CL H H CL CL CommonFixedPointandEndpointTheoremsforaCountableFamilyofMulti-ValuedMappings

Article

Common Fixed Point and Endpoint Theorems for a Countable Family of Multi-Valued Mappings

Hüseyin I¸sık¹ , Babak Mohammadi^2,* , Choonkil Park^3,* and Vahid Parvaneh⁴

1 Department of Mathematics, Faculty of Science and Arts, Mu¸s Alparslan University, 49250 Mu¸s, Turkey;

isikhuseyin76@gmail.com

2 Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran

3 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

4 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran;

zam.dalahoo@gmail.com

* Correspondence: bmohammadi@marandiau.ac.ir (B.M.); baak@hanyang.ac.kr (C.P.)

Received: 4 December 2019; Accepted: 24 December 2019; Published: 21 February 2020 Abstract:We prove some common fixed point and endpoint theorems for a countable infinite family of multi-valued mappings, as well as Allahyari et al. (2015) did for self-mappings. An example and an application to a system of integral equations are given to show the usability of the results.

Keywords:common fixed point; endpoint; infinite family; multi-valued mapping; (HS) property

1. Introduction

The study of common fixed point for a family of contraction mappings was initiated by ´Ciri´c in [1].

Recently, in 2015, Allahyari et al. [2] introduced some new type of contractions for a countable family of contraction self-mappings and studied common fixed point for them.

On the other hand, existence of a fixed point for multi-valued mappings has been important for many mathematicians. In 1969, Nadler [3] extended the Banach contraction principle to multi-valued mappings. After that, many authors generalized Nadler’s result in different ways (see, for instance [4–8]).

In 2012, Samet et al. [9] introduced the notion of α-admisssible mappings and a new type of contraction to a mapping T : X Ñ X called α-ψ-contractive mapping, that is, αpx,yqdpTx,Tyq ďψpdpx,yqqfor allx,yPX. This result generalized and improved many existing fixed point results. In the last few years, some authors have extended the notion ofα-admisssibility andα-ψ-contraction to multi-valued mappings (see, [10,11]). In addition, common fixed point for a finite family or countable family of multi-valued mappings has been studied by some researchers (see, for example [12–16]).

The aim of this paper is to extend the new type of common contractivity for a family of mappings, introduced by Allahyari et al. (2015), toα-admisssible multi-valued mappings.

LetpX,dqbe a metric space, 2^Xthe set of all nonempty subsets ofX, andCLpXqthe set of all nonempty closed subsets ofX. Assume thatHis the generalized Hausdorff metric onCLpXqdefined by

HpA,Bq “

# maxtsup_xPADpx,Bq, sup_yPBDpy,Aqu, if it exists,

8, otherwise, (1)

for all A,B P CLpXq, whereDpx,Bq “ inf_yPBdpx,yq. LetT : X Ñ 2^X is a multi-valued mapping. An elementxPXis said to be a fixed point ofTifxPTx, andxis called an endpoint ofTwheneverTx“ txu.

Mathematics2020,8, 292; doi:10.3390/math8020292 www.mdpi.com/journal/mathematics

2. Main Results

Now, we are ready to state and prove the main results of this study.

Definition 1. LetXbe an arbitrary space andα:XˆXÑ r0,8qbe a function. Assume thatT_n :XÑ2^X(n = 1,2,...) is a family of multi-valued mappings. We say thattT_nuisα-admissible whenever for each xPXand yPT_nx withαpx,yq ě1, we haveαpy,zq ě1for allzPT_n`1y.

Theorem 1. LetpX,dqbe a complete metric space and0ăa_i,j pi,j“1, 2, ...qwith a_i,i`1 ‰1for all i“1, 2, ...

satisfy:

(i) for each j, limiÑ8ai,j ă1;

(ii) ř8

n“1Ană 8, where An“śn i“1

a_i,i`1

1´a_i,i`1.

Letα:XˆXÑ r0,8qbe a given function andtT_nube a sequence of multi-valued operatorsT_n :XÑCLpXq(n = 1,2,...) such that

αpx,yqHpT_ix,T_jyq ďa_i,jrDpx,T_jyq `Dpy,T_ixqs, (2) for all x,yPX;i,j“1, 2, ...with x‰y and i‰j. Moreover, assume that the following assertions hold:

(iii) there exist x₀PXand x₁PT₁x₀with x₀‰x₁andαpx₀,x₁q ě1;

(iv) tT_nuisα-admissible;

(v) for each sequencetxnuinXwithαpxn,x_n`1q ě1for all n and xnÑx, we haveαpxn,xq ě1for all n.

Then eachTnhave a common fixed point inX.

Proof. Using(iii)and (2), we have

Dpx1,T₂x1q ď αpx0,x1qHpT₁x0,T₂x1q ď a1,2rDpx0,T₂x1q `Dpx1,T₁x0qs

“ a_1,2Dpx0,T₂x₁q

ď a_1,2rdpx₀,x₁q `Dpx₁,T₂x₁qs, which implies

Dpx1,T₂x1q ď a1,2

1´a_1,2dpx0,x1q ă a1,2

1´a_1,2pdpx0,x1q,

wherep ą1 is a fixed number. From the above inequality, there existsx2P T₂x1such thatdpx1,x2q ă

a_1,2

1´a_1,2pdpx0,x1q. SincetT_nuisα-admissible, we haveαpx1,x2q ě1. Similarly, Dpx2,T₃x2q ď a_2,3

1´a2,3dpx1,x2q ă a_2,3 1´a2,3

a_1,2

1´a1,2pdpx0,x1q, and so there existsx3 P T₃x2such thatdpx2,x3q ă _1´a^a2,3

2,3

a_1,2

1´a_1,2pdpx0,x1q. Continuing this process, we obtain a sequencetxnuinXsuch thatxn`1PT<sub>n`1</sub>xn,αpxn,xn`1q ě1, and

dpxn,x_n`1q ăAnpdpx0,x₁q, for alln“1, 2, ... . (3) For anyn,mPN^withnăm, from triangle inequality, we get

dpxn,xmq ď

m´1ÿ

k“n

dpx_k,x_k`1q ď

m´1ÿ

k“n

A_kpdpx₀,x₁q Ñ0

asn,mÑ 8. Therefore, we have shown thattxnuis a Cauchy sequence. SincepX,dqis complete, there existsx PXsuch thatxn Ñ x. From(v), we getαpxn,xq ě 1 for alln. Now, we shall show that xis a common fixed point ofTn. Letmbe an arbitrary positive integer. Then, for anynPN, we have

Dpx,T_mxq ď dpx,xnq `Dpxn,T_mxq

ď dpx,xnq `αpxn´1,xqHpTnxn´1,Tmxq

ď dpx,xnq `an,mrDpxn´1,T<sub>m</sub>xq `Dpx,T_nxn´1qs ď dpx,xnq `an,mrDpxn´1,T<sub>m</sub>xq `dpx,xnqs.

Takinglimin both sides of the above inequality, asnÑ 8, we get Dpx,Tmxq ď plimnÑ8an,mqDpx,Tmxq, which impliesDpx,T_mxq “0 and soxPT_mx.

Theorem 2. LetpX,dqbe a complete metric space and0ăai,j pi,j“1, 2, ...qwith ai,i`1 ‰1for all i“1, 2, ...

satisfy:

(i) for each (j), lim_iÑ8a_i,j ă1;

(ii) ř₈

n“1Ană 8where An“ś_n

i“1 a_i,i`1

1´a_i,i`1.

Letα:XˆXÑ r0,8qbe a given function andtT_nube a sequence of multi-valued operatorsT_n :XÑCLpXq(n = 1,2,...) such that

αpx,yqHpT_ix,T_jyq ďai,jmaxtdpx,yq,Dpx,T_ixq,Dpy,T_jyq,Dpx,T_jyq,Dpy,T_ixqu, (4) for all x,yPX;i,j“1, 2, ...with x‰y and i‰j. Moreover, assume that the following assertions hold:

(iii) there exist x0PXand x₁PT₁x0with x0‰x₁andαpx0,x₁q ě1;

(iv) tT_nuisα-admissible;

(v) for each sequencetxnuinXwithαpxn,xn`1q ě1for all n and xnÑx, we haveαpxn,xq ě1for all n.

Then eachT_nhave a common fixed point inX.

Proof. By(iii)and (4), we have

Dpx1,T₂x1q ď αpx0,x1qHpT₁x0,T₂x1q

ď a_1,2maxtdpx0,x₁q,Dpx0,T₁x0q,Dpx₁,T₂x₁q,Dpx0,T₂x₁q,Dpx₁,T₁x0qu ď a_1,2rdpx₀,x₁q `Dpx₁,T₂x₁qs,

which implies

Dpx1,T₂x1q ď a1,2

1´a_1,2dpx0,x1q ă a1,2

1´a_1,2pdpx0,x1q,

whichp ą1 is a fixed number. From the above inequality, there existsx2P T₂x1such thatdpx1,x2q ă

a_1,2

1´a_1,2pdpx0,x1q. Continuing in this manner and as in proof of Theorem1, we obtain a sequencetxnuwith

αpxn,xn`1q ě1 andxPXsuch thatxn Ñx. Using(v), we getαpxn,xq ě1 for alln. Next, we show thatx is a common fixed point ofT_n. Letmbe an arbitrary positive integer. Then, for anynPN^{, we have}

Dpx,T_mxq ď dpx,xnq `Dpxn,T_mxq

ď dpx,xnq `αpx_n´1,xqHpT_nx_n´1,T_mxq

ď dpx,xnq `an,mmaxtdpxn´1,xq,Dpxn´1,Tnxn´1q,Dpx,Tmxq, Dpxn´1,T_mxq,Dpx,T_nxn´1qu

ď dpx,xnq `an,mmaxtdpxn´1,xq,dpxn´1,xnq,Dpx,T_mxq,Dpxn´1,T_mxq,dpx,xnqu.

Taking lim as n Ñ 8, we obtain Dpx,T_mxq ď plimnÑ8an,mqDpx,T_mxq, which impliesDpx,T_mxq “0. This means thatxPT_mxand the proof is complete.

Theorem 3. LetpX,dqbe a complete metric space and0 ď ai,j, 0 ă bi,jpi,j “ 1, 2, ...qwith ai,i`1 ‰ 1for all i“1, 2, ...satisfy:

(i) for each j, lim_iÑ8a_i,j ă1and lim_iÑ8b_i,jă 8;

(ii) ř8

n“1Ană 8where An“śn i“1

b_i,i`1

1´a_i,i`1.

Letα:XˆXÑ r0,8qbe a given function andtT_nube a sequence of multi-valued operatorsT_n :XÑCLpXq(n = 1,2,...) such that

αpx,yqHpT_ix,T_jyq ďai,jDpy,T_jyqϕpDpx,T_ixq,dpx,yqq `bi,jdpx,yq, (5) for all x,yPX;i,j“1, 2, ...with x‰y and i‰j, whereϕ:r0,8q ˆ r0,8q Ñ r0,8qis a continuous function such thatϕpt,tq “1for all tP r0,8qand for any t₁,s₁,t₂,s₂P r0,8q,

t₁ďt₂,s₁“s₂ùñϕpt₁,s₁q ďϕpt₂,s₂q.

Moreover, assume that the following assertions hold:

(iii) there exist x0PXand x1PT₁x0with x0‰x1andαpx0,x1q ě1;

(iv) tT_nuisα-admissible;

(v) for each sequencetxnuinXwithαpxn,xn`1q ě1for all n and xnÑx, we haveαpxn,xq ě1for all n.

Then eachT_nhave a common fixed point inX.

Proof. By(iii)and (5), we have

Dpx1,T₂x1q ď αpx0,x1qHpT₁x0,T₂x1q

ď a_1,2Dpx₁,T₂x₁qϕpDpx0,T₁x0q,dpx0,x₁qq `b<sub>1,2</sub>dpx0,x<sub>1</sub>q ď a<sub>1,2</sub>Dpx<sub>1</sub>,T<sub>2</sub>x<sub>1</sub>qϕpdpx<sub>0</sub>,x<sub>1</sub>q,dpx<sub>0</sub>,x<sub>1</sub>qq `b_1,2dpx₀,x₁q ď a1,2Dpx1,T₂x1q `b1,2dpx0,x1q,

which gives us

Dpx1,T₂x1q ď b_1,2

1´a_1,2dpx0,x1q ă b_1,2

1´a_1,2pdpx0,x1q,

wherep ą1 is a fixed number. From the above inequality, there existsx₂P T₂x₁such thatdpx₁,x₂q ă

b_1,2

1´a_1,2pdpx0,x1q. Similarly,

Dpx₂,T₃x₂q ď b2,3

1´a2,3

dpx₁,x₂q ă b2,3

1´a2,3

b1,2

1´a1,2

pdpx₀,x₁q,

and so there existsx3 P T₃x2such thatdpx2,x3q ă _1´a^b2,3

2,3

b_1,2

1´a_1,2pdpx0,x1q. Continuing this process, we obtain a sequencetxnuinXsuch thatxn`1PT<sub>n`1</sub>xn,αpxn,xn`1q ě1, and

dpxn,xn`1q ăAnpdpx0,x₁q, for alln“1, 2, ... . (6) Again, as in the proof of Theorem1, we conclude thattxnuis a Cauchy sequence, and so there existsxPX such thatxnÑx. From the assumption(v), we getαpxn,xq ě1 for alln. To show thatxis a common fixed point ofT_n, letmbe an arbitrary positive integer. Then, for anynPN, we have

Dpx,Tmxq ď dpx,xnq `Dpxn,Tmxq ďdpx,xnq `αpxn´1,xqHpTnxn´1,Tmxq

ď dpx,xnq `an,mDpx,T<sub>m</sub>xqϕpDpxn´1,T<sub>n</sub>xn´1q,dpxn´1,xqq `bn,mdpxn´1,xq ď dpx,xnq `an,mDpx,T<sub>m</sub>xqϕpdpxn´1,xnq,dpxn´1,xqq `bn,mdpxn´1,xq.

Takinglimin both sides of the above inequality, asnÑ 8, we obtain Dpx,Tmxq ď plimnÑ8an,mqDpx,Tmxq.

We concludeDpx,T_mxq “0 and thusxPT_mx.

3. Common Endpoint Theorems

The notion of endpoints of multi-valued mappings has been studied by some researchers in the last decade (see for instance, [17–19]). In current section, we state and prove some common endpoint theorems for a sequence of multi-valued mappings with the contractions mentioned in Section2. We need the following definition.

Definition 2. LetT_n :XÑCLpXq(n = 1,2,...) be a sequence of multi-valued mappings. We say thattT_nuhas (HS) property whenever for each xPXthere exists yPT_nx such thatHpT_nx,T_{n`1</sub>yq ěsup<sub>bPTn`1y}dpy,bq.

Theorem 4. Let pX,dq be a complete metric space and0 ď a_i,j pi,j “ 1, 2, ...q with a_i,i`1 ‰ 1 for all i “ 1, 2, ...satisfy:

(i) for each (j), limiÑ8ai,j ă1;

(ii) ř8

n“1Ană 8where An“śn i“1

a_i,i`1

1´a_i,i`1.

Letα:XˆXÑ r0,8qbe a given function andtT_nube a sequence of multi-valued operatorsT_n :XÑCLpXq(n = 1,2,...) satisfying (HS) property such that

αpx,yqHpT_ix,T_jyq ďa_i,jrDpx,T_jyq `Dpy,T_ixqs, (7) for all x,yPX;i,j“1, 2, ...with x‰y and i‰j. Moreover, assume that the following assertions hold:

(iii) there exists x₀PXsuch that for any xPT₁x₀,we haveαpx₀,xq ě1;

(iv) tT_nuisα-admissible;

(v) for each sequencetxnuinXwithαpxn,xn`1q ě1for all n and xnÑx, we haveαpxn,xq ě1for all n.

Then eachT_nhave a common endpoint inX.

Proof. Since tT_nu has (HS) property, there exists x₁ P T₁x₀ such that HpT₁x0,T₂x1q ěsup_bPT2x1dpx1,bq. From(iii), we haveαpx0,x1q ě1. Similarly, there existsx2PT₂x1such

thatHpT₂x1,T₃x2q ěsup_bPT3x2dpx2,bq. SincetT_nuisα-admissible, soαpx1,x2q ě1. If we continue this process, we obtain a sequencetxnuinXsuch thatxnPT_nx_n´1,αpx_n´1,xnq ě1, and

HpT_nx_n´1,T_n`1xnq ě sup

bPT_n`1xn

dpxn,bq, (8)

for allně1. Then we have

dpxn,x_{n`1</sub>q ď sup<sub>bPTn`1xn}dpxn,bq ďαpx_n´1,xnqHpT_nx_n´1,T<sub>n`1</sub>xnq ď an,n`1rDpxn´1,T<sub>n`1</sub>xnq `Dpxn,T_nxn´1qs

ď a_{n,n`1</sub>rdpx<sub>n´1</sub>,x<sub>n`1}qs ďa<sub>n,n`1</sub>rdpx<sub>n´1</sub>,xnq `dpxn,x_n`1qs.

From the above inequality, we get

dpxn,xn`1q ď a<sub>n,n`1</sub>

1´an,n`1dpxn´1,xnq ď...ďAndpx0,x1q.

Hencetxnuis a Cauchy sequence, and so there existsx P X such thatxn Ñ x. From(v)we deduce αpxn,xq ě1 for alln. Now we show thatxis a common endpoint ofT_n. LetmPNbe arbitrary. Then, for anynPN, we have

Hptxu,T_mxq ď dpx,xnq `Hptxnu,T<sub>n`1</sub>xnq `αpxn,xqHpT<sub>n`1</sub>xn,T_mxq

ď dpx,xnq `αpxn´1,xnqHpT<sub>n</sub>xn´1,T<sub>n`1</sub>xnq `αpxn,xqHpT<sub>n`1</sub>xn,T_mxq ď dpx,xnq `an,n`1rDpxn´1,T<sub>n`1</sub>xnq `Dpxn,T_nxn´1qs

`a<sub>n`1,m</sub>rDpxn,T_mxq `Dpx,T<sub>n`1</sub>xnqs

ď dpx,xnq `a<sub>n,n`1</sub>rdpx_n´1,x<sub>n`1</sub>qs `a<sub>n`1,m</sub>rDpxn,T<sub>m</sub>xq `dpx,x_n`1qs.

TakinglimasnÑ 8, we obtain

Hptxu,T_mxq ď plimnÑ8an`1,mqDpx,T<sub>m</sub>xq ď plimnÑ8an`1,mqHptxu,T_mxq, which impliesHptxu,T_mxq “0 and soT_mx“ txu. Sincemwas arbitrary, the proof is complete.

Theorem 5. In the statement of Theorem4, if we add the extra conditionαpx,yq ě1for any common endpoints x,y ofTn, then the common endpoint ofTnis unique.

Proof. Let x,y be two common endpoints of T_n. Since ř₈

n“1An ă 8, there exists i0 P N such that

a_i0,i0`1

1´a_{i0,i0`1</sub> ă1, which impliesa<sub>i0,i0`1}ă1

2. Then, using (7), we get dpx,yq “ HpT_ix,T_i0`1yq

ď αpx,yqHpT_i0x,T_i0`1yq

ď a_{i0,i0`1</sub>rDpx,T<sub>i0`1}yq `Dpy,T_i0xqs

“ 2a_i0,i0`1dpx,yq, which impliesdpx,yq “0 and sox“y.

Example 1. Consider the spaceX“ r0, 1swith the usual metric dpx,yq “ |x´y|. Define a sequence of mappings T_n :XÑCLpXqby

T_npxq “

$

’&

’%

t1u, ¹₂ďxď1,

”2

3`<sub>n`2</sub>¹ , 1ı

, x“0, t0u, 0ăxă¹₂.

Also consider the constants ai,j “ ¹₃`<sub>|i´j|`6</sub>¹ . Then limiÑ8ai,j “ ¹₃ ă1, for all j PN. An “ś_n

i“1 a_i,i`1

1´a_i,i`1 “

p¹⁰₁₁qⁿ. Thusř8

n“1An “ř8

n“1p¹⁰₁₁qⁿ ă 8. Also let αpx,yq “

# 1, x,yP t0u Y”

1 2, 1ı

, 0, otherwise.

Now we show thatαpx,yqHpT_ix,T_jyq ďa_i,jrDpx,T_jyq `Dpy,T_ixqs, for all x,yPX. If0ăxă¹₂ or0ăyă ¹₂, thenαpx,yq “0and we have nothing to prove. Therefore, we may assume x,y P t0u Y”

1 2, 1ı

. We consider the following cases:

(1) x,yP

”1 2, 1ı

. In this case we haveαpx,yqHpT_ix,T_jyq “Hpt1u,t1uq “0ďa_i,jrDpx,T_jyq `Dpy,T_ixqs, for all x,yPX.

(2) xP

”1 2, 1ı

and y“0. In this case we have

αpx,yqHpT_ix,T_jyq “Hpt1u,

„2 3` 1

j`2, 1

 q

“ |1´ p2 3` 1

j`2q| “ 1 3´ 1

j`2 ď 1 3 ď p1

3` 1

|i´j| `6qp|x´ p2 3` 1

j`2q| ` |0´1|q

“a_i,jrDpx,T_jyq `Dpy,T_ixq.

(3) x“y“0,iăj. Then

αpx,yqHpT_ix,T_jyq “ |2 3` 1

j`2´ p2 3 ` 1

i`2q| “ 1

i`2´ 1

j`2 ď 1 i`2 ď p1

3` 1

|i´j| `6qp2 3` 1

i`2` p2 3 ` 1

j`2qq

“ai,jrDpx,T_jyq `Dpy,T_ixqs.

Also for x0“0and x₁ “1, we have x₁ P t1u “”

2

3`<sub>1`2</sub>¹ , 1ı

“T₁x0andαpx,yq “1ě1. It is easy to check thattTnuisα-admissible. Also, for any common endpoints x,y, we haveαpx,yq ě1. Thus, all of the conditions of Theorem4and Theorem5are satisfied. Therefore, the mappingsT_nhave a unique common endpoint. Here x“1is the unique common endpoint ofT_n.

Theorem 6. Let pX,dq be a complete metric space and0 ď a_i,j pi,j “ 1, 2, ...q with a_i,i`1 ‰ 1 for all i “ 1, 2, ...satisfy:

(i) for each (j), limiÑ8ai,j ă1;

(ii) ř8

n“1Ană 8where An“śn i“1

a_i,i`1

1´a_i,i`1.

Letα:XˆXÑ r0,8qbe a given function andtT_nube a sequence of multi-valued operatorsT_n :XÑCLpXq(n = 1,2,...) satisfying (HS) property such that

αpx,yqHpT_ix,T_jyq ďa_i,jmaxtdpx,yq,Dpx,T_ixq,Dpy,T_jyq,Dpx,T_jyq,Dpy,T_ixqu, (9) for all x,yPX;i,j“1, 2, ...with x‰y and i‰j. Moreover, assume that the following assertions hold:

(iii) there exists x0PXsuch that for any xPT₁x0,we haveαpx0,xq ě1;

(iv) tTnuisα-admissible;

(v) for each sequencetxnuinXwithαpxn,x_n`1q ě1for all n and xnÑx, we haveαpxn,xq ě1for all n.

Then eachTnhave a common endpoint inX.

Proof. As in the proof of Theorem4, there exists a sequencetxnuinXsuch thatxnPT_nxn´1,αpxn´1,xnq ě 1, and

HpT_nx_n´1,T_n`1xnq ě sup

bPT_n`1xn

dpxn,bq, for allně1. Then we have

dpxn,xn`1q ď sup<sub>bPTn`1xn</sub>dpxn,bq ďαpxn´1,xnqHpTnxn´1,T_{n`1</sub>xnq ď a<sub>n,n`1}maxtdpx_n´1,xnq,Dpx_n´1,T_nx_n´1q,Dpxn,T_n`1xnq,

Dpx_n´1,T<sub>n`1</sub>xnq,Dpxn,T<sub>n</sub>x<sub>n´1</sub>qu ď an,n`1rdpxn´1,xnq `dpxn,xn`1qs.

From the above inequality, we get

dpxn,x<sub>n`1</sub>q ď an,n`1

1´a_n,n`1dpx_n´1,xnq ď...ďAndpx0,x₁q.

Thus,txnuis a Cauchy sequence and so there existsxPXsuch thatxnÑxandαpxn,xq ě1 for alln. Now, we show thatxis a common endpoint ofT_n. LetmPNbe arbitrary. Then, for anynPN, we have

Hptxu,Tmxq ď dpx,xnq `Hptxnu,T<sub>n`1</sub>xnq `αpxn,xqHpT<sub>n`1</sub>xn,Tmxq

ď dpx,xnq `αpxn´1,xnqHpT<sub>n</sub>xn´1,T<sub>n`1</sub>xnq `αpxn,xqHpT<sub>n`1</sub>xn,T_mxq ď dpx,xnq `an,n`1rdpxn´1,xnq `dpxn,xn`1qs

`a<sub>n`1,m</sub>maxtdpxn,xq,Dpxn,T_{n`1</sub>xnq,Dpx,T<sub>m</sub>xq,Dpxn,T<sub>m</sub>xq,Dpx,T<sub>n`1}xnqu ď dpx,xnq `an,n`1rdpxn´1,xnq `dpxn,xn`1qs

`an`1,mmaxtdpxn,xq,Dpxn,xn`1q,Dpx,T<sub>m</sub>xq,Dpxn,T<sub>m</sub>xq,Dpx,xn`1qu.

Takinglimin both sides of the above inequality, asnÑ 8, we obtain

Hptxu,T_mxq ď plimnÑ8a_{n`1,m</sub>qDpx,T<sub>m</sub>xq ď plimnÑ8a<sub>n`1,m}qHptxu,T_mxq, which impliesHptxu,T_mxq “0 and soT_mx“ txu.

Theorem 7. With the conditions of Theorem6, if we add the extra conditionαpx,yq ě1for any common endpoints x,y ofT_n, then the common endpoint ofT_nis unique.

Proof. Letx,ybe two common endpoints ofT_n. Using (9), we get dpx,yq “ HpT_ix,T_jyq ďαpx,yqHpT_ix,T_jyq

ď a_i,jmaxtdpx,yq,Dpx,T_ixq,Dpy,T_jyq,Dpx,T_jyq,Dpy,T_ixqu

“ a_i,jdpx,yq.

Thus,dpx,yq ďlimiÑ8ai,jdpx,yq, which means thatdpx,yq “0 and hencex“y.

Theorem 8. LetpX,dqbe a complete metric space and0 ď ai,j, 0 ď bi,jpi,j “ 1, 2, ...qwith ai,i`1 ‰ 1for all i“1, 2, ...satisfy:

(i) for each (j), lim_iÑ8a_i,j ă1, lim_iÑ8b_i,jă1;

(ii) ř₈

n“1Ană 8where An“ś_n

i“1 b_i,i`1

1´a_i,i`1.

Letα:XˆXÑ r0,8qbe a given function andtT_nube a sequence of multi-valued operatorsT_n :XÑCLpXq(n = 1,2,...) satisfying (HS) property such that

αpx,yqHpT_ix,T_jyq ďai,jDpy,T_jyqϕpDpx,T_ixq,dpx,yqq `bi,jdpx,yq, (10) for all x,yPX;i,j“1, 2, ...with x‰y and i‰j, whereϕis as in Theorem3. Moreover, assume that the following assertions hold:

(iii) there exists x₀PXsuch that for any xPT₁x₀,we haveαpx₀,xq ě1;

(iv) tT_nuisα-admissible;

(v) for each sequencetxnuinXwithαpxn,xn`1q ě1for all n and xnÑx, we haveαpxn,xq ě1for all n.

Then eachT_nhave a common endpoint inX.

Proof. As in the proof of Theorem4, there exists a sequencetxnuinXsuch thatxnPT_nx_n´1,αpx_n´1,xnq ě 1, and

HpT_nxn´1,T_n`1xnq ě sup

bPT_n`1x_n

dpxn,bq, for allně1. Then we have

dpxn,x_n`1q ď sup_bPT

n`1xndpxn,bq

ď αpxn´1,xnqHpT_nxn´1,T_n`1xnq

ď an,n`1Dpxn,T<sub>n`1</sub>xnqϕpDpxn´1,T_nxn´1q,dpxn´1,xnqq `bn,n`1dpxn´1,xnq ď a_{n,n`1</sub>dpxn,x<sub>n`1}q `b<sub>n,n`1</sub>dpx_n´1,xnq.

From the above inequality, we get

dpxn,x<sub>n`1</sub>q ď bn,n`1

1´an,n`1

dpx_n´1,xnq ď...ďAndpx₀,x₁q.

As in proof of Theorem1, we conclude thattxnuis a Cauchy sequence, and so there existsx P Xsuch thatxnÑxandαpxn,xq ě1 for alln. To show thatxis a common endpoint ofT_n, consider an arbitrary natural numberm. Then, for anynPN, we have

Hptxu,T_mxq ď dpx,xnq `Hptxnu,T<sub>n`1</sub>xnq `αpxn,xqHpT<sub>n`1</sub>xn,T_mxq

ď dpx,xnq `αpxn´1,xnqHpTnxn´1,T<sub>n`1</sub>xnq `αpxn,xqHpT<sub>n`1</sub>xn,Tmxq ď dpx,xnq `an,n`1Dpxn,T_n`1xnqϕpDpxn´1,T_nxn´1q,dpxn´1,xnqq

`bn,n`1dpxn´1,xnq

`a<sub>n`1,m</sub>Dpx,T_mxqϕpDpxn,T<sub>n`1</sub>xnq,dpxn,xqq `b<sub>n`1,m</sub>dpxn,xq ď dpx,xnq `an,n`1dpxn,xn`1q `bn,n`1dpxn´1,xnq

`an`1,mDpx,T_mxqϕpdpxn,xn`1q,dpxn,xqq `bn`1,mdpxn,xq.

TakinglimasnÑ 8, we obtain

Hptxu,T_mxq ď plimnÑ8an`1,mqDpx,T<sub>m</sub>xq ď plimnÑ8a<sub>n`1,m</sub>qHptxu,T_mxq, which showsHptxu,T_mxq “0. ThusTmx“ txu.

Theorem 9. In the statement of Theorem8, if we add the extra conditionαpx,yq ě1for any common endpoints x,y ofT_n, then the common endpoint ofT_nis unique.

Proof. Letx,ybe two common endpoints ofTn. Using (10), we have dpx,yq “ HpT_ix,T_jyq

ď αpx,yqHpT_ix,T_jyq

ď ai,jDpy,T_jyqϕpDpx,T_ixq,dpx,yqq `bi,jdpx,yq

“ b_i,jdpx,yq.

Therefore,dpx,yq ďlim_iÑ8b_i,jdpx,yq. Hencedpx,yq “0, which means thatx“y.

4. Application to Integral Equations

TakeI “ r0,Ts. LetX :“ CpI,R^qbe the set of all real valued continuous functions with domainI.

Define the mericdonXwith

dpx,yq “sup

tPI

p|xptq ´yptq|q “ ||x´y||.

Consider the system of integral equation:

xptq “pptq ` ż_T

0

Gpt,sqFnps,xpsqqds, tPI, n“1, 2, 3, . . . . (11) Our hypotheses on the data are the following:

pAq p:IÑR^andFn :IˆR^ÑRare continuous, for allnPN^; pBq G:IˆIÑRis continuous and measurable atsPIfor alltPI;

pCq Gpt,sq ě0 for allt,sPIandş_T

0 Gpt,sqdsď1 for alltPI;

pDq there existsx₀PXsuch thatx₀ptq ďşT

0Gpt,sqF₁ps,x₀psqqds, for alltPI; pEq for any x P X with xptq ď şT

0Gpt,sqFnps,xpsqqds, for all t P I, then we have şT

0 Gpt,sqFnps,xpsqqdsďşT

0 Gpt,sqFn`1ps,şT

0Gps,τqFnpτ,xpτqqdτqds, for alltPI.

Let 0ďa_i,jpi,j“1, 2, ...qwitha_i,i`1‰1 for alli“1, 2, ... satisfy:

pFq for each (j),lim_iÑ8a_i,jă1;

pGq Σ⁸_n“1An ă 8whereAn “ś_n

i“1 a_i,i`1

1´a_i,i`1;

pHq for eachtPI,x,yPXwithxďy, andi‰j, we have

|F_ipt,xptqq ´F_jpt,yptqq| ďa_i,jp|xptq ´ş_T

0 Gpt,sqF_jps,ypsqqds|

`|yptq ´ş_T

0Gpt,sqFips,xpsqqds|q. (12) Theorem 10. Under the assumptionspAq–pHq,the system of integral Equation (11) has a solution inX.

Proof. DefineΥn :XÑXas

pΥnxqptq “pptq ` ż_T

0

Gpt,sqFnps,xpsqqds, tPI for allnPN. In addition, defineα:XˆXÑ r0,8qby

αpx,yq “

# 1, xptq ďyptq f or all tPI, 0, otherwise.

Letx,ybe two arbitrary elements ofX. Ifxęy, thenαpx,yq “0 and so inequality (2) holds, obviously.

Now, letxďy. Then

|pΥixqptq ´ pΥjyqptq| “ | żT

0

Gpt,sqpFips,xpsqq ´Fjps,ypsqqds|

ď ż_T

0

Gpt,sq|F_ips,xpsqq ´F_jps,ypsqq|ds ď

żT 0

Gpt,sqai,jp|xpsq ´ żT

0

Gps,τqFjpτ,ypτqqdτ|

` |ypsq ´ ż_T

0

Gps,τqF_ipτ,xpτqqdτ|qds ď

żT 0

Gpt,sqa_i,jp|xpsq ´ pΥjyqpsq| ` |ypsq ´Υixqpsq|qds ď

żT 0

Gpt,sqai,jp||x´ pΥjyq|| ` ||y´Υixqpsq||qds ďai,jp||x´Υjy|| ` ||y´Υix||q

for everytPI. Take sup in the above inequality to find that αpx,yqdpΥix,Υjyq “ ||Υx´Υy||

ďa_i,jp||x´Υjy|| ` ||y´Υix||q “a<sub>i,j</sub>pdpx,Υjyq `dpy,Υixqq.

The propertiespDqandpEqyield that propertiespiiiqandpivqof Theorem1are satisfied. Obviously, the propertypvqof Theorem1holds. Thus, by that theorem,tΥnuhave a common fixed point, that is, the system of integral Equation (11) having a solution.

Author Contributions:Writing–original draft, H.I., B.M., C.P. and V.P.; Writing–review and editing, H.I., B.M., C.P.

and V.P. All authors have read and agreed to the published version of the manuscript.

Funding:This research received no external funding.

Conflicts of Interest:The authors declare no conflict of interest.

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