Strong convergence theorems by hybrid methods for the split common null point problem in Banach spaces

Wataru Takahashi,Jen-Chih Yao

doi:10.1186/s13663-015-0324-3

Abstract

AbstractIn this paper, we consider the split common null point problem in Banach spaces. Then using the hybrid method and the shrinking projection method in mathematical programming, we prove strong convergence theorems for finding a solution of the split common null point problem in Banach spaces.

Highlights

Let H and H be two real Hilbert spaces
Defining U = A∗(I – PQ)A in the split feasibility problem, we see that U : H → H is an inverse strongly monotone operator [ ], where A∗ is the adjoint operator of A and PQ is the metric projection of H onto Q
If D ∩ A– Q is nonempty, z ∈ D ∩ A– Q is equivalent to z = PD I – λA∗(I – PQ)A z, ( . )

Summary

Introduction

Let H and H be two real Hilbert spaces. Let D and Q be nonempty, closed, and convex subsets of H and H , respectively. Lemma ([ ]) Let E be a smooth, strictly convex, and reflexive Banach space. Theorem ([ ]) Let E be a uniformly convex and smooth Banach space and let J be the duality mapping of E into E∗. Let E be a uniformly convex Banach space with a Gâteaux differentiable norm and let A be a maximal monotone operator of E into E∗. Let E be a uniformly convex and smooth Banach space E and let Jr be the metric resolvent of A for r >. For a sequence {Cn} of nonempty, closed, and convex subsets of a Banach space E, define s-LinCn and w-LsnCn as follows: x ∈ s-LinCn if and only if there exists {xn} ⊂ E such that {xn} converges strongly to x and xn ∈ Cn for all n ∈ N. It is easy to show that if {Cn} is nonincreasing with respect to inclusion, {Cn} converges to

Cn in the sense of

From the monotonicity of B we see that

Then since