and ασ,ÿ=0, otherwise.


## 6. Suzuki Type Endpoint Results inF-MS

In 2008, Suzuki [26] obtained a fixed point result as generalization of the Banach fixed point theorem. In this section, we derive endpoint results for rational Suzuki type ζ-multivalued contraction in F-MS as consequence of our result.Corollary 21.
LetM,dF be a complete F-MS, ζ∈Ψ and T:M⟶CBM such that dFσ,Tσ≤dFσ,ÿ+ζdFσ,ÿ implies
(50)HFTσ,Tÿ≤ζRFσ,ÿ,where RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ/1+dFσ,ÿ, for all σ,ÿ∈M and T satisfies the property (BS). If T is continuous, then T has an endpoint.Proof.
Defineα:M×M⟶0,∞ by
(51)ασ,ÿ=1,dFσ,Tσ≤dFσ,ÿ+ζdFσ,ÿ,0,otherwise.It is easy to check thatT is α-admissible.Also, for every σ0∈M and σ1∈Tσ0, we have dFσ0,Tσ0≤dFσ0,σ1≤dFσ0,σ1+ζdFσ0,σ1. Hence, ασ0,σ1=1. It is very simple to check that
(52)ασ,ÿHFTσ,Tÿ≤ζRσ,ÿ,where RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ/1+dFσ,ÿ, for all σ,ÿ∈M. Therefore, by Theorem 10, T has an endpoint.Corollary 22.
Suppose thatM,dF be a complete F-MS, r∈0,1 and T:M⟶CBM such that 1/1+rdFσ,Tσ≤dFσ,ÿ implies that
(53)HFTσ,Tÿ≤rRFσ,ÿ,where RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ/1+dFσ,ÿ, for all σ,ÿ∈M and T enjoys property (BS). If T is continuous, then T has an endpoint.


## 7. Application to Nonlinear Integral Equations

Let CB (ℝ) represents the set of all nonempty closed and bounded subsets of ℝ and B=CI,ℝ be the space of all real-valued continuous functions on 0,1. Clearly, B equiped with the F-metric dF:B×B⟶0,+∞ given by
(54)dFσ,ÿ=eσ−ÿ,ifσ=ÿ,0,otherwise,where
(55)σ−ÿ=supt∈Iσt−ÿt,is a F-complete F-metric space (see [15]).Now, we consider the integral equation(56)σt=0tKt,s,σsdF+gt,t∈I, where σ∈B,K:0,1×0,1×ℝ⟶CBℝ and g:0,1⟶ℝ is continuous.Theorem 23.
Suppose that these conditions hold:(i)
for allσ∈B,K:0,1×0,1×ℝ⟶CBℝ is such that Kt,s,σs is continuous in 0,1×0,1(ii)
there existsI:0,1×0,1⟶ℝ which is continuous that satisfy the property inft∈I0tIt,sds=τ>0 such that for any σ,ℏ∈B and each kσt,s∈Kσt,s,σs, there exists kℏt,s∈Kσt,s,ℏs such that
(57)kσt,s−kℏt,s≤maxσs−ℏs,σs−kσt,sℏs−kℏt,s/1+σs−ℏs−It,sfor all t,s∈0,1.Then, the integral equation (56) has at least one solution in B.Proof.
Suppose that multivalued mappingT:B⟶CBB defined by
(58)Tσ=ϖ∈B:ϖt∈gt+∫0tKt,s,σsdFs,t∈a,b,for all σ∈B. Evidently, each endpoint of T is a solution of (56).Next, consider the set-valued operatorKσt,s:0,1×0,1⟶CBℝ, defined by
(59)Kσt,s=Kt,s,σs.Then, by Michael’s selection theorem,∃kσt,s:0,1×0,1⟶ℝ such that kσt,s∈Kσt,s for each t,s∈0,1. This implies that gt+∫0tkρt,sdFs∈Tσ. Hence, Tσ=∅. Next, we prove that the multivalued function T satisfies all the conditions of Theorem 10. Let σ,ÿ∈B and ρt∈Tσ. Then, ∃kσt,s∈Kσt,s for each t,s∈0,1 such that
(60)ρt=gt+∫0tkσt,sdFs,for t∈0,1. On the other side, by assumption (ii), ∃kℏt,s∈Kσt,s such that (57) holds. Now, by taking
(61)ϖt=gt+∫0tkℏt,sdFs,we get
(62)ϖt=gt+∫0tKt,s,ℏsdFs=Tℏ,for t∈0,1,
(63)dFρ,ϖ=eρ−ϖ≤esupt∈0,1∫0tkσt,sds−∫0tkℏt,sds≤esupt∈0,1∫0tkσt,s−kℏt,sds≤esupt∈0,1∫0tmaxσs−ÿs,σs−kσt,sℏs−kℏt,s/1+σs−ℏs−It,sds=esupt∈0,1∫0tmaxσs−ℏs,σs−kσt,sℏs−kℏt,s/1+σs−ℏsds−∫0tIt,sds≤emaxσs−ℏs,σs−kσt,sℏs−kℏt,s/1+σs−ℏs−inft∈0,1∫0tIt,sds≤emaxσs−ℏs,σs−kσt,sℏs−kℏt,s/1+σs−ℏs−τ≤emaxσs−ℏs,σs−kσt,sℏs−kℏt,s/1+σs−ℏs·e−τ=ζRFσ,ÿ,where ζt=e−τt. By interchanging the roles of σ and ÿ, we get that
(64)HFTσ,Tÿ≤ζRFσ,ÿ,where RFσ,ÿ=maxdFσ,ÿ,dFσ,TσdFÿ,Tÿ/1+dFσ,ÿ, for all σ,ÿ∈B. Taking ασ,ÿ=1, for all σ,ÿ∈B, all of the conditions of Theorem 10 are satisfied, and thus, T has an endpoint, which is a solution of integral equation (56).



---

*Source: 3739382-2022-06-06.xml*