Abstract
In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems.
Highlights
We present two algorithms to find the solutions to variational inclusion problems in Hilbert spaces
The variational inclusion problems have always been a topic discussed by a large number of scholars
Our work in this paper is based on the work conducted by Tan et al combined with the multistep inertial method and the Krasnosel’skiı–Mann algorithm for solving the variational inclusion problems in a real Hilbert space
Summary
In a real Hilbert space H with inner product h·, ·i and induced norm k · k, we assume that G : H → 2 H is a set-valued mapping while F : H → H is a single-valued mapping. We consider the following variational inclusion problem: find an element x ∗ ∈ H such that 0 ∈ Fx ∗ + Gx ∗. Proposed a modified forward–backward splitting algorithm (Algorithm 1) about null points of maximal monotone mappings. This algorithm is weakly convergent under some conditions. Inertial shrinking projection method (Algorithm 4) by combining the two algorithms (Algorithms 1 and 2) with two classes of hybrid projection methods to solve the variational inclusion problem in Hilbert spaces, as follows: Algorithm 3: Inertial hybrid projection algorithm. They proved these two algorithms are strongly convergent under certain conditions
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