Common fixed point theorems for multimaps in locally convex spaces and some applications

Abstract

In this paper some fixed point theorems for multimaps on locally convex spaces have been introduced. As their applications, some well-known fixed point theorems will be deduced. Also, we see a notable result in differential equations.

AMS Subject Classification:46A55, 46B99.

1 Introduction

Fixed point theorems have been studied by many authors. Probably the oldest fixed point theorem is the contraction mapping principle. From this theorem, the study of fixed point theorems arose in different settings. More than a thousand of fixed point theorems have been studied and proved by many authors. Among them, there are some notable cases such as Browerâ€™s, Schauder-Tychonovâ€™s, Markov-Kakutaniâ€™s and many others. Although many fixed point theorems have been flourishing for both single maps and groups of maps in metric and Banach spaces, only a few have been reported in general topological vector spaces (for example, see [1, 2] and [3]). Recently, set-valued maps (or multimaps) have been very important tools in non-linear analysis; e.g., see [4] and [5]. Some existence results have been generalized and extended to multimaps in general spaces (as an example, see [6]).

In this paper, some new generalized fixed point theorems for multimaps in locally convex spaces have been proved, and as their results, we imply some notable theorems such as Markov-Kakutani and Schauder fixed point theorems. Finally, we see an application of the proved results in differential equations.

2 Preliminaries

Let $\left(X,\mathrm{Ï„}\right)$ be a Hausdorff locally convex space. A family $\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}$ of seminorms on X is called an associated family for Ï„ if the family $\left\{\mathrm{Î³}U:\mathrm{Î³}>0\right\}$ forms a base of neighborhoods of zero for Ï„, where $U={â‹‚}_{i=1}^{n}{U}_{{\mathrm{Î±}}_{i}}$ and ${U}_{{\mathrm{Î±}}_{i}}=\left\{x:{p}_{{\mathrm{Î±}}_{i}}\left(x\right)<1\right\}$. A family $\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}$ of seminorms on X is called an augmented associated family for Ï„ if $\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}$ is an associated family such that $max\left\{{p}_{\mathrm{Î±}},{p}_{\mathrm{Î²}}\right\}âˆˆ\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}$ for any $\mathrm{Î±},\mathrm{Î²}âˆˆI$. We will denote by $\mathcal{A}\left(\mathrm{Ï„}\right)$ and ${\mathcal{A}}^{âˆ—}\left(\mathrm{Ï„}\right)$ the associated and augmented associated seminorms $\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}$, respectively. As a well-known result, there always exists a family $\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}$ of seminorms on X such that $\left\{{p}_{\mathrm{Î±}}:\mathrm{Î±}âˆˆI\right\}={\mathcal{A}}^{âˆ—}\left(\mathrm{Ï„}\right)$ (see [[7], p.203]). A subset M of X is Ï„-bounded in X if and only if each ${p}_{\mathrm{Î±}}$ is bounded on M.

A subset $AâŠ†X$ is said to be star-shaped if there exists an element $xâˆˆA$ such that $tx+\left(1âˆ’t\right)yâˆˆA$ for all $tâˆˆ\left[0,1\right]$ and for all $yâˆˆA$. Such an element is called a star-point of A. The set of all star-points of A is called the star-core of A.

Let C be a nonempty convex subset of X and $qâˆˆC$. A self-mapping T on C is said to be affine if

$T\left(\mathrm{Î»}x+\left(1âˆ’\mathrm{Î»}\right)y\right)=\mathrm{Î»}T\left(x\right)+\left(1âˆ’\mathrm{Î»}\right)T\left(y\right)$

for all $x,yâˆˆC$ and $\mathrm{Î»}âˆˆ\left(0,1\right)$. Also, T is called to be affine with respect to q if

$T\left(\mathrm{Î»}x+\left(1âˆ’\mathrm{Î»}\right)q\right)=\mathrm{Î»}T\left(x\right)+\left(1âˆ’\mathrm{Î»}\right)T\left(q\right)$

for all $xâˆˆC$ and $\mathrm{Î»}âˆˆ\left(0,1\right)$. There is an example of an affine mapping with respect to a point which is not affine (see [8]).

Let A be a subset of X. We will denote by ${2}^{A}$ the family of all subsets of A. Let X, Y be topological vector spaces and let $T:XâŸ¶{2}^{Y}$ be a multimap. T is said to be convex [9] if for each ${x}_{1},{x}_{2}âˆˆX$ and each $\mathrm{Î»}âˆˆ\left[0,1\right]$, we have

$\mathrm{Î»}T\left({x}_{1}\right)+\left(1âˆ’\mathrm{Î»}\right)T\left({x}_{2}\right)âŠ†T\left(\mathrm{Î»}{x}_{1}+\left(1âˆ’\mathrm{Î»}\right){x}_{2}\right).$

If T is single-valued, then the convexity of T is exactly the same as the affine condition (see [10]). For $AâŠ†X$, we will denote

T is said to be upper semicontinuous if for each $xâˆˆX$ and each open set V in Y with $T\left(x\right)âŠ‚V$, there exists an open neighborhood U of x such that $T\left(y\right)âŠ‚V$ for some $yâˆˆU$. Let $\mathcal{A}:=\left\{Tâˆ£T:XâŸ¶{2}^{X}\right\}$ be a family of multimaps on X into ${2}^{X}$. $\mathcal{A}$ is a commutating family of multimaps if for each $xâˆˆX$ and every pair S, T in $\mathcal{A}$, we have $S\left(T\left(x\right)\right)=T\left(S\left(x\right)\right)$. Also, we denote the fixed point set $\mathcal{F}\left(T\right)$ as $\mathcal{F}\left(T\right):=\left\{xâˆˆX:xâˆˆT\left(x\right)\right\}$.

Theorem 2.1 ([11])

Let K be a compact and star-shaped subset of a topological linear space X. Then every decreasing chain of nonempty, compact and star-shaped subsets of K has a nonempty intersection that is compact and star-shaped.

Lemma 2.2 ([11])

Suppose that K is a compact star-shaped subset of a topological linear space X and A is the corresponding star-core of K. Then A is a compact convex subset of A.

Lemma 2.3 ([10])

Let X be a nonempty, compact and convex subset of a Hausdorff locally convex space and let $T:XâŸ¶{2}^{X}$ be an upper semicontinuous multimap such that for each $xâˆˆX$, $T\left(x\right)$ is nonempty, closed and convex. Then there exists a point $\stackrel{Â¯}{x}âˆˆX$ such that $\stackrel{Â¯}{x}âˆˆT\left(\stackrel{Â¯}{x}\right)$.

Theorem 2.4 (Schauder [12])

Let X be a nonempty, compact and convex subset of a Hausdorff locally convex space V. If $T:XâŸ¶X$ is a continuous map, then T has a fixed point.

3 Main results

3.1 Fixed point results

Definition 3.1 Let C be a convex subset of a locally convex space X. For $qâˆˆC$ and $T:CâŸ¶{2}^{C}$, T is said to be q-convex if we have the following condition:

$T\left(\mathrm{Î»}q+\left(1âˆ’\mathrm{Î»}\right)x\right)âŠ‡\mathrm{Î»}T\left(q\right)+\left(1âˆ’\mathrm{Î»}\right)T\left(x\right)$

for all $xâˆˆC$ and $\mathrm{Î»}âˆˆ\left[0,1\right]$.

To establish the main results, we need the following lemma.

Lemma 3.2 Let C be a star-shaped subset of a topological vector space X and $T:CâŸ¶{2}^{C}$ be onto and q-convex with each q in the star-core of C. If S is the star-core of C, then S is invariant under T.

Proof Suppose that ${x}_{o}âˆˆS$. We will show that $T\left({x}_{0}\right)âˆˆS$. Let $tâˆˆT\left({x}_{0}\right)$, $yâˆˆC$ and $0â‰¤\mathrm{Î»}â‰¤1$. By hypothesis, there exists $xâˆˆC$ such that $yâˆˆT\left(x\right)$. Since ${x}_{0}$ is a star point of C, then $\mathrm{Î»}{x}_{0}+\left(1âˆ’\mathrm{Î»}\right)xâˆˆC$. But T is q-convex on S and C is invariant under T. Then we have

$\begin{array}{rcl}\mathrm{Î»}t+\left(1âˆ’\mathrm{Î»}\right)y& âˆˆ& \mathrm{Î»}T\left({x}_{0}\right)+\left(1âˆ’\mathrm{Î»}\right)T\left(x\right)\\ âŠ†& T\left(\mathrm{Î»}{x}_{0}+\left(1âˆ’\mathrm{Î»}\right)x\right)\\ âˆˆ& T\left(C\right)âŠ†C.\end{array}$

Hence S is invariant under T.â€ƒâ–¡

Now the main result of this section will be proved.

Theorem 3.3 Let X be a nonempty compact and star-shaped subset of a locally convex Hausdorff space V, C be the star-core of X and $\mathcal{A}$ be a commutating family of upper semicontinuous multimaps such that for each $Tâˆˆ\mathcal{A}$, $\mathcal{F}\left(T\right)âŠ†C$. Also, suppose that for each $xâˆˆX$, $T\left(x\right)$ is a nonempty, closed and star-shaped subset of X and T is p-convex for each $pâˆˆC$. Moreover, suppose that for each $xâˆˆ\mathcal{F}\left(T\right)$, we have the following condition:

$yâˆˆS\left(x\right)\phantom{\rule{1em}{0ex}}â‡’\phantom{\rule{1em}{0ex}}S\left(T\left(x\right)\right)âŠ†T\left(y\right),\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}S,Tâˆˆ\mathcal{A}.$
(1)

Then there is a point $\stackrel{Â¯}{x}âˆˆX$ such that for each $Tâˆˆ\mathcal{A}$, we have $\stackrel{Â¯}{x}âˆˆ\mathcal{F}\left(T\right)$.

Proof First, we try to confine the domain to a convex set. To do this, we apply Theorem 2.1, Lemma 3.2 and Zornâ€™s lemma to obtain a set $MâŠ†X$ such that M is minimal with respect to being nonempty, compact, star-shaped and invariant under each $Sâˆˆ\mathcal{A}$. We show that for each $Sâˆˆ\mathcal{A}$, $S\left(M\right)=M$. On the contrary, suppose $Tâˆˆ\mathcal{A}$ is such that $T\left(M\right)âŠ‚M$. Let $T\left(M\right)=N$. Since T is p-convex, then it preserves the star-shapeness. Hence N is star-shaped. Also, N is nonempty and compact. Suppose that $xâˆˆN=T\left(M\right)$. Then there exist some $yâˆˆM$ such that $x=T\left(y\right)$. Since $\mathcal{A}$ is commutative, for each $Sâˆˆ\mathcal{A}$, we have $S\left(x\right)=S\left(T\left(y\right)\right)=T\left(S\left(y\right)\right)âˆˆT\left(M\right)=N$. Hence, for each $Sâˆˆ\mathcal{A}$, $S\left(N\right)âŠ†N$. So, $NâŠ‚M$ is a nonempty, compact and star-shaped subset of X that is invariant under each $Sâˆˆ\mathcal{A}$. But this is a contradiction to the minimality of M, and so for each $Sâˆˆ\mathcal{A}$, we have $S\left(M\right)=M$. Let M be the star-core of X. In view of Lemma 2.2, M is a nonempty, compact and convex subset of X that is invariant under any $Sâˆˆ\mathcal{A}$. So, we can replace M by X and hence we may suppose that X is nonempty, compact and convex. For each $Tâˆˆ\mathcal{A}$, by hypothesis and Lemma 2.3, T is nonempty and closed in X. Suppose that ${x}_{1},{x}_{2}âˆˆ\mathcal{F}\left(T\right)$ and $\mathrm{Î»}âˆˆ\left[0,1\right]$. By p-convexity of T, we have

$\mathrm{Î»}{x}_{1}+\left(1âˆ’\mathrm{Î»}\right){x}_{2}âˆˆ\mathrm{Î»}T\left({x}_{1}\right)+\left(1âˆ’\mathrm{Î»}\right)T\left({x}_{2}\right)âŠ†T\left(\mathrm{Î»}{x}_{1}+\left(1âˆ’\mathrm{Î»}\right){x}_{2}\right).$

Hence $\mathcal{F}\left(T\right)$ is convex. We claim that for each $S,Tâˆˆ\mathcal{A}$, $\mathcal{F}\left(S\right)âˆ©\mathcal{F}\left(T\right)$ is nonempty, closed and convex in X. Consider the multimap $S:\mathcal{F}\left(T\right)âŸ¶{2}^{X}$ which is the restriction of S on $\mathcal{F}\left(T\right)$. We will show that for each $xâˆˆ\mathcal{F}\left(T\right)$, $S\left(x\right)âŠ†\mathcal{F}\left(T\right)$. Suppose there exists a point $yâˆˆS\left(x\right)$ such that $yâˆ‰T\left(y\right)$. According to commutativity in $\mathcal{A}$ and condition (1), we have

$yâˆˆS\left(x\right)âŠ†S\left(T\left(x\right)\right)=T\left(S\left(x\right)\right)âŠ†T\left(y\right),$

and this is a contradiction. Then the map $S:\mathcal{F}\left(T\right)âŸ¶{2}^{\mathcal{F}\left(T\right)}$ is an upper semicontinuous multimap such that for each $xâˆˆ\mathcal{F}\left(T\right)$, $F\left(x\right)$ is nonempty, closed and convex. Now by Lemma 2.3, there exists an element $\stackrel{Â¯}{x}âˆˆ\mathcal{F}\left(T\right)$ such that $\stackrel{Â¯}{x}âˆˆS\left(\stackrel{Â¯}{x}\right)$. Similarly, for each triple $\left\{S,T,W\right\}âŠ‚\mathcal{A}$, by considering the multimap $W:\mathcal{F}\left(T\right)âˆ©\mathcal{F}\left(S\right)âŸ¶{2}^{X}$, we claim that for each $xâˆˆ\mathcal{F}\left(T\right)âˆ©\mathcal{F}\left(S\right)$, $W\left(x\right)âŠ‚\mathcal{F}\left(T\right)âˆ©\mathcal{F}\left(S\right)$. To prove the claim, suppose that $yâˆˆW\left(x\right)$ such that $yâˆ‰T\left(y\right)$ or $yâˆ‰S\left(y\right)$. Let $yâˆ‰S\left(y\right)$. By commutativity of $\mathcal{A}$ and by equation (1), we have

$yâˆˆW\left(x\right)âŠ†yâˆˆW\left(T\left(x\right)\right)=T\left(W\left(x\right)\right)âŠ†T\left(y\right),$

which is a contradiction. So, the multimap $W:\mathcal{F}\left(T\right)âˆ©\mathcal{F}\left(S\right)âŸ¶{2}^{\mathcal{F}\left(T\right)âˆ©\mathcal{F}\left(S\right)}$ is upper semicontinuous, and for each x, $W\left(x\right)$ is nonempty, closed and convex. Therefore, there exists $\stackrel{Â¯}{x}âˆˆ\mathcal{F}\left(T\right)âˆ©\mathcal{F}\left(T\right)$ such that $\stackrel{Â¯}{x}âˆˆW\left(\stackrel{Â¯}{x}\right)$. By induction, we obtain a family $\left\{\mathcal{F}:Tâˆˆ\mathcal{A}\right\}$ in X which has the finite intersection property. We know that each â„± is nonempty and compact. So, there exists $\stackrel{Â¯}{x}âˆˆX$ such that $\stackrel{Â¯}{x}âˆˆ{â‹‚}_{Tâˆˆ\mathcal{A}}T$.â€ƒâ–¡

In the sequel, we see some important corollaries that can be implied as the results of the above theorem.

In Theorem 3.3, when $\mathcal{A}$ is a commutating family of single-valued maps, then the convexity assumption on $\mathcal{A}$ is actually the affine condition. Also, we can easily see that equation (1) automatically holds. So, we have the following results.

Corollary 3.4 (Markov-Kakutani)

Let X be a nonempty, compact and convex subset of a Hausdorff locally convex space and let $\mathcal{A}$ be a commutating family of affine mappings. Then there exists an element $\stackrel{Â¯}{x}âˆˆX$ such that $f\left(\stackrel{Â¯}{x}\right)=\stackrel{Â¯}{x}$ for each $fâˆˆ\mathcal{A}$.

Corollary 3.5 ([11])

Let K be a compact star-shaped subset of a locally convex space V. Suppose F is a commutative family of continuous affine mappings of K into itself. Then F has a common fixed point in K.

If we apply the technique introduced in the first part of proof in Theorem 3.3, we have the following extension of Lemma 2.3.

Corollary 3.6 Let X be a nonempty, compact and star-shaped subset of a Hausdorff locally convex space V and let $T:XâŸ¶{2}^{X}$ be an upper semicontinuous multimap such that for each $xâˆˆX$, $T\left(x\right)$ is nonempty, closed and convex. Then there exists a point $\stackrel{Â¯}{x}âˆˆX$ such that $\stackrel{Â¯}{x}âˆˆT\left(\stackrel{Â¯}{x}\right)$.

Also, if T is a single-valued map, we have the following extension of the Schauder fixed point theorem.

Corollary 3.7 Let X be a nonempty, compact and star-shaped subset of a Hausdorff locally convex space V. If $T:XâŸ¶X$ is a continuous map, then T has a fixed point.

3.2 Applications for the Cauchy problem

Suppose that $\left(X,\mathrm{Ï„}\right)$ is a Hausdorff locally convex space and $\left(Y,d\right)$ is a complete metric space. Take a set $AâŠ†X$, a mapping $C:AâŸ¶C\left(A\right)âŠ†Y$ and a mapping T defined at least on the set $AÃ—C\left(A\right)$ and having the range in X. We are interested in the existence of a point $xâˆˆA$ such that

$x=T\left(x,C\left(x\right)\right).$
(2)

Definition 3.8 The family $\left\{T\left(â‹\dots ,y\right):yâˆˆY\right\}$ is said to be equicontractive if there exists a $kâˆˆ\left[0,1\right)$ such that

${p}_{\mathrm{Î±}}\left(T\left({x}_{1},y\right)âˆ’T\left({x}_{2},y\right)\right)â‰¤k{p}_{\mathrm{Î±}}\left({x}_{1}âˆ’{x}_{2}\right)$

for each ${p}_{\mathrm{Î±}}âˆˆ{A}^{âˆ—}\left(\mathrm{Ï„}\right)$ and for all $\left({x}_{1},y\right)$, $\left({x}_{2},y\right)$ in the domain of T.

Theorem 3.9 Suppose that A is a compact star-shaped subset of a Hausdorff locally convex space X and $C:AâŸ¶Y$ is a continuous operator. Let

$T:AÃ—C\left(A\right)âŸ¶A$
(3)

be a continuous operator such that the family $\left\{T\left(â‹\dots ,y\right):yâˆˆC\left(A\right)\right\}$ is equicontractive. Then operator equation (2) admits a solution in A.

Proof Suppose $yâˆˆC\left(A\right)$ is arbitrary. Since the operator

$xâŸ¶T\left(x,y\right):AâŸ¶A$

is a contraction, there exists a unique point $x=F\left(y\right)âˆˆA$ that satisfies the operator equation

$T\left(F\left(y\right),y\right)=F\left(y\right).$

Now we will show that the mapping $yâŸ¶F\left(y\right):C\left(A\right)âŸ¶A$ is continuous. To do this, suppose that $\left({y}_{\mathrm{Î²}}\right)$ is a net in $C\left(A\right)$ such that ${y}_{\mathrm{Î²}}âŸ¶{y}_{0}$. Then for each ${p}_{\mathrm{Î±}}âˆˆ{A}^{âˆ—}\left(\mathrm{Ï„}\right)$, we have

$\begin{array}{rcl}{p}_{\mathrm{Î±}}\left(F\left({y}_{\mathrm{Î²}}\right)âˆ’F\left({y}_{0}\right)\right)& =& {p}_{\mathrm{Î±}}\left(T\left(F\left({y}_{\mathrm{Î²}}\right),{y}_{\mathrm{Î²}}\right)âˆ’T\left(F\left({y}_{0}\right),{y}_{0}\right)\right)\\ â‰¤& {p}_{\mathrm{Î±}}\left(T\left(F\left({y}_{\mathrm{Î²}}\right),{y}_{\mathrm{Î²}}\right)âˆ’T\left(F\left({y}_{0}\right),{y}_{\mathrm{Î²}}\right)\right)\\ +{p}_{\mathrm{Î±}}\left(T\left(F\left({y}_{0}\right),{y}_{\mathrm{Î²}}\right)âˆ’T\left(F\left({y}_{0}\right),{y}_{0}\right)\right)\\ â‰¤& k{p}_{\mathrm{Î±}}\left(F\left({y}_{\mathrm{Î²}}\right)âˆ’F\left({y}_{0}\right)\right)+{p}_{\mathrm{Î±}}\left(T\left(F\left({y}_{0}\right),{y}_{\mathrm{Î²}}\right)âˆ’T\left(F\left({y}_{0}\right),{y}_{0}\right)\right).\end{array}$

Therefore, we have

${p}_{\mathrm{Î±}}\left(F\left({y}_{\mathrm{Î²}}\right)âˆ’F\left({y}_{0}\right)\right)â‰¤{\left(1âˆ’k\right)}^{âˆ’1}{p}_{\mathrm{Î±}}\left(T\left(F\left({y}_{0}\right),{y}_{\mathrm{Î²}}\right)âˆ’T\left(F\left({y}_{0}\right),{y}_{0}\right)\right),$

and this proves the continuity of F. Now, the operator FoC is a self-mapping on A that is continuous and affine. Hence, by Corollary 3.7 there exists a point $\stackrel{Â¯}{x}âˆˆA$ such that $F\left(C\left(\stackrel{Â¯}{x}\right)\right)=\stackrel{Â¯}{x}$. But this means that

$T\left(\stackrel{Â¯}{x},C\left(\stackrel{Â¯}{x}\right)\right)=T\left(F\left(C\left(\stackrel{Â¯}{x}\right)\right),C\left(\stackrel{Â¯}{x}\right)\right)=F\left(C\left(\stackrel{Â¯}{x}\right)\right)=\stackrel{Â¯}{x}.$

â€ƒâ–¡

Before continuing, we replace equation (3) by the following:

$T\left(x,C\left(x\right)\right)âˆˆA,\phantom{\rule{1em}{0ex}}xâˆˆA.$
(4)

In the sequel, an application of the above theorem will be presented, but first we remind some preliminaries.

Let I be an interval of the real line of the form $\left[0,t\right)$, where t is a positive number or âˆž. Let X be a Banach space ${C}_{b}\left(I,\mathbb{R}\right)$ of all continuous and bounded functions $x:IâŸ¶\mathbb{R}$ endowed with a sup-norm $âˆ¥â‹\dots âˆ¥$. By $B\left(o,r\right)$, we will denote the ball in X centered at zero and having radius r (>0). Consider the following initial value problem (Cauchy problem):

$\left\{\begin{array}{c}{x}^{â€²}\left(t\right)=g\left(t\right)+F\left(t\right)x\left(t\right){\left[{âˆ«}_{0}^{t}D\left(tâˆ’s\right)x\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]}^{\mathrm{Î·}},\phantom{\rule{1em}{0ex}}tâˆˆI=\left[0,t\right)\hfill \\ x\left(0\right)=0,\hfill \end{array}$
(5)

in the real scalar case, where $\mathrm{Î·}>0$. We will show the existence of a solution $xâˆˆX$ for equation (5). Suppose that $g,F,Dâˆˆ{L}_{1}\left(I,\mathbb{R}\right)$ and let ${âˆ¥â‹\dots âˆ¥}_{1}$ stand for the usual ${L}_{1}$-(semi)-norm. Also, assume that ${âˆ¥Fâˆ¥}_{1}{âˆ¥Dâˆ¥}_{1}>0$. For any fixed ${K}_{1}âˆˆ\left(0,{âˆ¥Fâˆ¥}_{1}^{\frac{âˆ’1}{\mathrm{Î·}}}{âˆ¥Dâˆ¥}_{1}^{âˆ’1}\right)$, set $N={âˆ¥Fâˆ¥}_{1}^{\mathrm{Î·}}{âˆ¥Dâˆ¥}_{1}^{\mathrm{Î·}}$ and ${M}_{1}={K}_{1}\left(1âˆ’{âˆ¥Fâˆ¥}_{1}N\right)$. By these assumptions, we have the following theorem.

Theorem 3.10 Let the above conditions hold. If $gâˆˆ{L}_{1}\left(I,\mathbb{R}\right)$ and ${âˆ¥gâˆ¥}_{1}â‰¤{M}_{1}$, then equation (5) admits a (global) solution $xâˆˆB\left(0,{K}_{1}\right)$.

Proof If we define the operator $C:XâŸ¶X$ by

$\left(Cx\right)\left(t\right)={\left[{âˆ«}_{0}^{t}D\left(tâˆ’s\right)x\left(s\right)\right]}^{\mathrm{Î·}},$

then C is continuous. Suppose that $A=\left\{xâˆˆX:âˆ¥xâˆ¥â‰¤{K}_{1}\right\}$. A simple calculation shows that C maps $B\left(0,{K}_{1}\right)$ into the ball $B\left(0,N\right)$. Now we consider the operator $T:AÃ—B\left(0,N\right)âŸ¶X$ as defined by

$T\left(x,y\right)\left(t\right)={âˆ«}_{0}^{t}g\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{âˆ«}_{0}^{t}F\left(s\right)x\left(s\right)y\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}tâˆˆI.$

We will show that T has the condition of Theorem 3.9. For each $xâˆˆA$, the operator $xâŸ¶T\left(x,C\left(x\right)\right)$ is continuous and if $y=C\left(x\right)$, then for each $tâˆˆI$, we have

$|T\left(x,y\right)\left(t\right)|â‰¤{âˆ¥gâˆ¥}_{1}+{K}_{1}{âˆ¥Fâˆ¥}_{1}Nâ‰¤{K}_{1}\left(1âˆ’{âˆ¥Fâˆ¥}_{1}N\right)+{K}_{1}{âˆ¥Fâˆ¥}_{1}N={K}_{1}.$

Hence for each $xâˆˆA$, we have $T\left(x,C\left(x\right)\right)âˆˆA$. Also, for each ${x}_{1},{x}_{2}âˆˆA$, $yâˆˆC\left(A\right)$ and $tâˆˆI$, we have

$|T\left({x}_{1},y\right)\left(t\right)âˆ’T\left({x}_{2},y\right)\left(t\right)|â‰¤{âˆ«}_{0}^{t}|F\left(s\right)||{x}_{1}\left(s\right)âˆ’{x}_{2}\left(s\right)||y\left(s\right)|\phantom{\rule{0.2em}{0ex}}dsâ‰¤âˆ¥{x}_{1}âˆ’{x}_{2}âˆ¥,$

where $k={âˆ¥{F}_{1}âˆ¥}_{1}N<1$. This proves the equicontractivity condition and hence the proof is complete.â€ƒâ–¡

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Correspondence to Choonkil Park.

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The authors declare that they have no competing interests.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Goudarzi, H., Azadi Kenary, H. & Park, C. Common fixed point theorems for multimaps in locally convex spaces and some applications. Fixed Point Theory Appl 2013, 37 (2013). https://doi.org/10.1186/1687-1812-2013-37