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# On the Krasnoselskii-type fixed point theorems for the sum of continuous and asymptotically nonexpansive mappings in Banach spaces

- Areerat Arunchai
^{1}and - Somyot Plubtieng
^{1}Email author

**2011**:28

https://doi.org/10.1186/1029-242X-2011-28

© Arunchai and Plubtieng; licensee Springer. 2011

**Received:**30 January 2011**Accepted:**29 July 2011**Published:**29 July 2011

## Abstract

In this article, we prove some results concerning the Krasnoselskii theorem on fixed points for the sum *A* + *B* of a weakly-strongly continuous mapping and an asymptotically nonexpansive mapping in Banach spaces. Our results encompass a number of previously known generalizations of the theorem.

## Keywords

- Krasnoselskii's fixed point theorem
- asymptotically nonexpansive mapping
- weakly-strongly continuous mapping
- uniformly asymptotically regular
- measure of weak noncompactness

## 1 Introduction

As is well known, Krasnoselskii's fixed point theorem has a wide range of applications to nonlinear integral equations of mixed type (see [1]). It has also been extensively employed to address differential and functional differential equations. His theorem actually combines both the Banach contraction principle and the Schauder fixed point theorem, and is useful in establishing existence theorems for perturbed operator equations. Since then, there have appeared a large number of articles contributing generalizations or modifications of the Krasnoselskii fixed point theorem and their applications (see [2]-[21]).

The study of asymptotically nonexpansive mappings concerning the existence of fixed points have become attractive to the authors working in nonlinear analysis. Goebel and Kirk [22] introduced the concept of asymptotically nonexpansive mappings in Banach spaces and proved a theorem on the existence of fixed points for such mappings in uniformly convex Banach spaces. In 1971, Cain and Nashed [23] generalized to locally convex spaces a well known fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The class of asymptotically nonexpansive mappings includes properly the class of nonexpansive mappings as well as the class of contraction mappings. Recently, Vijayaraju [21] proved by using the same method some results concerning the existence of fixed points for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in locally convex spaces. Very recently, Agarwal et al. [1] proved some existence theorems of a fixed point for the sum of a weakly-strongly continuous mapping and a nonexpansive mapping on a Banach space and under Krasnoselskii-, Leray Schauder-, and Furi-Pera-type conditions.

Motivated and inspired by Agarwal et al. [1] and Vijayaraju [21], in this article we will prove some new generalized forms of the Krasnoselskii theorem on fixed points for the sum *A* + *B* of a weakly-strongly continuous mapping and an asymptotically nonexpansive mapping in Banach spaces. These results encompass a number of previously known generalizations of the theorem.

## 2 Preliminaries

Let *M* be a nonempty subset of a Banach space *X* and *T* : *M* → *X* be a mapping. We say that *T* is *weakly-strongly continuous* if for each sequence {*x*_{
n
}} in *M* which converges weakly to *x* in *M*, the sequence {*Tx*_{
n
}} converges strongly to *Tx*. The mapping *T* is *nonexpansive* if ||*Tx* -*Ty*|| ≤ ||*x* - *y*|| for all *x*, *y* ∈ *M*, and *T* is *asymptotically nonexpansive* (see [22]) if there exists a sequence {*k*_{
n
}} with *k*_{
n
} ≥ 1 for all *n* and lim_{n→∞}*k*_{
n
}= 1 such that ||*T*^{
n
}*x* - *T*^{
n
}*y*|| ≤ *k*_{
n
}||*x* - *y*|| for all *n* ≥ 1 and *x, y* ∈ *M*.

**Definition 2.1**. [21] If

*B*and

*A*map

*M*into

*X*, then

*B*is called a

*uniformly asymptotically regular*with respect to

*A*if, for each

*ε >*0 there exists

*n*

_{0}∈ ℕ, such that

for all *n* ≥ *n*_{0} and all *x* ∈ *M*.

*X*be a Banach space, Ω(

*X*) is the collection of all nonempty bounded subsets of

*X*, and is the subset of Ω(

*X*) consisting of all weak compact subsets of

*X*. Let

*B*

_{ r }denote the closed ball in

*X*countered at 0 with radius

*r >*0. In [24], De Blasi introduced the following mapping

*ω*: Ω(

*X*) → [0, ∞) defined by

for all *M* ∈ Ω(*X*). For completeness, we recall some properties of *ω*(·) needed below (for the proofs we refer the reader to [24]).

**Lemma 2.2**. [24] *Let M*_{1} *and M*_{2} ∈ Ω(*X*), *then we have*

(*i*) *If M*_{1} ⊆ *M*_{2}, *then ω*(*M*_{1}) ≤ *ω*(*M*_{2}).

(*ii*) *ω*(*M*_{1}) = 0 *if and only if M*_{1} *is relatively weakly compact*.

(*iii*)
, *where*
*is the weak closure of M*_{1}.

(*iv*) *ω*(*λM*_{1}) = |*λ*|*ω*(*M*_{1}) *for all λ* ∈ ℝ.

(*v*) *ω*(*co*(*M*_{1})) = *w*(*M*_{1}).

(*vi*) *ω*(*M*_{1} + *M*_{2}) ≤ *ω*(*M*_{1}) + *ω*(*M*_{2}).

(*vii*) *If* (*M*_{
n
})_{n≥1}*is a decreasing sequence of nonempty, bounded and weakly closed subsets of X with* lim_{n→∞}*ω*(*M*_{
n
}) = 0, *then*
*and*
, *i*.*e*.,
*is relatively weakly compact*.

Throughout this article, a measure of weak noncompactness will be a mapping *ψ* : Ω(*X*) → [0, *∞*) which satisfies the assumptions (i)-(vii) cited in Lemma 2.2.

**Definition 2.3**. [25] Let *M* be a closed subset of *X* and *I*, *T* : *M* → *M* be two mappings. A mapping *T* is said to be *demiclosed* at the zero, if for each sequence {*x*_{
n
}} in *M*, the conditions *x*_{
n
} → *x*_{0} weakly and *Tx*_{
n
} → 0 strongly imply *Tx*_{0} = 0.

**Lemma 2.4**. [26]-[29] *Let X be a uniformly convex Banach space, M be a nonempty closed convex subset of X, and let T* : *M* → *M be an asymptotically nonexpansive mapping with F*(*T*) ≠ ∅. *Then I* - *T is demiclosed at zero, i.e., for each sequence* {*x*_{
n
}} *in M, if* {*x*_{
n
}} *converges weakly to q* ∈ *M and* {(*I* - *T*)*x*_{
n
}} *converges strongly to* 0, *then* (*I* - *T*)*q* = 0.

**Definition 2.5**. [1, 13] Let *X* be a Banach space and let *ψ* be a measure of weak noncompactness on *X*. A mapping *B* : *D*(*B*) ⊆ *X* → *X* is said to be *ψ-contractive* if it maps bounded sets into bounded sets and there is a *β* ∈ [0, 1) such that *ψ*(*B*(*S*)) ≤ *βψ*(*S*) for all bounded sets *S* ⊆ *D*(*B*). The mapping *B* : *D*(*B*) ⊆ *X* → *X* is said to be *ψ-condensing* if it maps bounded sets into bounded sets and *ψ*(*B*(*S*)) *< ψ*(*S*) whenever *S* is a bounded subset of *D*(*B*) such that *ψ*(*S*) *>* 0.

Let
be a nonlinear operator from
into *X*. In the next section, we will use the following two conditions:

*x*

_{ n })

_{n∈ℕ}is a weakly convergent sequence in , then has a strongly convergent subsequence in

*X*.

*x*

_{ n })

_{n∈ℕ}is a weakly convergent sequence in , then has a weakly convergent subsequence in

*X*.

*Remark*2.6. 1. Operators satisfying or are not necessarily weakly continuous (see [12, 19, 30]).

- 2.
- 3.
A mapping satisfies if and only if it maps relatively weakly compact sets into relatively weakly compact ones (use the Eberlein-

*Š*mulian theorem [31]). - 4.
A mapping satisfies if and only if it maps relatively weakly compact sets into relatively compact ones.

- 5.

The following fixed point theorems are crucial for our purposes.

**Lemma 2.7**. [12] *Let M be a nonempty closed bounded convex subset of a Banach space X. Suppose that A* : *M* → *X and B* : *X* → *X satisfying:*

(*i*) *A is continuous, AM is relatively weakly compact and A satisfies*
,

(*ii*) *B is a strict contraction satisfying*
,

(*iii*) *Ax* + *By* ∈ *M for all x*, *y* ∈ *M*.

*Then, there is an x* ∈ *M such that Ax* + *Bx* = *x*.

**Lemma 2.8**. [20] *Let M be a nonempty closed bounded convex subset of a Banach space X. Suppose that A* : *M* → *X and B* : *M → X are sequentially weakly continuous such that:*

(*i*) *AM is relatively weakly compact*,

(*ii*) *B is a strict contraction*,

(*iii*) *Ax* + *By* ∈ *M for all x*, *y* ∈ *M*.

*Then, there is an x* ∈ *M such that Ax* + *Bx* = *x*.

**Lemma 2.9**. [1]

*Let X be a Banach space and let ψ be measure of weak noncompactness on X. Let Q and C be closed, bounded, convex subset of X with Q*⊆

*C. In addition, let U be a weakly open subset of Q with*0 ∈

*U, and*

*a weakly sequentially continuous and ψ-condensing mapping. Then either*

*here ∂*_{
Q
}*U is the weak boundary of U in Q*.

*In addition, suppose that B verifies*
. *Then for each bounded subset S of X, we have ψ*(*BS*) ≤ *kψ*(*S*);

*here,ψ is the De Blasi measure of weak noncompactness*.

**Lemma 2.11**. [15, 32] *Let X be a Banach space with C* ⊆ *X closed and convex. Assume U is a relatively open subset of C with* 0 ∈ *U,* *bounded and* *a condensing mapping. Then, either F has a fixed point in* *or there is a point u* ∈ *∂U and λ* ∈ (0,1) *with u* = *λF*(*u*)*; here*
*and ∂U denote the closure of U in C and the boundary of U in C, respectively*.

**Lemma 2.12**. [15, 32] *Let X be a Banach space and Q a closed convex bounded subset of X with* 0 ∈ *Q*. *In addition, assume F* : *Q* → *X a condensing mapping with if*
*is a sequence in ∂Q*× [0, 1] *converging to* (*x*, *λ*) *with X* = *λF*(*x*) *and* 0 *< λ <* 1, *then λ*_{
j
}*F* (*x*_{
j
}) ∈ *Q for j sufficiently large, holding. Then F has a fixed point*.

## 3 Main results

Now, we are ready to state and prove the main result of this section.

**Theorem 3.1**. *Let M be a nonempty bounded closed convex subset of a Banach space X. Let A* : *M* → *X and B* : *M* → *M satisfy the following:*

(*i*) *A is weakly-strongly continuous, and AM is relatively weakly compact*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞) *satisfying*
,

(*iii*) *if* (*x*_{
n
}) *is a sequence of M such that* ((*I* - *B*)*x*_{
n
}) *is weakly convergent, then the sequence* (*x*_{
n
}) *has a weakly convergent subsequence*,

(*iv*) *I* - *B is demiclosed*,

(*v*) *B*^{
n
}*x* + *Ay* ∈ *M for all x*, *y* ∈ *M and n* = 1, 2,...,

(*vi*) *B is uniformly asymptotically regular with respect to A*.

*Then, there is an x* ∈ *M such that Ax* + *Bx* = *x*.

*a*

_{ n }

*B*

^{ n }is contraction on

*M*. Therefore, by Lemma 2.7, there is an

*x*

_{ n }∈

*M*such that

*a*

_{ n }→ 1 as

*n*→ ∞ and

*M*is bounded and

*B*

^{ n }

*x*+

*Ay*∈

*M*for all

*x*,

*y*∈

*M*. Since

*B*is uniformly asymptotically regular with respect to

*A*, it follows that

*AM*is weakly compact and passing eventually to a subsequence, we may assume that {

*Ax*

_{ n }} converges weakly to some

*y*∈

*M*. By (3.7), we have

By hypothesis (*iii*), the sequence {*x*_{
n
}} has a subsequence
which converges weakly to some *x* ∈ *M*. Since *A* is weakly-strongly continuous,
converges strongly to *Ax*.

Hence, by the demiclosedness of *I* - *B*, we have *Ax* + *Bx* = *x*.

To complete the proof, it remains to consider the case 0 ∉ *M*. In such a case, let us fix any element *x*_{0} ∈ *M* and let *M*_{0} = {*x* - *x*_{0}, *x* ∈ *M* }. Define two mappings *A*_{0} : *M*_{0} → *X* and *B*_{0} : *M*_{0} → *M* by
and
, for *x* ∈ *M*. By the result of the first case for *A*_{0} and *B*_{0}, we have an *x* ∈ *M* such that *A*_{0}(*x* - *x*_{0}) + *B*_{0}(*x* - *x*_{0}) = *x* - *x*_{0}. Hence *Ax* + *Bx* = *x*. □

**Corollary 3.2**. *Let M be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let A* : *M* → *X and B* : *M* → *M satisfy the following:*

(*i*) *A is weakly-strongly continuous*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞),

(*iii*) *B*^{
n
}*x* + *Ay* ∈ *M for all x*, *y* ∈ *M, and n* = 1, 2,...,

(*iv*) *B is uniformly asymptotically regular with respect to A*.

*Then, there is an x* ∈ *M such that Ax* + *Bx* = *x*.

Our next result is the following:

**Theorem 3.3**. *Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A* : *M* → *X and B* : *M* → *M are two weakly sequentially continuous mappings that satisfy the following:*

(*i*) *AM is relatively weakly compact*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞),

(*iii*) *if* (*x*_{
n
}) *is a sequence of M such that* ((*I* - *B*)*x*_{
n
}) *is weakly convergent, then the sequence* (*x*_{
n
}) *has a weakly convergent subsequence*,

(*iv*) *B*^{
n
}*x* + *Ay* ∈ *M for all x*, *y* ∈ *M, and n* = 1, 2,...,

(*v*) *B is uniformly asymptotically regular with respect to A*.

*Then, there is an x* ∈ *M such that Ax* + *Bx* = *x*.

*Proof*. Without loss of generality, we may assume that 0 ∈

*M*. Let for all

*n*∈ ℕ. By hypothesis (

*iv*), we have

*AM*is weakly compact and passing eventually to a subsequence, we may assume that {

*Ax*

_{ n }} converges weakly to some

*y*∈

*M*. Hence, by (3.16)

By hypothesis (*iii*), the sequence {*x*_{
n
}} has a subsequence
which converges weakly to some *x* ∈ *M*. Since *A* and *B* are weakly sequentially continuous,
converges weakly to *Ax*, and
converges weakly to *Bx*. Hence, *Ax* + *Bx* = *x*. □

**Theorem 3.4**. *Let Q and C be closed bounded convex subset of a Banach space X with Q* ⊆ *C. In addition, let U be a weakly open subset of Q with* 0 ∈ *U*,
*and B* : *X* → *X are two weakly sequentially continuous mappings satisfying the following:*

(*i*)
*is a relatively weakly compact*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞),

(*iii*) *if* (*x*_{
n
}) *is a sequence of M such that* ((*I* - *B*)*x*_{
n
}) *is weakly convergent, then the sequence* (*x*_{
n
}) *has a weakly convergent subsequence*,

(*iv*) *B*^{
n
}*x* + *Ay* ∈ *C for all*
, *and n* = 1, 2,...,

(*v*) *B is uniformly asymptotically regular with respect to A*.

*here, ∂*_{
Q
}*U is the weak boundary of U in Q*.

*Proof*. Let for all

*n*∈ ℕ. We first show that the mapping

*F*

_{ n }=

*a*

_{ n }

*A*+

*a*

_{ n }

*B*

^{ n }is

*ψ*-contractive with constant

*a*

_{ n }. To see that, let

*S*be a bounded subset of . Using the homogeneity and the subadditivity of the De Blasi measure of weak noncompactness, we obtain

*F*

_{ n }is

*ψ*-contractive with constant

*a*

_{ n }. Moreover, taking into account that 0 ∈

*U*and using assumption (

*iv*), we infer that

*F*

_{ n }map into C. Next, we suppose that (3.19) does not occur, and

*F*

_{ n }does not have a fixed point on

*∂*

_{ Q }

*U*(otherwise we are finished since (3.18) occurs). If there exists a

*u*∈

*∂*

_{ Q }

*U*, and

*λ*∈ (0, 1) with

*u*=

*λF*

_{ n }

*u*then

*u*=

*λa*

_{ n }

*Au*+

*λa*

_{ n }

*B*

^{ n }

*u*. It is impossible since

*λa*

_{ n }∈ (0, 1). By Lemma 2.9, there exists such that

*AM*is weakly compact and passing eventually to a subsequence, we may assume that {

*Ax*

_{ n }} converges weakly to some . Thus, we have

By hypothesis (*iii*), the sequence {*x*_{
n
}} has a subsequence
which converges weakly to some
. Since *A* and *B* are weakly sequentially continuous,
converges weakly to *Ax*, and
converges weakly to *Bx*. Hence, *Ax* + *Bx* = *x*. □

**Theorem 3.5**. *Let U be a bounded open convex set in a Banach space X with* 0 ∈ *U. Suppose*
*and B* : *X* → *X are continuous mappings satisfying the following:*

(*i*)
*is compact, and A is weakly-strongly continuous*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞), *and I* - *B is demiclosed*,

(*iii*) *if* (*x*_{
n
}) *is a sequence of*
*such that* ((*I* - *B*)*x*_{
n
}) *is weakly convergent, then the sequence* (*x*_{
n
}) *has a weakly convergent subsequence*,

(*iv*) *B is uniformly asymptotically regular with respect to A*.

*Proof*. Suppose (3.27) does not occur and let for all

*n*∈ ℕ. The mapping

*F*

_{ n }:=

*a*

_{ n }

*A*+

*a*

_{ n }

*B*

^{ n }is the sum of a compact and a strict contraction. This implies that

*F*

_{ n }is a condensing mapping (see [13]). By Lemma 2.11, we deduce that there is an such that

*AM*is weakly compact and passing eventually to a subsequence, we may assume that {

*Ax*

_{ n }} converges weakly to some . This implies that

By hypothesis (*iii*), the sequence {*x*_{
n
}} has a subsequence
which converges weakly to some
. Since *A* is weakly-strongly continuous,
converges strongly to *Ax*.

By the demiclosedness of *I* - *B*, we have *Ax* + *Bx* = *x*. □

**Corollary 3.6**. *Let U be a bounded open convex set in a uniformly convex Banach space X with* 0 ∈ *U. Suppose*
*and B* : *X* → *X are continuous mappings satisfying the following*.

(*i*)
*is compact, and A is weakly-strongly continuous*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞),

(*iii*) *B is uniformly asymptotically regular with respect to A*.

**Theorem 3.7**. *Let Q be a closed convex bounded set in a Banach space X with* 0 ∈ *Q. Suppose A* : *Q* → *X and B* : *X* → *X are continuous mappings satisfying the following:*

(*i*) *A*(*Q*) *is compact, and A is weakly-strongly continuous*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞), *and I* - *B is demiclosed*,

(*iii*) *if* (*x*_{
n
}) *is a sequence of*
*such that* ((*I* - *B*)*x*_{
n
}) *is weakly convergent, then the sequence* (*x*_{
n
}) *has a weakly convergent subsequence*,

(*iv*) *if*
*is a sequence of ∂Q* × [0, 1] *converging to* (*x*, *λ*) *with X* = *λAx* + *λB*^{
n
}*x and* 0 ≤ *λ* < 1, *then λ*_{
j
}*Ax*_{
j
} + *λ*_{
j
}*B*^{
n
}*x*_{
j
} ∈ *Q for j sufficiently large*,

(*v*) *B is uniformly asymptotically regular with respect to A*.

*Then, A* + *B has a fixed point in Q*.

*Proof*. We first define

*F*

_{ n }:=

*a*

_{ n }

*A*+

*a*

_{ n }

*B*

^{ n }, where for all

*n*∈ ℕ. Since

*F*

_{ n }is the sum of a compact mapping and a strict contraction mapping, it follows that

*F*

_{ n }is a condensing mapping. For any let fixed

*n*, we have is a sequence of

*∂Q*× [0, 1] converging to (

*y*,

*λ*) with

*y*=

*λF*

_{ n }(

*y*) and 0 ≤

*λ <*1. Then

*y*=

*a*

_{ n }

*λAy*+

*a*

_{ n }

*λB*

^{ n }

*y*. From assumption (

*iv*), it follows that

*a*

_{ n }

*λ*

_{ j }

*Ay*

_{ j }+

*a*

_{ n }

*λ*

_{ j }

*B*

^{ n }

*y*

_{ j }∈

*Q*for

*j*sufficiently large. Applying Lemma 2.12 to

*F*

_{ n }, we deduce that there is an

*x*

_{ n }∈

*Q*such that

By hypothesis (*iii*), the sequence {*x*_{
n
}} has a subsequence
which converges weakly to some *x* ∈ *Q*. Since *A* is weakly-strongly continuous,
converges strongly to *Ax*.

Hence, by the demiclosedness of *I* - *B*, we have *Ax* + *Bx* = *x*. □

**Corollary 3.8**. *Let Q be a closed convex bounded set in a uniformly convex Banach space X with* 0 ∈ *Q. Suppose A* : *Q* → *X and B* : *X* → *X are continuous mappings satisfying the following:*

(*i*) *A*(*Q*) *is compact and A is weakly-strongly continuous*,

(*ii*) *B is an asymptotically nonexpansive mapping with a sequence* (*k*_{
n
}) ⊂ [1, ∞),

(*iii*) *if*
*is a sequence of ∂Q* × [0, 1] *converging to* (*x*, *λ*) *with X* = *λAx* + *λB*^{
n
}*x and* 0 ≤ *λ* < 1, *then λ*_{
j
}*Ax*_{
j
} + *λ*_{
j
}*B*^{
n
}*x*_{
j
} ∈ *Q for j sufficiently large*,

(*iv*) *B is uniformly asymptotically regular with respect to A*.

*Then, A* + *B has a fixed point in Q*.

## Declarations

### Acknowledgements

The authors would like to thank the referee for the insightful comments and suggestions. The first author would like to thanks The Thailand Research Fund for financial support and the second author is also supported by the Royal Golden Jubilee Program under Grant PHD/0282/2550, Thailand. Moreover, the second author the Thailand Research Fund for financial support under Grant BRG5280016.

## Authors’ Affiliations

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