Generalized Resolvent Operator Research Articles

On level sets of norm of generalized resolvent of operators pencils

We prove that the generalized resolvent operator defined in a Hilbert space cannot remain constant on any open subset of the resolvent set. Under certain conditions we also prove the same result for a complex uniformly convex Banach space. These results extend the known ones.

Total asymptotically nonexpansive mappings and generalized variational-like inclusion problems in semi-inner product spaces

ABSTRACT This paper focuses on investigating the problem of finding a common element of the set of solutions of a generalized nonlinear implicit variational-like inclusion problem involving an -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. and the set of fixed points of a total asymptotically nonexpansive mapping. To achieve such a purpose, a new iterative algorithm is constructed. Applying the concepts of graph convergence and generalized resolvent operator associated with an -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a point belonging to the intersection of the two sets mentioned above is proved.

Generalized resolvent operator involving 𝒢(·,·)-co-monotone mapping for solving generalized variational inclusion problem

Abstract In this paper, we define a generalized resolvent operator involving a 𝒢 ⁢ ( ⋅ , ⋅ ) {\mathcal{G}(\,\cdot\,,\cdot\,)} -co-monotone mapping for solving a generalized variational inclusion problem. We show that the resolvent operator under consideration is single-valued as well as Lipschitz continuous. An iterative algorithm is constructed to obtain the approximate solution of our problem. An example is provided to illustrate the theoretical findings.

$H(.,.,.,.)$-$varphi$-$eta$-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.

Cayley inclusion problem with its corresponding generalized resolvent equation problem in uniformly smooth Banach spaces

ABSTRACT A new inclusion problem is introduced using generalized Cayley operator and we call it Cayley inclusion problem. We also study its corresponding resolvent equation problem. By using a generalized resolvent operator and generalized Yosida approximation operator, first we establish a fixed point formulation for Cayley inclusion problem. An algorithm is defined to find the solution of Cayley inclusion problem. An existence and convergence result is proved. Secondly, we have shown the equivalence of Cayley inclusion problem with a resolvent equation. We define an iterative algorithm with some of its equivalent forms for solving resolvent equation problem. A numerical example is constructed and a convergence graph is shown by using MATLAB program.

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.

Existence of solution and iterative approximation of a system of generalized variational-like inclusion problems in semi-inner product spaces

In this paper, we consider the system of generalized variational-like inclusion problems in semi-inner product spaces. We define a class of (H,?)-?-monotone operators and its associated class of generalized resolvent operators. Further, using generalized resolvent operator technique, we give the existence of solution of the generalized variational-like inclusion problems. Furthermore, we suggest an iterative algorithm and give the convergence analysis of the sequences generated by the iterative algorithm. The results presented in this paper extend and unify the related known results in the literature.

On fluttering modes for aircraft wing model in subsonic air flow.

The paper deals with unstable aeroelastic modes for aircraft wing model in subsonic, incompressible, inviscid air flow. In recent author's papers asymptotic, spectral and stability analysis of the model has been carried out. The model is governed by a system of two coupled integrodifferential equations and a two-parameter family of boundary conditions modelling action of self-straining actuators. The Laplace transform of the solution is given in terms of the 'generalized resolvent operator', which is a meromorphic operator-valued function of the spectral parameter λ, whose poles are called the aeroelastic modes. The residues at these poles are constructed from the corresponding mode shapes. The spectral characteristics of the model are asymptotically close to the ones of a simpler system, which is called the reduced model. For the reduced model, the following result is shown: for each value of subsonic speed, there exists a radius such that all aeroelastic modes located outside the circle of this radius centred at zero are stable. Unstable modes, whose number is always finite, can occur only inside this 'circle of instability'. Explicit estimate of the 'instability radius' in terms of model parameters is given.

New general systems of set-valued variational inclusions involving relative ( A , η ) -maximal monotone operators in Hilbert spaces

The purpose of this paper is to introduce and study a class of new general systems of set-valued variational inclusions involving relative -maximal monotone operators in Hilbert spaces. By using the generalized resolvent operator technique associated with relative -maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms. The results presented in this paper improve and extend some known results in the literature.

Graph-convergent analysis of over-relaxed ( A , η , m ) -proximal point iterative methods with errors for general nonlinear operator equations

In this paper, we introduce and study a new class of over-relaxed (A,η,m)-proximal point iterative methods with errors for solving general nonlinear operator equations in Hilbert spaces. By using Liu’s inequality and the generalized resolvent operator technique associated with (A,η,m)-monotone operators, we also prove the existence of solution for the nonlinear operator inclusions and discuss the graph-convergent analysis of iterative sequences generated by the algorithm. Furthermore, we give some examples and an application for solving the open question (2) due to Li and Lan (Adv. Nonlinear Var. Inequal. 15(1):99-109, 2012). The numerical simulation examples are given to illustrate the validity of our results. MSC:49J40, 47H05, 65B05.

Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces

This paper deals with the existence of solutions for a class of nonlinear implicit variational inclusion problems in semi-inner product spaces. We construct an iterative algorithm for approximating the solution for the class of implicit variational inclusions problems involving A-monotone and H-monotone operators by using the generalized resolvent operator technique.

General hybrid [formula omitted]-proximal point algorithm frameworks for finding common solutions of nonlinear operator equations and fixed point problems

Abstract In this paper, we introduce and study a new general class of hybrid ( A , η , m ) -proximal point algorithm frameworks for finding the common solutions of nonlinear operator equations and fixed point problems of Lipschitz continuous operators in Hilbert spaces. Further, by using the generalized resolvent operator technique associated with ( A , η , m ) -maximal monotone operators, we discuss the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm frameworks. Finally, using software Matlab 7.0, the numerical simulation examples are given to illustrate the validity of main results presented in this paper.

Graphical approximation of common solutions to generalized nonlinear relaxed cocoercive operator equation systems with (A, η)-accretive mappings

Abstract In this paper, we develop a new perturbed iterative algorithm framework with errors based on the variational graphical convergence of operator sequences with (A, η)-accretive mappings in Banach space. By using the generalized resolvent operator technique associated with (A, η)-accretive mappings, we also prove the existence of solutions for a class of generalized nonlinear relaxed cocoercive operator equation systems and the variational convergence of the sequence generated by the perturbed iterative algorithm in q-uniformly smooth Banach spaces. The obtained results improve and generalize some well-known results in recent literatures. 2000 Mathematics Subject Classification: 47H05; 49J40

On Over-Relaxed Proximal Point Algorithms for Generalized Nonlinear Operator Equation with (A,η,m)-Monotonicity Framework

In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the generalized resolvent operator technique associated with the (A,η,m)-monotone operators, the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm are discussed. Our results improve and generalize the corresponding results in the literature.

On general over-relaxed proximal point algorithm and applications

A general framework to the over-relaxed proximal point algorithm based on H-maximal monotonicity is introduced, and then it is applied to the approximation solvability of a general class of nonlinear inclusion problems based on the generalized resolvent operator technique. This form of the proximal point algorithm seems to be more application-oriented to inclusion problems.

The Over-Relaxed A-Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework

The purpose of this paper is (1) a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the ( , )-accretive mapping is introduced, and (2) it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.

VARIATIONAL-LIKE INCLUSION SYSTEMS VIA GENERAL MONOTONE OPERATORS WITH CONVERGENCE ANALYSIS

Abstract. In this paper using Lipschitz continuity of the resolvent oper-ator associated with general H-maximal m-relaxed -monotone operators,existence and uniqueness of the solution of a variational inclusion systemis proved. Also, an iterative algorithm and its convergence analysis isgiven. 1. IntroductionThe concept of general H-maximal m-relaxed -monotone operator (so-called the general G--monotone mapping in [3]) as a generalization of thegeneral A-monotone mapping [3, 8, 13, 14], the general (H;)-monotone op-erator [5, 6], general H-monotone operator [20] in Banach spaces, and alsoas a generalization of the (A;)-maximal m-relaxed monotone operator [2], A-maximal m-relaxed monotone operator [1, 17, 19], G--monotone operator [22],(A;)-monotone operator [18], A-monotone operator [16], (H;)-monotone op-erator [12], H-monotone operator [7, 11], maximal -monotone operator [10]and classical maximal monotone operator [21] in Hilbert spaces, is introducedand considered in [4]. At the mentioned paper the authors provided some ex-amples and also they studied many properties of general H-maximal m-relaxed-monotone operators. Further, the generalized resolvent operator associatedwith this type of monotone operators has been de ned and some results aboutLipschitz continuity of this type of monotone operators has been established.At the present paper, rst we recall some notions, de nitions, and results aboutmonotone operators and their generalized versions. Using Lipschitz continu-ity of the resolvent operator associated with general H-maximal m-relaxed-monotone operators, existence and uniqueness of the solution of a variationalinclusion system is proved. Further, we construct an iterative algorithm andthe convergence analysis of this algorithm is given.

General Proximal Point Algorithmic Models and Nonlinear Variational Inclusions Involving RMM Mappings

The proximal point algorithms based on relative $A$-maximal monotonicity (RMM) is introduced, and then it is applied to the approximation solvability of a general class of nonlinear inclusion problems using the generalized resolvent operator technique. This algorithm seems to be more application-oriented to solving nonlinear inclusion problems. Furthermore, the obtained result could be applied to generalize the Douglas-Rachford splitting method to the case of RMM mapping based on the generalized proximal point algorithm.

A general framework for the over-relaxed A-proximal point algorithm and applications to inclusion problems

First, a general framework for the over-relaxed A -proximal point algorithm based on the A -maximal monotonicity is introduced, and second it is applied to the approximation solvability of a general class of nonlinear inclusion problems using the generalized resolvent operator technique. The over-relaxed A -proximal point algorithm is of interest in the sense that it is quite application-oriented, but nontrivial in nature. The results obtained are general in nature.

Sensitivity analysis for cocoercively monotone variational inclusions and (A,η)-maximal monotonicity

Sensitivity analysis for cocoercively monotone variational inclusions based on the generalized resolvent operator technique is discussed. The notion of the cocoercive monotonicity unifies most of the existing notions, such as cocoercivity, strong monotonicity, relaxed monotonicity, relaxed cocoercivity, and others. As a result, the obtained results are general in nature.