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Convergence results of multi-valued nonexpansive mappings in Banach spaces

Abstract

The purpose of this paper is to establish two new iteration schemes as follows: x n = α n x n 1 +(1 α n ) y n , y n T x n , n1, x n = β n u+ α n x n 1 +(1 α n β n ) y n , y n T x n , n1, for a multi-valued nonexpansive mapping T in a uniformly convex Banach space and prove that { x n } and { x n } converge strongly to a fixed point of T under some suitable conditions, respectively. Moreover, a gap in Sahu (Nonlinear Anal. 37:401-407, 1999) is found and revised.

MSC:47H04, 47H10, 47J25.

1 Introduction

Let E be a Banach space, K be a nonempty, closed, and convex subset of E. We denote the family of all subsets of E by 2 E , the family of nonempty, closed, and bounded subsets of E by CB(E), the family of nonempty compact subsets of E by C(E), and the family of nonempty, compact, and convex subsets of E by CC(E). Let H(,) be Hausdorff metric on CB(E), i.e.,

H(A,B)=max { sup x A d ( x , B ) , sup x B d ( x , A ) } A,BCB(E),
(1.1)

where d(x,B)=inf{xy:yB}.

A multi-valued mapping T:KCB(K) is called nonexpansive (respectively, contractive), if for any x,yK, we have

H ( T x , T y ) x y ( respectively , H ( T x , T y ) k x y  for some  k [ 0 , 1 ) ) .
(1.2)

A point x is called a fixed point of T if xTx. In this paper, F(T) stands for the fixed point set of the mapping T.

In 1968, Markin [1] firstly established the nonexpansive multi-valued convergence results in Hilbert space. Banach’s contraction principle was extended to a multi-valued contraction by Nadler [2] in 1969.

In 1974, a breakthrough was achieved by TC Lim using Edelstein’s method of asymptotic centers [3].

Theorem 1.1 (Lim [4])

Let K be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space E and T:KC(K) a multi-valued nonexpansive mapping. Then T has a fixed point.

In 1990, Kirk and Massa [5] obtained another important result for the multi-valued nonexpansive mappings.

Theorem 1.2 (Kirk and Massa [5])

Let K be a nonempty, bounded, closed, and convex subset of a Banach space E and T:KCC(K) a multi-valued nonexpansive mapping. Suppose that the asymptotic center in K of each bounded sequence of E is nonempty and compact. Then T has a fixed point.

In 1999, Sahu [6] obtained the strong convergence theorems of the nonexpansive type and non-self multi-valued mappings for the following iteration process:

x n = t n x 0 +(1 t n ) u n ,n0,
(1.3)

where u n T x n , x 0 K, t n (0,1), and lim n t n =0. Unfortunately, a gap exists in the proof of Theorem 1 in p.405 of [6]; there are the following inequalities:

x n u n , J ( x n v ) = x n v + v u n , J ( x n v ) x n v 2 u n v x n v x n v 2 x n v 2 = 0 .
(1.4)

Clearly, if the above inequality holds, then the inequality u n v x n v must be assumed, for all vF(T), but this inequality does not hold generally, based on the definition of the Hausdorff metric H(,) on CB(E), for all vT(v).

Remark 1.1 To revise the gap we have found in Theorem 1 of [6], in this paper, we change the fixed point set of T (F(T):={xK,xTx}) into F(T):={xK,x=Tx}. The above problem is solved easily. Indeed,

x n u n , J ( x n v ) = x n v + v u n , J ( x n v ) x n v 2 u n v x n v x n v 2 H ( T x n , T v ) x n v x n v 2 x n v 2 = 0 .
(1.5)

In 2001, Xu [7] extended Theorem 1.2 to a multi-valued nonexpansive nonself-mapping and obtained the fixed theorems. Some recent fixed point results for multi-valued nonexpansive mappings can be found in [813] and the references therein.

Motivated by Sahu [6] and the above results, we propose two new iteration processes (1.6) and (1.7) and we study them in this paper. Let K be a nonempty, bounded, closed, and convex subset of E, and u and x 0 be fixed elements of K. We have

x n = α n x n 1 +(1 α n ) y n , y n T x n ,n1,
(1.6)
x n = β n u+ α n x n 1 +(1 α n β n ) y n , y n T x n ,n1,
(1.7)

where α n , β n (0,1) satisfy certain conditions, and we prove some strongly convergence theorems for the multi-valued nonexpansive mappings in Banach spaces. The results presented in this paper mainly extend and improve the corresponding results of Sahu [6] on the iteration algorithms.

2 Preliminaries

Let E be a real uniformly convex Banach space and let J denote the normalized duality mapping from E to 2 E defined by

J(x)= { f E , x , f = x f , x = f } ,xE,
(2.1)

where E denotes the dual space of E and , denotes the generalized duality pair. It is well known that if E is smooth or if E is strictly convex, then J is single-valued.

Recall that the norm of Banach space E is said to be Gâteaux differentiable (or E is said to be smooth), if the limit

lim t 0 x + t y x t
(2.2)

exists for each x, y on the unit sphere S(E) of E. Moreover, if for each y in S(E) the limit defined by (2.2) is uniformly attained for x in S(E), we say that the norm of E is uniformly Gâteaux differentiable. It is also well known that if E has a uniformly Gâteaux differentiable norm, then the duality mapping J is norm-to-weak star uniformly continuous on each bounded subset of E.

A Banach space E is called uniformly convex, if for each ϵ>0 there is a δ>0 such that for x,yE with x,y1, and xyϵ, x+y2(1δ) holds. The modulus of convexity of E is defined by

δ E (ϵ)=inf { 1 1 2 ( x + y ) : x , y 1 , x y ϵ } ,

for all ϵ[0,2]. E is said to be uniformly convex if δ E (0)=0, and δ(ϵ)>0 for all 0<ϵ2.

Throughout this paper, we write x n x (respectively, x n x) to indicate that the sequence x n weakly (respectively, weak*) converges to x, and as usual x n x will symbolize strong convergence. In order to show our main results, the following definitions and lemmas are needed.

Let LIM be a continuous linear functional on l satisfying LIM=1=LIM(1). Then we know that LIM is a mean on N if and only if

inf{ a n ;nN}LIM(a)sup{ a n ;nN}

for every a=( a 1 , a 2 ,) l . According to time and circumstances, we use LIM n ( a n ) instead of LIM(a). A mean LIM on N is called a Banach limit if

LIM n ( a n )= LIM n ( a n + 1 )

for every a=( a 1 , a 2 ,) l . Furthermore, we have the following results [14, 15].

Lemma 2.1 ([[14], Lemma 1])

Let C be a nonempty, closed, and convex subset of a Banach space E with uniformly Gâteaux differentiable norm. Let { x n } be a bounded sequence of E and let LIM n be a mean LIM on N and zC. Then

LIM n x n z 2 =min LIM n x n y 2

if and only if

LIM n y z , J ( x n z ) 0

for all yC.

Definition 2.1 A multi-valued mapping T:KCB(K) is said to satisfy Condition I if there is a nondecreasing function f:[0,)[0,) with f(0)=0, f(r)>0 for r(0,) such that

d(x,Tx)f ( d ( x , F ( T ) ) ) for all xK.

An example of mappings that satisfy Condition I can be found in reference [16].

3 Strong convergence theorems

Theorem 3.1 Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty closed convex subset of E, T:KC(K) be a multi-valued nonexpansive mapping. Assume that F(T) and T(y)={y} for each yF(T). Let { x n } be an implicit Mann type iteration defined by (1.6), where α n (0,1) and lim n α n =0, then the sequence { x n } converges strongly to a fixed point of T.

Proof Firstly, let pF(T), n1, we show that lim n x n p exists and { x n } is bounded. Using (2.1), we obtain

x n p 2 = α n x n 1 + ( 1 α n ) y n p , j ( x n p ) = ( 1 α n ) y n p , j ( x n p ) + α n x n 1 p , j ( x n p ) ( 1 α n ) y n p x n p + α n x n 1 p x n p ( 1 α n ) H ( T x n , T p ) x n p + α n x n 1 p x n p ( 1 α n ) x n p 2 + α n x n 1 p x n p ,

so

x n p 2 x n 1 p x n p.
(3.1)

If x n p=0, then lim n x n p=0 apparently holds.

If x n p>0, from (3.1) we have

x n p x n 1 p,

we find that { x n p} is a decreasing sequence, so

lim n x n p

exists. Hence { x n } is bounded, and so is { y n }.

Secondly, we show that lim n x n y n =0.

It follows from (1.6) and from { x n } and y n T x n being bounded, that there exists a real number M 1 >0 such that

x n 1 y n M 1 , x n y n = α n x n 1 y n α n M 1 .

Since lim n α n =0, we have

x n y n 0,as n,

therefore

lim n x n y n =0.
(3.2)

We define ϕ:K[0,) by ϕ(x):= LIM n x n x 2 for each xK, since E is uniformly convex (hence reflexive) and ϕ is continuous, convex, and ϕ(x) as x, ϕ attains its infimum over K (see [17]). Let

M:= { z K : ϕ ( z ) = min x K ϕ ( x ) } .

Then M is nonempty, bounded, closed, and convex (see [[15], Theorem 1.3.11]).

Next, we show that M is singleton.

Since M and { x n } are bounded, there exists R>0 such that M, { x n } B R (0) for all n0. Then, by inequality (3.12) of [18] for y 1 , y 2 M we have

x n 1 2 ( y 1 + y 2 ) 2 1 2 x n y 1 2 + 1 2 x n y 2 2 1 4 g ( y 1 y 2 ) .

If y 1 y 2 , we have

r:=ϕ ( 1 2 ( y 1 + y 2 ) ) 1 2 ϕ( y 1 )+ 1 2 ϕ( y 2 ) 1 4 g ( y 1 y 2 ) <r.

This is a contradiction. Therefore, M has a unique element.

Now we show that zM is the fixed point of T. Since T is compact valued, we have some w n Tz for zM such that

y n w n =d( y n ,Tz)H(T x n ,Tz) x n z;

indeed, w n = P T z ( y n ), since Tz is a compact set, and we take lim n w n =wTz.

It follows from the above inequality that

LIM n x n w LIM n x n y n + LIM n y n w n + LIM n w n w = LIM n y n w n LIM n x n z ,

and hence wM, it follows that M is singleton, so w=zTz.

Therefore, zF(T) and so Tz={z} by the assumption, and F(T) is nonempty.

On the other hand, for vF(T), we have

x n y n , J ( x n v ) = x n v + v y n , J ( x n v ) = x n v 2 y n v , J ( x n v ) x n v 2 y n v x n v x n v 2 H ( T x n , T v ) x n v x n v 2 x n v x n v = 0 .

It follows from (1.6) that

0 x n y n , J ( x n v ) = α n x n 1 y n , J ( x n v )
(3.3)

and

x n x n 1 , J ( x n v ) = x n y n , J ( x n v ) + y n x n 1 , J ( x n v ) x n y n x n v + y n x n 1 , J ( x n v ) .
(3.4)

Hence, from (3.2), (3.3), and (3.4), we obtain

LIM n x n x n 1 , J ( x n v ) =0,vF(T).
(3.5)

Finally, we show that LIM n x n 1 v,J( x n v)0. Let s(0,1), then by Lemma 1 of [19] we get

x n z s ( x n 1 z ) 2 x n z 2 + 2 s ( x n 1 z ) , j ( x n z s ( x n 1 z ) ) = x n z 2 2 s x n 1 z , j ( x n z ) 2 s x n 1 z , j ( x n z s ( x n 1 z ) ) j ( x n z ) .

Let ε>0 be arbitrary, then since J is norm-to-weak uniformly continuous on bounded subsets of E, there exists δ>0 such that for all s(0,δ), we have

LIM n x n 1 z , j ( x n z ) 1 2 s [ LIM n x n z 2 LIM n x n z s ( x n 1 z ) 2 ] + ε < ε ,

since zM, and it is a minimizer of ϕ over K. Now, since ε is arbitrary this implies that

LIM n x n 1 z , j ( x n z ) 0.
(3.6)

Combining inequalities (3.5) and (3.6) we get

LIM n x n z , j ( x n z ) = LIM n x n z 2 0.

Therefore, there is a subsequence { x n j } of { x n } which converges strongly to z.

Since lim n x n p exists pF(T), we get

lim n x n z=0.

The proof is completed. □

Theorem 3.2 Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K a nonempty, closed, and convex subset of E, T:KCB(K) be a multi-valued nonexpansive mapping that satisfies Condition I, assume F(T) and T(y)={y} for each yF(T). Let { x n } be an implicit Mann type iteration defined by (1.6), where α n (0,1) and lim n α n =0, then the sequence { x n } converges strongly to a fixed point of T.

Proof It follows from the proof of Theorem 3.1 that

lim n d( x n ,T x n )=0.

Then Condition I implies that

lim n d ( x n , F ( T ) ) =0.

The remainder of the proof is the same as Theorem 2.4 of [20]. □

Theorem 3.3 Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K be a nonempty, closed, and convex subset of E, T:KC(K) be a multi-valued nonexpansive mapping. Assume that F(T) and T(y)={y} for each yF(T). Let { x n } be the modified implicit Mann type iteration defined by (1.7), where α n , β n (0,1) and lim n α n β n = lim n β n =0. Then the sequence { x n } converges strongly to a fixed point of T.

Proof Firstly, let λ n = β n 1 α n and z n = λ n u+(1 λ n ) y n , then

x n = α n x n 1 +(1 α n ) z n .
(3.7)

Let pF(T), n1; we show that { x n } is bounded. Using (2.1), we obtain

x n p 2 = α n x n 1 + ( 1 α n ) z n p , j ( x n p ) = ( 1 α n ) z n p , j ( x n p ) + α n x n 1 p , j ( x n p ) ( 1 α n ) λ n ( u p ) , j ( x n p ) + ( 1 α n ) ( 1 λ n ) y n p , j ( x n p ) + α n x n 1 p x n p ( 1 α n ) ( λ n u p x n p + ( 1 λ n ) H ( T x n , T p ) x n p ) + α n x n 1 p x n p β n u p x n p + ( 1 α n β n ) x n p 2 + α n x n 1 p x n p ,

so

x n p 2 β n α n + β n up x n p + α n α n + β n x n 1 p x n p .
(3.8)

If x n p=0, then { x n } is apparently bounded.

If x n p>0, from (3.7) we have

x n p β n α n + β n u p + α n α n + β n x n 1 p max { x n 1 p , u p } max { x 0 p , u p } .

Thus, { x n } is bounded.

Secondly, we show that lim n x n y n =0.

It follows from (1.7) and since { x n } and y n T x n are bounded, there exists a real number M 2 >0 such that

x n 1 z n M 2 , x n z n = α n x n 1 z n α n M 2 .

Since lim n α n =0, we have

x n z n 0as n,

therefore

lim n x n z n =0.

Since

x n y n x n z n + z n y n = x n z n + β n 1 α n u y n ,

then

lim n x n y n =0.
(3.9)

We define ϕ:K[0,) by ϕ(x):= LIM n x n x 2 for each xK. Since E uniformly convex (hence reflexive) and ϕ is continuous, convex and ϕ(x) as x, ϕ attains its infimum over K (see, e.g., [17]). Let

M:= { z K : ϕ ( z ) = min x K ϕ ( x ) } .

Then M is nonempty, and it is also bounded, closed, and convex (see [[15], Theorem 1.3.11]).

In the same way as the proof of Theorem 3.1, we see that M is also singleton, and it follows from (3.9) that z is the fixed point of T.

By Lemma 2.1, we have

LIM n x z , J ( x n z ) 0,

for all xK. In particular, we have

LIM n u z , J ( x n z ) 0.
(3.10)

It follows from the proof of (3.6) of Theorem 3.1 that (3.6) also holds here. Thus

x n z 2 = β n ( u z ) + α n ( x n 1 z ) + ( 1 α n β n ) ( y n z ) , J ( x n z ) β n u z , J ( x n z ) + α n x n 1 z , J ( x n z ) + ( 1 α n β n ) { H ( T x n , T z ) x n z } β n u z , J ( x n z ) + α n x n 1 z , J ( x n z ) + ( 1 α n β n ) x n z 2 .

Hence

x n z 2 β n α n + β n u z , J ( x n z ) + α n α n + β n x n 1 z , J ( x n z ) u z , J ( x n z ) + α n α n + β n ( x n 1 z , J ( x n z ) u z , J ( x n z ) ) u z , J ( x n z ) + α n α n + β n M 0 ,

where M 0 is a constant such that | x n 1 u,J( x n z)| M 0 . Thus

LIM n x n z 2 LIM n u z , J ( x n z ) + LIM n α n β n α n β n + 1 M 0 0 .

Therefore, there is a subsequence { x n j } of { x n } which converges strongly to z. To complete the proof, suppose there is another subsequence { x n k } of { x n } which converges strongly to  z ¯ . Then z ¯ is a fixed point of T by (3.9) and z ¯ K because K is closed. It then follows from (3.10) that

z ¯ z , J ( z ¯ z ) = z ¯ z 2 0.

This proves the strong convergence of { x n } to zF(T).

The proof is completed. □

4 Numerical examples

Now, we give two real numerical examples in which the conditions satisfy the ones of Theorem 3.1 and Theorem 3.3.

Example 4.1 Let E=R, K=[1,1], T(x)= 1 5 x, xE, whichis nonexpansive, α n = 1 n for every nN. Then { x n } is the sequence generated by

x n = 1 n x n 1 + ( 1 1 n ) 1 5 x n ,
(4.1)

and x n 0 as n, where 0 is the fixed point of T.

Example 4.2 Let E=R, K=[1,1], T(x)= 1 2 x, xE, which is nonexpansive, α n = 1 n 100 , β n = 1 n 50 for every nN. Then { x n } is the sequence generated by

x n = 1 n 50 + 1 n 100 x n 1 + ( 1 1 n 100 1 n 50 ) 1 2 x n ,
(4.2)

and x n 0 as n, where 0 is the fixed point of T.

Remark 4.1 We can prove Example 4.1 and Example 4.2 by Theorem 3.1 and Theorem 3.3, respectively, and we show two numerical experiments (Figure 1 and Figure 2) which can explain that the sequence { x n } strongly converges to 0.

Figure 1
figure 1

x(1)=1 , iteration steps n=100 .

Figure 2
figure 2

x(1)=0.2 , iteration steps n=1,000 .

Remark 4.2 From the above numerical examples, we can see that the convergence results in this paper are important. The main reason is that the convergence of the two iteration schemes in this paper can easily be implemented by the software of Matlab 7.0, so they can be applied for numerical calculations in practical problems.

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Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (No. K5051370004), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2014JQ1020), the National Science Foundation of China (No. 61202178 and No. 61373174), the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11426167).

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He, H., Liu, S. & Chen, R. Convergence results of multi-valued nonexpansive mappings in Banach spaces. J Inequal Appl 2014, 483 (2014). https://doi.org/10.1186/1029-242X-2014-483

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Keywords

  • multi-valued nonexpansive mappings
  • fixed point
  • strong convergence
  • implicit iteration