A notion os 2-compactibility is introduced for noncommuting mappings in Menger space to obtain fi... more A notion os 2-compactibility is introduced for noncommuting mappings in Menger space to obtain fixed point results. Further a result in random normed space and a result on best approximation is obtained as application of the above notion.
Journal of Applied Mathematics and Computing, 2021
Monotone inclusion problems are crucial to solve engineering problems and problems arising in dif... more Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.
The present paper deals with the convergence of two modified Mann type of iteration schemes for a... more The present paper deals with the convergence of two modified Mann type of iteration schemes for a single and a finite family of mappings to the fixed and common fixed point, respectively, of a single and a finite family of quasi-nonexpansive mappings on a uniformly convex Banach space. An example is added in support of our main result. The results obtained generalize the earlier results of Rhoades (J Math Anal Appl 56:741–750, [6]), Som et al. (Proc Nat Acad Sci (India) 70(A)(II):185–189, [8]), and others in turn.
The objective of this work is to design a new iterative method based on Armijo's type-modifie... more The objective of this work is to design a new iterative method based on Armijo's type-modified extragradient method for solving the inclusion problem (A+B)-1 (0), where A is a maximal monotone vector field and B is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the rate of convergence. A convergence theorem is established for the proposed extragradient method, which significantly improves existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field's monotonicity can be removed.
Numerical Functional Analysis and Optimization, 2021
Abstract A large number of nonlinear and optimization problems can be reduced to altering point p... more Abstract A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems.
Journal of Applied and Numerical Optimization, 2020
In this paper, we develop S-iterative techniques in terms of exponential mappings for finding fix... more In this paper, we develop S-iterative techniques in terms of exponential mappings for finding fixed points of nonexpansive mappings on Hadamard manifolds. To demonstrate numerical applicability, we show our proposed methods are faster than the existing results in the literature. For the applications, we focus on the common singularities of set-valued monotone vector fields and convex feasibility problems.
This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for ap... more This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.
In this paper, we introduce and study the generalized η-D-gap function for an equilibrium problem... more In this paper, we introduce and study the generalized η-D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized η-D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized η-D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples.
In this paper, the existence of fixed point of expansive type mappings is studied in modular func... more In this paper, the existence of fixed point of expansive type mappings is studied in modular function spaces.
ABSTRACT. In this paper, we design an iterative algorithm that converges strongly to common fixed... more ABSTRACT. In this paper, we design an iterative algorithm that converges strongly to common fixed points of a sequence of asymptotically nonexpansive mappings in a reflexive Banach space with weakly continuous duality mapping. Our results show that the uniform ...
It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877-898], that the proximal point... more It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877-898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with non-summable error sequence is also discussed.
It is well known that the rate of convergence of S-iteration process introduced by Agarwal
et al... more It is well known that the rate of convergence of S-iteration process introduced by Agarwal
et al. [J. Nonlinear Convex Anal., 8 (1) (2007), 61-79] is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce a parallel S-iteration process for finding altering points of nonlinear operators. We apply our algorithms to solve a system of operator equations in Banach space setting. This work also includes convergence analysis of hybrid steepest-descent-like method and hybrid Newton-like method in the context of altering points.
The main strategy of this paper is intended to speed up the convergence of the inertial Mann iter... more The main strategy of this paper is intended to speed up the convergence of the inertial Mann iterative method and further speed up it through the normal S-iterative method for a certain class of nonexpansive type operators that are linked with variational inequality problems. Our new convergence theory permits us to settle down the difficulty of unification of Korpelevich's extragradient method, Tseng's extragardient method, and subgardient extragardient method for solving variational inequality problems through an auxiliary algorithmic operator, which is associated with seed operator. The paper establishes an interesting fact that the relaxed inertial normal S-iterative extragradient methods do influence much more on convergence behaviour. Finally, the numerical experiments are carried out to illustrate that the relaxed inertial iterative methods, in particular the relaxed inertial normal S-iterative extragradient methods, may have a number of advantages over other methods ...
We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms... more We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of nonexpansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward-backward-type algorithm and a Douglas-Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Furthe...
Every asymptotically nonexpansive mapping is uniformly continuous, but this fact is not true for ... more Every asymptotically nonexpansive mapping is uniformly continuous, but this fact is not true for asymptotically S-nonexpansive mappings in general. In a Banach space X, by constructing a sequence {xn} defined in Browder’s technique for a demicontinuous asymptotically S-nonexpansive semigroup T = {T (t) : t ≥ 0} of mappings from C ⊂ X into itself with function k(·), we prove that the common fixed point set F (S) ∩ ∩t>0F (T (t)) is the sunny nonexpansive retract of F (S), where F (S) is the fixed point set of a weakly continuous mapping S form C into itself. Under the assumption on uniform convexity of X, we prove that the common fixed point set F (S)∩∩t>0F (T (t)) is closed and convex.
Journal of Applied Mathematics and Computing, 2020
The article introduces a new algorithm for solving a class of variational inequality problems for... more The article introduces a new algorithm for solving a class of variational inequality problems for monotone operators and system of nonlinear variational inequalities problems for two inverse strongly monotone operators. We describe how to incorporate the extragradient like technique based on altering points technique with inertial effects. A weak convergence theorem is established for the proposed algorithm. Numerical examples are performed to illustrate the numerical efficiency of the algorithm and compare with other algorithms.
A notion os 2-compactibility is introduced for noncommuting mappings in Menger space to obtain fi... more A notion os 2-compactibility is introduced for noncommuting mappings in Menger space to obtain fixed point results. Further a result in random normed space and a result on best approximation is obtained as application of the above notion.
Journal of Applied Mathematics and Computing, 2021
Monotone inclusion problems are crucial to solve engineering problems and problems arising in dif... more Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.
The present paper deals with the convergence of two modified Mann type of iteration schemes for a... more The present paper deals with the convergence of two modified Mann type of iteration schemes for a single and a finite family of mappings to the fixed and common fixed point, respectively, of a single and a finite family of quasi-nonexpansive mappings on a uniformly convex Banach space. An example is added in support of our main result. The results obtained generalize the earlier results of Rhoades (J Math Anal Appl 56:741–750, [6]), Som et al. (Proc Nat Acad Sci (India) 70(A)(II):185–189, [8]), and others in turn.
The objective of this work is to design a new iterative method based on Armijo's type-modifie... more The objective of this work is to design a new iterative method based on Armijo's type-modified extragradient method for solving the inclusion problem (A+B)-1 (0), where A is a maximal monotone vector field and B is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the rate of convergence. A convergence theorem is established for the proposed extragradient method, which significantly improves existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field's monotonicity can be removed.
Numerical Functional Analysis and Optimization, 2021
Abstract A large number of nonlinear and optimization problems can be reduced to altering point p... more Abstract A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems.
Journal of Applied and Numerical Optimization, 2020
In this paper, we develop S-iterative techniques in terms of exponential mappings for finding fix... more In this paper, we develop S-iterative techniques in terms of exponential mappings for finding fixed points of nonexpansive mappings on Hadamard manifolds. To demonstrate numerical applicability, we show our proposed methods are faster than the existing results in the literature. For the applications, we focus on the common singularities of set-valued monotone vector fields and convex feasibility problems.
This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for ap... more This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.
In this paper, we introduce and study the generalized η-D-gap function for an equilibrium problem... more In this paper, we introduce and study the generalized η-D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized η-D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized η-D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples.
In this paper, the existence of fixed point of expansive type mappings is studied in modular func... more In this paper, the existence of fixed point of expansive type mappings is studied in modular function spaces.
ABSTRACT. In this paper, we design an iterative algorithm that converges strongly to common fixed... more ABSTRACT. In this paper, we design an iterative algorithm that converges strongly to common fixed points of a sequence of asymptotically nonexpansive mappings in a reflexive Banach space with weakly continuous duality mapping. Our results show that the uniform ...
It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877-898], that the proximal point... more It is known, by Rockafellar [SIAM J. Control Optim., 14 (1976), 877-898], that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi [Optimization, 37(1996), 239-252] introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with non-summable error sequence is also discussed.
It is well known that the rate of convergence of S-iteration process introduced by Agarwal
et al... more It is well known that the rate of convergence of S-iteration process introduced by Agarwal
et al. [J. Nonlinear Convex Anal., 8 (1) (2007), 61-79] is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce a parallel S-iteration process for finding altering points of nonlinear operators. We apply our algorithms to solve a system of operator equations in Banach space setting. This work also includes convergence analysis of hybrid steepest-descent-like method and hybrid Newton-like method in the context of altering points.
The main strategy of this paper is intended to speed up the convergence of the inertial Mann iter... more The main strategy of this paper is intended to speed up the convergence of the inertial Mann iterative method and further speed up it through the normal S-iterative method for a certain class of nonexpansive type operators that are linked with variational inequality problems. Our new convergence theory permits us to settle down the difficulty of unification of Korpelevich's extragradient method, Tseng's extragardient method, and subgardient extragardient method for solving variational inequality problems through an auxiliary algorithmic operator, which is associated with seed operator. The paper establishes an interesting fact that the relaxed inertial normal S-iterative extragradient methods do influence much more on convergence behaviour. Finally, the numerical experiments are carried out to illustrate that the relaxed inertial iterative methods, in particular the relaxed inertial normal S-iterative extragradient methods, may have a number of advantages over other methods ...
We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms... more We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of nonexpansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward-backward-type algorithm and a Douglas-Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Furthe...
Every asymptotically nonexpansive mapping is uniformly continuous, but this fact is not true for ... more Every asymptotically nonexpansive mapping is uniformly continuous, but this fact is not true for asymptotically S-nonexpansive mappings in general. In a Banach space X, by constructing a sequence {xn} defined in Browder’s technique for a demicontinuous asymptotically S-nonexpansive semigroup T = {T (t) : t ≥ 0} of mappings from C ⊂ X into itself with function k(·), we prove that the common fixed point set F (S) ∩ ∩t>0F (T (t)) is the sunny nonexpansive retract of F (S), where F (S) is the fixed point set of a weakly continuous mapping S form C into itself. Under the assumption on uniform convexity of X, we prove that the common fixed point set F (S)∩∩t>0F (T (t)) is closed and convex.
Journal of Applied Mathematics and Computing, 2020
The article introduces a new algorithm for solving a class of variational inequality problems for... more The article introduces a new algorithm for solving a class of variational inequality problems for monotone operators and system of nonlinear variational inequalities problems for two inverse strongly monotone operators. We describe how to incorporate the extragradient like technique based on altering points technique with inertial effects. A weak convergence theorem is established for the proposed algorithm. Numerical examples are performed to illustrate the numerical efficiency of the algorithm and compare with other algorithms.
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Papers by D. R. Sahu
et al. [J. Nonlinear Convex Anal., 8 (1) (2007), 61-79] is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce a parallel S-iteration process for finding altering points of nonlinear operators. We apply our algorithms to solve a system of operator equations in Banach space setting. This work also includes convergence analysis of hybrid steepest-descent-like method and hybrid Newton-like method in the context of altering points.
et al. [J. Nonlinear Convex Anal., 8 (1) (2007), 61-79] is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce a parallel S-iteration process for finding altering points of nonlinear operators. We apply our algorithms to solve a system of operator equations in Banach space setting. This work also includes convergence analysis of hybrid steepest-descent-like method and hybrid Newton-like method in the context of altering points.