Abstract
The purpose of this paper is first to introduce and study the general split equality variational inclusion problems and the general split equality optimization problems in the setting of infinite-dimensional Hilbert spaces and then propose a new simultaneous iterative algorithm. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm converging strongly to a solution for these two kinds of problems are proved. As special cases, we shall utilize our results to study the split feasibility problems, the split equality equilibrium problems, and the split optimization problems. The results presented in the paper not only extend and improve the corresponding recent results announced by many authors, but they also provide an affirmative answer to an open question raised by Moudafi in his recent work.
Highlights
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H, respectively
Assuming that the (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∗, 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 (GSEVIP) ( . ) and (GSEOP) ( . ), in Sections and, we propose a new simultaneous type iterative algorithm
Summary
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H , respectively. Ax∗ = By∗, where H , H , and H are three real Hilbert spaces, A : H → H and B : H → H are two linear and bounded operators, hi : H → R and gi : H → R are two countable families of proper, convex, and lower semicontinuous functions. The so-called split equality equilibrium problem with respective to h, g, and D (SEEP(h, g, D)) is to find x∗ ∈ D, y∗ ∈ D such that h x∗, u ≥ , ∀u ∈ D, g y∗, v ≥ , ∀v ∈ D and Ax∗ = By∗, where A, B : D → D are two linear and bounded operators. Let H and H be two real Hilbert spaces, A : H → H be a linear and bounded operators, h : H → R and g : H → R be two proper convex and lower semicontinuous functions. The results presented in the paper extend and improve the corresponding results announced by Moudafi et al [ , , ], Eslamian and Latif [ ], Chen et al [ ], Censor et al [ , – , ], Chuang [ ], Naraghirad [ ], Chang and Wang [ ], Ansari and Rehan [ ], and some others
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