Resolvent Operator Technique Research Articles

Resolvent operator technique and iterative algorithms for system of generalized nonlinear variational inclusions and fixed point problems: Variational convergence with an application

This article is devoted to investigate the problem of finding a common point of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational inclusions involving P-?-accretive mappings. For finding such an element, a new iterative algorithm is suggested. The concepts of graph convergence and the resolvent operator associated with a P-?-accretive mapping are used and a new equivalence relationship between graph convergence and resolvent operators convergence of a sequence of P-?-accretive mappings is established. As an application of the obtained equivalence relationship, we prove the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common element of the above two sets. These results are new, and can be viewed as a refinement and improvement of some known results in the literature.

System of generalized variational-like inclusions involving (P,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{(P,\\eta )}$$\\end{document}-accretive mapping and fixed point problems in real Banach spaces

This paper attempts to prove the Lipschitz continuity of the resolvent operator associated with a (P,η)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(P,\\eta )$$\\end{document}-accretive mapping and compute an estimate of its Lipschitz constant. This is done under some new appropriate conditions that are imposed on the parameter and mappings involved in it; with the goal of approximating a common element of the solution set of a system of generalized variational-like inclusions and the fixed point set of a total asymptotically nonexpansive mapping in the framework of real Banach spaces. A new iterative algorithm based on the resolvent operator technique is proposed. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the two sets mentioned above. The final section is dedicated to investigating and analyzing the notion of a generalized H(., .)-accretive mapping introduced and studied by Kazmi et al. (Appl Math Comput 217:9679–9688, 2011). In this section, we provide some comments based on the relevant results presented in their work.

A study of the effective Hamiltonian method for decay dynamics* *Project supported by the National Natural Science Foundation of China (11964010, 11564013 and 11464014), the Natural Science Foundation of Hunan Province (2020JJ4495), the Scientific Research Fund of Hunan Provincial Education Department (22A0377 and 21A0333), and the Jishou University Innovation Foundation for Postgraduate (Jdy20038).

The decay dynamic of an excited quantum emitter (QE) is one of the most important contents in quantum optics. It has been widely applied in the field of quantum computing and quantum state manipulation. When the electromagnetic environment is described by several pseudomodes, the effective Hamiltonian method based on the multi-mode Jaynes–Cummings model provides a clear physical picture and a simple and convenient way to solve the decay dynamics. However, in previous studies, only the resonant modes are taken into account, while the non-resonant contributions are ignored. In this work, we study the applicability and accuracy of the effective Hamiltonian method for the decay dynamics. We consider different coupling strengths between a two-level QE and a gold nanosphere. The results for dynamics by the resolvent operator technique are used as a reference. Numerical results show that the effective Hamiltonian method provides accurate results when the two-level QE is resonant with the plasmon. However, when the detuning is large, the effective Hamiltonian method is not accurate. In addition, the effective Hamiltonian method cannot be applied when there is a bound state between the QE and the plasmon. These results are of great significance to the study of the decay dynamics in micro-nano structures described by quasi-normal modes.

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Abstract This paper deals with the control problems governed by fractional impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder’s fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution, phase space, and approximate controllability of fractional order impulsive systems in Banach spaces. Finally, we investigate the approximate controllability of the fractional order heat equation with non-instantaneous impulses and delay by using the developed results.

A new system of generalized nonlinear variational inclusion problems in semi-inner product spaces

In this work we reflect a new system of generalized nonlinear variational inclusion problems in 2-uniformly smooth Banach spaces. By using resolvent operator technique, we offer an iterative algorithm for figuring out the approximate solution of the said system. The motive of this paper is to review the convergence analysis of a system of generalized nonlinear variational inclusion problems in 2-uniformly smooth Banach spaces. The proposition used in this paper can be considered as an extension of propositions for examining the existence of solution for various classes of variational inclusions considered and studied by many authors in 2-uniformly smooth Banach spaces.

General Variational Inclusions and Nonexpansive Mappings

In this paper, we introduce a new class of variational inclusions involving three operators. We suggest and analyze three-step iterations for finding the common element of the set of fixed points of a nonexpansive mappings and the set of the solutions of the variational inclusions using the resolvent operator technique. We also study the convergence criteria of three-step iterative method under some mild conditions. Inertial type methods are suggested and investigated for general variational inclusions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.

$H(.,.,.,.)$-$varphi$-$eta$-cocoercive Operator with an Application to Variational Inclusions

In this work, we study generalized $ H(.,.,.,.)$-$varphi$-$eta-$ cocoercive operator to find the solution of variational like inclusion involving an infinite family of set-valued mappings in semi-inner product spaces via resolvent equation approach. Furthermore, we established an equivalence between the set-valued variational-like inclusion problem and fixed point problem by employing generalized resolvent operator technique involving generalized $H(.,.,.,.)$-$varphi$-$eta$-cocoercive operator. Using the equivalent formulation of set-valued variational-like inclusion problem and resolvent equation problem, an iterative algorithm is developed that approximate the uniqueness of solution of the resolvent equation problem.

Algorithmic and analytical approach of solutions of a system of generalized multi-valued nonlinear variational inclusions

The main contributions of this paper is twofold. First, our primary concern is to suggest a new iterative algorithm using the P-?-proximal-point mapping technique and Nadler?s technique for finding the approximate solutions of a system of generalized multi-valued nonlinear variational-like inclusions. Under some appropriate conditions imposed on the parameters and mappings involved in the system of generalized multi-valued nonlinear variational-like inclusions, the strong convergence of the sequences generated by our proposed iterative algorithm to a solution of the aforesaid system is proved. Second, the H(.,.)-?-cocoercive mapping considered in [R. Ahmad, M. Dilshad, M. Akram, Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach sapces, Filomat 26(5)(2012) 897- 908] is investigated and analyzed, and the fact that under the assumptions imposed on H(., .)-?-cocoercive mapping, every H(.,.)-?-cocoercive mapping is P-?-accretive and is not a new one is pointed out. At the same time, some important comments on H(.,.)-?-cocoercive mapping and the results given in the above-mentioned paper are stated.

Three-step Iterative Algorithm with Error Terms of Convergence and Stability Analysis for New NOMVIP in Ordered Banach Spaces

This article undertakes to study a NOMVIP involving XNOR-operation and solved by employing a proposed three-step iterative algorithm in ordered Banach Space. Under suitable conditions, we obtain the strong convergence and existence results of NOMVIP involving XNOR-operation by applying the resolvent operator technique with XNOR and XOR operations and discuss the stability of the proposed algorithm. Finally, we provide a numerical example to confirm the convergence of the suggested algorithm in support of our considered problem which gives the grantee that all the proposed conditions of our main result have been formulated by using MATLAB programming.

(A(., .), η)-MONOTONE MAPPINGS AND A SYSTEM OF GENERALIZED SET-VALUED VARIATIONAL-LIKE INCLUSION PROBLEM

In this paper, we introduce and study a system of variational inclusions called system of generalized set-valued variational-like inclusion problem involving (A(., .), η)-monotone mappings in real Banach spaces. By means of resolvent operator technique, we prove the existence of solutions for this system of variational inclusions. Further, we suggest an iterative algorithm for finding the approximate solution of this system and discuss the convergence criteria of the sequences generated by the iterative algorithm under some suitable conditions.

Iterative Algorithms for a Generalized System of Mixed Variational-Like Inclusion Problems and Altering Points Problem

In this article, we introduce and study a generalized system of mixed variational-like inclusion problems involving αβ-symmetric η-monotone mappings. We use the resolvent operator technique to calculate the approximate common solution of the generalized system of variational-like inclusion problems involving αβ-symmetric η-monotone mappings and a fixed point problem for nonlinear Lipchitz mappings. We study strong convergence analysis of the sequences generated by proposed Mann type iterative algorithms. Moreover, we consider an altering points problem associated with a generalized system of variational-like inclusion problems. To calculate the approximate solution of our system, we proposed a parallel S-iterative algorithm and study the convergence analysis of the sequences generated by proposed parallel S-iterative algorithms by using the technique of altering points problem. The results presented in this paper may be viewed as generalizations and refinements of the results existing in the literature.

Global attractiveness and exponential stability for impulsive fractional neutral stochastic evolution equations driven by fBm

This paper is concerned with a class of fractional neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion (fBm). First, by means of the resolvent operator technique and contraction mapping principle, we can directly show the existence and uniqueness result of mild solution for the aforementioned system. Then we develop a new impulsive-integral inequality to obtain the global attracting set and pth moment exponential stability for this type of equation. Worthy of note is that this powerful inequality after little modification is applicable to the case with delayed impulses. Moreover, sufficient conditions which guarantee the pth moment exponential stability for some pertinent systems are stated without proof. In the end, an example is worked out to illustrate the theoretical results.

Stability and convergence analysis for a new class of GNYIP involving XOR-operation in ordered positive Hilbert spaces

In the setting of real ordered positive Hilbert spaces, a new class of general nonlinear ordered Yosida inclusion problem involving ? operation has been considered and solved by employing a perturbed two step-iterative algorithm. The stability and convergence analysis of solution of new class of Yosida inclusion problem involving ? operation has been substantiated by applying a new resolvent operator and Yosida approximation operator method with XOR-operation technique. The iterative algorithm and results demonstrated in this article have witnessed, a significant improvement for many previously known results of this domain. Further, we give a numerical example in support of our main result by using MATLprogramming.

An Iterative Algorithm for the Nonlinear MC2 Model with Variational Inequality Method

The multiple criteria and multiple constraint level (MC 2 ) model is a useful tool to deal with the decision programming problems, which concern multiple decision makers and uncertain resource constraint levels. In this paper, by regarding the nonlinear MC 2 problems as a class of mixed implicit variational inequalities, we develop an iterative algorithm to solve the nonlinear MC 2 problems through the resolvent operator technique. The convergence of the generated iterative sequence is analyzed and discussed by a calculation example, and the stability of Algorithm 1 is also verified by error propagation. By comparing with two other MC 2 -algorithms, Algorithm 1 performs well in terms of number of iterations and computation complexity.

Approximation solvability for a system of implicit nonlinear variational inclusions with Η-monotone operators

AbstractIn this paper, we introduce and study a new system of variational inclusions which is called a system of nonlinear implicit variational inclusion problems with A-monotone and H-monotone operators in semi-inner product spaces. We define the resolvent operator associated with A-monotone and H-monotone operators and prove its Lipschitz continuity. Using resolvent operator technique, we prove the existence and uniqueness of solution for this new system of variational inclusions. Moreover, we suggest an iterative algorithm for approximating the solution of this system and discuss the convergence analysis of the sequences generated by the iterative algorithm under some suitable conditions.

Parametric generalized mixed multi-valued implicit quasi-variational inclusion problems

In this paper, by using a resolvent operator technique of maximal monotone mappings and the property of afixed-point set ofmulti-valued contractivemapping, we studythe behavior andsensitivity analysisof a solution set for a parametric generalized mixed multi-valued implicit quasi-variational inclusion problem in Hilbert space. Further, under some suitable conditions, we discuss the Lipschitz continuity (or continuity) of the solution set with respect to the parameter. By exploiting the technique of this paper, one can generalize and improve many known results in the literature.

Solvability for a system of generalized nonlinear ordered variational inclusions in ordered Banach spaces

In this paper, we deal a resolvent operator technique is applied to address a system of generalized nonlinear ordered variational inclusions in real ordered Banach spaces and derived an algorithm for a solution of the considered system. Here, we prove an existence result for the solution of the system of generalized nonlinear ordered variational inclusions and discuss convergence of sequences suggested by the algorithms.

System of nonlinear variational inclusion problems with $(A,\eta)$-maximal monotonicity in Banach spaces

This paper deals with a new system of nonlinear variational inclusion problems involving $(A,\eta)$-maximal relaxed monotone and relative $(A,\eta)$-maximal monotone mappings in 2-uniformly smooth Banach spaces. Using the generalized resolvent operator technique, the approximation solvability of the proposed problem is investigated. An iterative algorithm is constructed to approximate the solution of the problem. Convergence analysis of the proposed algorithm is investigated. Similar results are also proved for other system of variational inclusion problems involving relative $(A,\eta)$-maximal monotone mappings and $(H,\eta)$-maximal monotone mappings.

Approximation solution for system of generalized ordered variational inclusions with \u2295 operator in ordered Banach space

The resolvent operator approach is applied to address a system of generalized ordered variational inclusions with ⊕ operator in real ordered Banach space. With the help of the resolvent operator technique, Li et al. (J. Inequal. Appl. 2013:514, 2013; Fixed Point Theory Appl. 2014:122, 2014; Fixed Point Theory Appl. 2014:146, 2014; Appl. Math. Lett. 25:1384-1388, 2012; Fixed Point Theory Appl. 2013:241, 2013; Eur. J. Oper. Res. 16(1):1-8, 2011; Fixed Point Theory Appl. 2014:79, 2014; Nonlinear Anal. Forum 13(2):205-214, 2008; Nonlinear Anal. Forum 14: 89-97, 2009) derived an iterative algorithm for approximating a solution of the considered system. Here, we prove an existence result for the solution of the system of generalized ordered variational inclusions and deal with a convergence scheme for the algorithms under some appropriate conditions. Some special cases are also discussed.

A Perturbed Iterative Algorithm for Split General Mixed Variational Inequality Problem

In this paper, we introduce a split general mixed variational inequality problem, which is a natural extension of a split variational inequality problem, split general quasi-variational inequality problem in Hilbert spaces. Using the resolvent operator technique, we propose a perturbed iterative algorithm for the split general mixed variational inequality problem. Further, we discuss the convergence criteria of the iterative algorithm. The results presented in this paper extend and improve many previously known results in this area.