Krasnosel’skii type fixed point theorems under weak topology features

Abstract

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let $M$ be a nonempty bounded closed convex subset of a Banach space $X$. Suppose that $A:M\to X$ and $B:X\to X$ are two weakly sequentially continuous mappings satisfying:

• $AM$ is relatively weakly compact;

• $B$ is a strict contraction;

• $(x=Bx+Ay,y\in M)\Rightarrow x\in M$.

Then $A+B$ has at least one fixed point in $M$.

This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.

Introduction

In 1958, M. A. Krasnosel’skii [1], [2] proved the following fixed point theorem which is an important supplement to both the Schauder fixed point theorem and the Banach contraction principle.

Theorem 1.1

Let $M$ be a nonempty bounded closed convex subset of a Banach space $X$ . If $A:M\to X$ and $B:M\to X$ are mappings satisfying:

• $A$ is compact and continuous;

• $B$ is a strict contraction mapping;

• $AM+BM\subset M$ ;

then $A+B$ has at least one fixed point in $M$ .

As is well known, Theorem 1.1 has a wide range of applications to nonlinear integral equations of mixed type. It has also been extensively used in differential and functional differential equations. Several attempts have been made in the literature to prove the analogue of Theorem 1.1 for the weak topology. In [3], Barroso established a version of Theorem 1.1 using the weak topology of a Banach space. His result requires the weak continuity and the weak compactness of $A$ while $B$ must be a linear operator satisfying the condition $\Vert B^p\Vert <1$ for some integer $p\ge 1$. The proof is modeled on the Schauder–Tychonoff fixed point theorem and uses the weak continuity of ${(I-B)}^{-1}$. In a more recent paper [4], Barroso and Teixeira established a fixed point theorem for the sum $A+B$ of a weakly sequentially continuous mapping $A$ and a weakly sequentially continuous strict contraction $B$. The result requires the weak compactness of ${(I-B)}^{-1}A$. This condition will be relaxed in the present paper by merely assuming the weak compactness of $A$. This is effected in part by means of the conjunction of the technique of measures of weak noncompactness with a generalization of the Schauder–Tychonoff fixed point principle owing to Arino, Gautier and Penot [5]. It is our object here to improve all of the aforementioned results by proving an analogue of Theorem 1.1 for the weak topology (see Theorem 2.1). This result can be thought as an extension of the well known Arino–Gautier–Penot fixed point theorem [5] and the Cain–Nashed fixed point theorem in a Banach space setting [6]. It is also an extension of Burton’s variant of Theorem 1.1 (see [7, Theorem 2]). Later on, our main result will be exploited to derive some fixed point theorems for the sum of a weakly compact and a nonexpansive mapping, which encompass a number of previously known results in the literature.

For the remainder of this section we gather some notations and preliminary facts. Let $X$ be a Banach space, let $\mathcal{B}\left(X\right)$ denote the collection of all nonempty bounded subsets of $X$ and $\mathcal{W}\left(X\right)$ the subset of $\mathcal{B}\left(X\right)$ consisting of all weakly compact subsets of $X$. Also, let $B_r$ denote the closed ball centered at 0 with radius $r$.

In our considerations the following definition will play an important role.

Definition 1.2 [8]

A function $\mu :\mathcal{B}\left(X\right)\to {\mathbb{R}}_{+}$ is said to be a measure of weak noncompactness if it satisfies the following conditions:

• The family $ker\left(\mu \right)=\{M\in \mathcal{B}\left(X\right):\mu \left(M\right)=0\}$ is nonempty and $ker\left(\mu \right)$ is contained in the set of relatively weakly compact sets of $X$.

• $M_1\subseteq M_2\Rightarrow \mu \left(M_1\right)\le \mu \left(M_2\right)$.

• $\mu \left(\overline{co}\left(M\right)\right)=\mu \left(M\right)$, where $\overline{co}\left(M\right)$ is the closed convex hull of $M$.

• $\mu (\lambda M_1+(1-\lambda )M_2)\le \lambda \mu \left(M_1\right)+(1-\lambda )\mu \left(M_2\right)$ for $\lambda \in [0,1]$.

• If ${\left(M_n\right)}_{n\ge 1}$ is a sequence of nonempty weakly closed subsets of $X$ with $M_1$ bounded and $M_1\supseteq M_2\supseteq \cdots \supseteq M_n\supseteq \cdots$ such that ${lim}_{n\to \infty }\mu \left(M_n\right)=0$, then $M_{\infty }≔{\bigcap }_{n=1}^{\infty }{M}_n$ is nonempty.

The family $ker\mu$ described in (1) is said to be the kernel of the measure of weak noncompactness $\mu$. Note that the intersection set $M_{\infty }$ from (5) belongs to $ker\mu$ since $\mu \left(M_{\infty }\right)\le \mu \left(M_n\right)$ for every $n$ and ${lim}_{n\to \infty }\mu \left(M_n\right)=0$. Also, It can be easily verified that the measure $\mu$ satisfies

$$\mu \left({\overline{M}}^w\right)=\mu \left(M\right)$$

where ${\overline{M}}^w$ is the weak closure of $M$.

The first important example of a measure of weak noncompactness has been defined by De Blasi [9] as follows :

$$w\left(M\right)=inf\{r>0:\text{ there exists }W\in \mathcal{W}\left(X\right)\text{ with }M\subseteq W+B_r\}\text{,}$$

for each $M\in \mathcal{B}\left(X\right)$.

This measure of weak noncompactness enjoys several other nice properties [9] such as the subadditivity

$$w(M_1+M_2)\le w\left(M_1\right)+w\left(M_2\right)\text{,}$$

and the homogeneity

$$w\left(\lambda M\right)=\left|\lambda \right|w\left(M\right)\text{.}$$

A particularly important additional property of the De Blasi measure of weak noncompactness is

$$w\left(M\right)=0\text{if and only if }M\text{ is relatively weakly compact .}$$

For further purposes, the following definition will also be needed.

Definition 1.3

Let $X$ be a Banach space. A mapping $B:X\to X$ is said to be nonexpansive (or sometimes called a contraction) if

$$\Vert Bx-By\Vert \le \Vert x-y\Vert$$

for all $x,y\in X$.