Abstract
AbstractThe purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces.We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.
Highlights
Let H be a Hilbert space with the inner product ⟨·, ·⟩ and the induced norm || · ||
The purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces
We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces
Summary
Let H be a Hilbert space with the inner product ⟨·, ·⟩ and the induced norm || · ||. Let C be a nonempty, closed and convex subset of H, Θ : C × C → R be a nonlinear bifunction, h : C → H be a nonlinear mapping, and φ : C → R ∪ {+∞} be a proper convex lower semicontinuous function. The authors in [38] proposed the following iterative algorithm to solve the problem of finding a common element in SEP(Θ1, Θ2) and a fixed point of a nonspreading multi-valued mapping S : C → K(C). Rizvi [39] introduced the following algorithm for solving (11), as well as fixed point problems for a nonexpansive mapping S in real Hilbert spaces. Motivated by the above works, it is our aim in this paper to study the SGMEP (4) and introduce a new iterative algorithm for approximating a common solution of (4) and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by modifying the accelerated Mann algorithm (8), combined with a modified viscosity approximation method of (6) to obtain a new faster iterative algorithm for finding a common solution of (4) and a fixed point of nonspreading mappings in real Hilbert spaces. Our result is interesting and compliments many recent results previously obtained in this direction in the literature
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