Abstract
The purpose of this paper is to introduce and study the general split equality problem and general split equality fixed point problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converges strongly to a solution of the general split equality fixed point problem and the general split equality problem for quasi-nonexpansive mappings in Hilbert spaces. As an application, we shall utilize our results to study the null point problem of maximal monotone operators, the split feasibility problem, and the equality equilibrium problem. The results presented in the paper extend and improve the corresponding results announced by Moudafi et al. (Nonlinear Anal. 79:117-121, 2013; Trans. Math. Program. Appl. 1:1-11, 2013), Eslamian and Latif (Abstr. Appl. Anal. 2013:805104, 2013) and Chen et al. (Fixed Point Theory Appl. 2014:35, 2014), Censor and Elfving (Numer. Algorithms 8:221-239, 1994), Censor and Segal (J. Convex Anal. 16:587-600, 2009) and some others.
Highlights
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H, respectively
Assuming that the split feasibility problem (SFP) is consistent, it is not hard to see that x∗ ∈ C solves SFP if and only if it solves the fixed point equation x∗ = PC I – γ A∗(I – PQ)A x∗, Chang and Agarwal Journal of Inequalities and Applications 2014, 2014:367 http://www.journalofinequalitiesandapplications.com/content/2014/1/367 where PC and PQ are the metric projection from H onto C and from H onto Q, respectively, γ > is a positive constant and A∗ is the adjoint of A
For solving the GSEFP ( . ) and general split equality problem: (GSEP) ( . ), in Sections and, we propose an algorithm for finding the solutions of the general split equality fixed point problem and general split equality problem in a Hilbert space
Summary
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H , respectively. Moudafi [ ] introduced the following split equality problem (SEP): to find x ∈ C, y ∈ Q such that Ax = By, where A : H → H and B : H → H are two bounded linear operators. In [ ], Moudafi and Al-Shemas introduced the following split equality fixed point problem: find x ∈ C := F(S), y ∈ Q := F(T) such that Ax = By, where S : H → H and T : H → H are two firmly quasi-nonexpansive mappings, F(S) and F(T) denote the fixed point sets of S and T, respectively. Eslamian and Latif [ ] and Chen et al [ ] introduced and studied some kinds of general split feasibility problem and split equality problem in real Hilbert spaces, and under suitable conditions some strong convergence theorems are proved. In Section we utilize our results to study the split feasibility problem, the null point problem of maximal monotone operators, and the equality equilibrium problem
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