Abstract
We study the regularization method for solving the variational inclusion problem of the sum of two monotone operators in Hilbert spaces. The strong convergence theorem is then established under some relaxed conditions which mainly improves and recovers that of Qin et al. (Fixed Point Theory Appl. 2014:75, 2014). We also apply our main result to the convex minimization problem, the fixed point problem and the variational inequality problem. Finally we provide numerical examples for supporting the main result.
Highlights
Let C be a nonempty subset of a real Hilbert space H
We study the problem of finding xsuch that
Some typical problems arising in various branches of science, applied sciences, economics, and engineering such as machine learning, image restoration, and signal recovery can be viewed as this form
Summary
Let C be a nonempty subset of a real Hilbert space H. The inverse of B, denoted by B– , is defined by x ∈ B– y if and only if y ∈ Bx. We study the problem of finding xsuch that. ∈ Ax + Bx, where A : C → H is an operator and B : D(B) ⊂ H → H is a set-valued operator. This problem is called the variational inclusion problem. It includes, as special cases, the variational inequality problem, the split feasibility problem, the linear inverse problem, and the following convex minimization problem: min F(x) + G(x), x∈H where F : H → R is a smooth convex function, and G : H → R is a non-smooth convex function.
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