Abstract
In this paper, we consider the split feasibility problem in Banach spaces. By applying the shrinking projection method, we propose an iterative method for solving this problem. It is shown that the algorithm under two different choices of the stepsizes is strongly convergent to a solution of the problem.
Highlights
In this paper, we consider the split feasibility problem [1]
E split feasibility problem was first treated in Euclidean spaces and recently was extended to more general framework including Hilbert spaces and Banach spaces
Our aim of this paper is to continue the above works by constructing new iterative methods in Banach spaces
Summary
We consider the split feasibility problem [1]. It is very useful in dealing with problems arising from various applied disciplines (see, e.g., [2,3,4,5,6]). E split feasibility problem was first treated in Euclidean spaces and recently was extended to more general framework including Hilbert spaces and Banach spaces. Where rn > 0 is a properly chosen stepsize, A∗ is the conjugate of A, I is the identity operator, and PC, PQ denote the metric projections onto the respective sets. Wang [10] recently proposed a new method, which generates a sequence as rnJX∗ JX X: 〈zn −. Our aim of this paper is to continue the above works by constructing new iterative methods in Banach spaces. By applying ideas (6) and (7), we introduce a new iterative algorithm and propose two different choices of the stepsize. We show that if the spaces involved are smooth and uniformly convex, the algorithm converges strongly under both choices of the stepsize. It is worth noting that one choice of the stepsize does not need any a priori knowledge of the operator norm
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