The Split Common Fixed Point Problem for Nonexpans

The Split Common Fixed Point Problem for Nonexpansive

Mapping and Quasi--Strict Pseudo-contraction Mapping

in Banach SpaceXuejiao Zi, Lili Yang, Shaorong YangDepartment of General Education, The College of Arts and

Sciences•KunmingAbstract: In this article, we obtain a solution of split common fixed point problem for

nonexpansive mapping and quasi--strict pseudo-contraction mapping by proposing an iterative

scheme in Banach space. The strong convergence theorem of the sequence produced by our

iterative scheme is ds: Split common fixed point problem; Nonexpansive mapping; Quasi--strict pseudo-contraction mapping; Banach spaceDOI: 10.47297/taposatWSP2633-456910.202102021. Introduction

The split feasibility problem (SFP) was proposed for modeling inverse problems [1]. It is

considered the problem as follows: find a pointand . (1.1)

Where is a bounded linear operator. and be nonempty closed and convex

subsets of finite-dimensional Hilbert spaces and , respectively.

and are replaced by the fixed point sets of nonlinear operators

, respectively. (The fixed point sets of

S and

Y are represented by

Then (1.1) is called the split common fixed problem (SCFPP): find a point with and

If

and

and)

. (1.2)Many conclusions of weak and strong convergence theorems on SFP in Hilbert space have

been obtained by continuous research of authors, see for instance[5-9]. Last several years, some

studies of SFP have attempted to generalize from Hilbert space to Banach space, for example, in

[10-12], authors introduced different iterative schemes to find solution of SFP and gained the strong

convergence theorems with the condition of semi-compactness on the operators. In 2018, Ma et

al.[13] introduced iterative scheme converges strongly to a solution of following split feasibility

and fixed point problem absence the assumption of semi-compactness on the operator:

find a point such that and . (1.3)Funded project: This work was supported by the Scientific Reserch Foundation of College of Arts and Sciences

Yunnan Normal University (No. 18KJYYB011).42

Vol.2 No.2 2021of

Where and be two real Banach space and

Q be a nonempty closed and convex subset

be a quasi-- nonexpansive mapping.

In this article, enlightened by the research of [13], our main purpose is to use shrinking

projective method to construct an iterative scheme and find a solution of the split common fixed

point problem (1.2) in Banach space. The strong convergence theorem of the sequence produced

by the constructed scheme is proved without the assumption of semi-compactness on the

operators.2. PreliminariesThroughout this article, we suppose

U be a real Banach space.

is called strictly convex if

modulus of convexity of

U

is a function asfor all

is the dual space of

with . The

for anyany . . We say

U is uniformly convex if

as , and for

The modulus of smoothness of

U

is a function

If as , we say U is uniformly smooth. Suppose that be a invariant

real number and

for

is known as -uniformly smooth if there is a constant and

. Every -uniformly smooth Banach space is uniformly smooth is

. denotes the generalized duality pairing between

U and

is defined byThe normalized mapping

L is uniformly norm-to-norm continuous on each bounded subset of

U when

U is uniformly

smooth Banach spaces.

Let

U is a strictly convex, smooth and reflexive Banach space. Consider the following

function defined in [14,15]:It is evident from (2.4) thatand

43

Theory and Practice of Science and TechnologyandLet

C be a closed and convex subset of

U, In

[14], the mapping

described the generalized projection, is defined byThat is

.Lemma 2.1([15]) Suppose be a nonempty closed convex subset of strictly convex, smooth

and reflective Banach space

U . Then we have:, where

, which is

is the unique solution to the minimization problem

Definition 2.2 Let

C be a nonempty closed convex subset of smooth Banach spac

U a

mapping is called:(i) Nonexpansive, if

(ii) quasi-with

Definition 2.3 A mapping

and as

Lemma 2.4([16]) Let

Banach space

U. If

and

-strict pseudo-contraction mapping, if and there is a constant

.is known as closed if for any sequence

, then .and either or is bounded, then .

with

be two sequences of a uniformly convex and smooth

Lemma 2.5([17]) Let

C be a nonempty, closed and convex subset of strictly convex, smooth,

and reflective Banach space

U let and . Then

Lemma 2.6([18]) Suppose

U be a 2-uniformly smooth Banach space and the best smoothness

constants . Then we have that:3. Main ResultsTheorem 3.1 Suppose be a 2-uniformly convex and 2-uniformly smooth real Banach

space, its the best smoothness constant Let be a strictly convex, smooth and reflective

Banach space. and be the normalized mapping of and , respectly. Let be a

closed quasi--strict pseudo-contraction mapping and be a nonexpansive

mapping and Let be a bounded linear operator with its adjoint

Let and be a sequence generated by:44

Vol.2 No.2 2021with the following conditions hold:Proof. The proof is shown in following five 1:We prove that

For

for any

and are closed and convex for any

Let is closed and convex for is closed and convex based on

，due toWe get

Since

we obtainis closed and convex of

is closed and convex. Let is closed and convex for

.