Abstract
Two new inertial-type extragradient methods are proposed to find a numerical common solution to the variational inequality problem involving a pseudomonotone and Lipschitz continuous operator, as well as the fixed point problem in real Hilbert spaces with a ρ-demicontractive mapping. These inertial-type iterative methods use self-adaptive step size rules that do not require previous knowledge of the Lipschitz constant. We also show that the proposed methods strongly converge to a solution of the variational inequality and fixed point problems under appropriate standard test conditions. Finally, we present several numerical examples to show the effectiveness and validation of the proposed methods.
Highlights
Assume that Y is a nonempty, closed, and convex subset of a real Hilbert space X with the inner product h·, ·i and the induced norm k · k
The main contribution of this study is to investigate the convergence analysis of the iterative schemes for solving variational inequality and fixed point problems in real Hilbert spaces
The reason and inspiration for investigating such a common solution problem is its potential applicability to mathematical models whose constraints can be stated as fixed point problems
Summary
Assume that Y is a nonempty, closed, and convex subset of a real Hilbert space X with the inner product h·, ·i and the induced norm k · k. The main contribution of this study is to investigate the convergence analysis of the iterative schemes for solving variational inequality and fixed point problems in real Hilbert spaces. The reason and inspiration for investigating such a common solution problem is its potential applicability to mathematical models whose constraints can be stated as fixed point problems. This is especially relevant in applications such as signal processing, composite minimization, optimum control, and image restoration; see, for example, [1,2,3,4,5]. Let us take a look at both of the problems highlighted by this research. We look at the classic variational inequality problem [6,7] which is expressed as follows: Find r ∗ ∈ Y such that =(r ∗ ), y − r ∗ ≥ 0, ∀ y ∈ Y
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