In this article we introduce an inertial iterative scheme with self-adaptive step size for finding a common solution of split common null point problem for a finite family of maximal monotone operators and fixed point...
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In this article we introduce an inertial iterative scheme with self-adaptive step size for finding a common solution of split common null point problem for a finite family of maximal monotone operators and fixed point problem for a finite family of multivalued demicontractive mappings between a Banach space and Hilbert space. Strong convergence result is obtained for the proposed algorithm. The self-adaptive step size ensures no requirement for a prior knowledge or estimate of the norm of the operator. The inertial term introduced in the algorithm is efficient, it helps to avoid imposing some strong conditions usually used for inertial-type algorithms by many authors. We give some applications of our results to game theory, split equilibrium and minimum-norm problems. Numerical experiment is also presented to demonstrate the efficiency of our proposed method as well as comparing with other existing method in the literature. Our results improve and generalize many well known results in this direction in the literature.
We investigate an inertial algorithm of gradient type in connection with the minimization of a non-convex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradien...
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We investigate an inertial algorithm of gradient type in connection with the minimization of a non-convex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradient method. We prove some abstract convergence results which applied to our numerical scheme allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. Further, we obtain convergence rates for the generated sequences and the objective function values formulated in terms of the Lojasiewicz exponent of a regularization of the objective function. Finally, some numerical experiments are presented in order to compare our numerical scheme and some algorithms well known in the literature.
In this work, we establish the closedness and convexity of the set of fixed points of equally continuous and asymptotically demicontractive mapping in the intermediate sense. We proposed an inertial hybrid projection ...
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In this work, we establish the closedness and convexity of the set of fixed points of equally continuous and asymptotically demicontractive mapping in the intermediate sense. We proposed an inertial hybrid projection technique for determining an approximate common solution to three significant problems. The first is the system of generalized mixed equilibrium problems with relaxed eta-zeta$$ \eta -\zeta $$ monotone mappings, the second is the problem of fixed points of a countable family of equally continuous and asymptotically demicontractive mappings in the intermediate sense, and the third is of determining a point in a null space of a countable family of inverse strongly monotone mappings in Hilbert space. Based on these problems, we formulate a theorem and establish its strong convergence to their common solution. Additionally, we studied the applications of our algorithm to variational inequality problems and convex optimization problems. Finally, we numerically demonstrate the efficiency and robustness of our scheme. Several results available in the literature can be obtained as special cases of our result.
In this paper, we introduce a self-adaptive inertial iterative algorithm for solving a split null point problem with maximal monotone operators and a common fixed point problem of a finite family of multivaluted demic...
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In this paper, we introduce a self-adaptive inertial iterative algorithm for solving a split null point problem with maximal monotone operators and a common fixed point problem of a finite family of multivaluted demicontractive mappings and an infinite family of strict pseudo-contractive mappings between a Banach space and a Hilbert space. Next, we prove a strong convergence result for the proposed algorithm under some mild conditions on the control parameters without a priori estimate of the norm of the linear operator involved. Then, we give a numerical example to illustrate the effectiveness of the algorithm.
In this paper, we study the problem of finding the solution of monotone variational inclusion problem (MVIP) with constraint of common fixed point problem (CFPP) of strict pseudocontractions. We propose a new viscosit...
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In this paper, we study the problem of finding the solution of monotone variational inclusion problem (MVIP) with constraint of common fixed point problem (CFPP) of strict pseudocontractions. We propose a new viscosity method, which combines the inertial technique with self-adaptive step size strategy for approximating the solution of the problem in the framework of Hilbert spaces. Unlike several of the existing results in the literature, our proposed method does not require the co-coerciveness and Lipschitz continuity assumptions of the associated single-valued operator. Also, our method does not involve any linesearch technique which could be time-consuming, rather we employ a self-adaptive step size technique that generates a nonmonotonic sequence of step sizes. Moreover, we prove strong convergence result for our algorithm under some mild conditions and apply our result to study other optimization problems. We present several numerical experiments to demonstrate the computational advantage of our proposed method over the existing methods in the literature. Our result complements several of the existing results in the current literature in this direction.
In this paper, we introduce a new inertial-viscosity approximation method for solving a split generalized equilibrium problem and common fixed point problem in real Hilbert spaces. The algorithm is designed such that ...
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In this paper, we introduce a new inertial-viscosity approximation method for solving a split generalized equilibrium problem and common fixed point problem in real Hilbert spaces. The algorithm is designed such that its convergence does not require the norm of the bounded linear operator underlying the split equilibrium problem. Moreover, a strong convergence result is proved under mild conditions in real Hilbert spaces. Furthermore, we give some numerical examples to show the efficiency and accuracy of the proposed method and we also compare the performance of our algorithm with other related methods in the literature.
In this paper, we propose a new modified inertial simultaneous algorithm of common fixed point prob-lems for a finite family of demicontractive mappings and obtain some strong convergence results in real Hilbert space...
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In this paper, we propose a new modified inertial simultaneous algorithm of common fixed point prob-lems for a finite family of demicontractive mappings and obtain some strong convergence results in real Hilbert spaces. Meanwhile, we also give a numerical example to demonstrate the efficiency of our proposed algorithm. Our results improve and extend some corresponding known results.
Let H be a real Hilbert space. Let F:H -> 2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{u...
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Let H be a real Hilbert space. Let F:H -> 2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F:H\rightarrow 2<^>{H}$\end{document} and K:H -> 2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K:H\rightarrow 2<^>{H}$\end{document} be two maximal monotone and bounded operators. Suppose the Hammerstein inclusion 0 is an element of u+KFu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\in u+KFu$\end{document} has a solution. We construct an inertial-type algorithm and show its strong convergence to a solution of the inclusion. As far as we know, this is the first inertial-type algorithm for Hammerstein inclusions in Hilbert spaces. We also give numerical examples to compare the new algorithm with some existing ones in the literature.
In this paper, we study an inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive maps in a uniformly convex and uniformly smooth real Banach space. We prove a stro...
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In this paper, we present hybrid inertial proximal algorithms for the split variational inclusion problems in Hilbert spaces, and provide convergence theorems for the proposed algorithms. In fact, an inertial type alg...
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In this paper, we present hybrid inertial proximal algorithms for the split variational inclusion problems in Hilbert spaces, and provide convergence theorems for the proposed algorithms. In fact, an inertial type algorithm was proposed as an acceleration process. As application, we study split minimization problem, split feasibility problem, relaxed split feasibility problem and linear inverse problem in real Hilbert spaces. Finally, numerical results are given for our main results.
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