Variational Inclusions Research Articles

Graph Convergence and Iterative Approximation of Solution of a Set‐Valued Variational Inclusions

In this article, first, we introduce a class of proximal‐point mapping associated with generalized αiβj‐Hp(.,.,…)‐accretive mapping. Further, we discuss the graph convergence of generalized αiβj‐Hp(.,.,…)‐accretive mapping. As an application, we consider a set‐valued variational inclusion problem (SVIP) in real Banach spaces. Furthermore, we propose an iterative scheme involving the above class of proximal‐point mapping to find a solution of SVIP and discuss its convergence under some convenient assumptions. An example is constructed and demonstrated few graphics in support of our main results.

New inertial approximation schemes for general quasi variational inclusions

In this article, we introduce and consider some new classes of general quasi variational inclusions, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We prove that the general quasi variational inclusions are equivalent to the fixed point problems. This alternative formulation is used to discuss the existence of a solution as well as to propose some iterative methods. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inclusions include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inclusions for further research activities.

Algorithms of common solutions to modified generalized system of variational inclusion problem and hierarchical fixed point problem

This manuscript deals with two problems : the first one is a new problem of the system of variational inclusion that is called modified generalized system of variational inclusion problem(MGSVIP) and the other one is a hierarchical fixed point problem in the framework of real Hilbert space. We establish the important lemma that show the relation between fixed point of nonlinear mapping and solution of MGSVIP for proving the main theorem. To approximate the common solution of these problems, we design an iterative scheme under suitable conditions on parameters. A strong convergence result for the proposed iterative scheme is proved. Applying our main result, we prove strong convergence theorems of the modification system of variational inequalities problem and variational inclusion problem. Moreover, we give the numerical example for supporting our results.

Tseng‐Type Algorithms for the Split Variational Inclusion

In this paper, we study iterative methods for solving the split variational inclusion in Hilbert spaces. We consider a Tseng‐type algorithm with self‐adaptive step sizes for finding a solution of the split variational inclusion. We show the weak convergence of the proposed sequence under some mild assumptions.

Convergence analysis of general parallel $ S $-iteration process for system of mixed generalized Cayley variational inclusions

<abstract><p>This work is concentrated on the study of a system of mixed generalized Cayley variational inclusions. Parallel Mann iteration process is defined in order to achieve the solution. We define an altering point problem which is equivalent to our system and then we construct general parallel $ S $-iteration process. Finally, we discuss convergence criteria and provide an example.</p></abstract>

System of generalized nonlinear variational-like inclusion problems in 2-uniformly smooth Banach spaces

In this manuscript, we introduce and study the existence of a solution of a system of generalized nonlinear variational-like inclusion problems in 2-uniformly smooth Banach spaces by using $H(.,.)$-$eta$-proximal mapping. The method used in this paper can be considered as an extension of methods for studying the existence of solutions of various classes of variational inclusions considered and studied by many authors in 2-uniformly smooth Banach spaces. Some important results, theorems and the existence of solution of the proposed system of generalized nonlinear variational-like inclusion problems have been derived.

Convergence analysis for split hierachical monotone variational inclusion problem in Hilbert spaces

Abstract In this paper, we introduce a new iterative algorithm for approximating a common solution of Split Hierarchical Monotone Variational Inclusion Problem (SHMVIP) and Fixed Point Problem (FPP) of k-strictly pseudocontractive mappings in real Hilbert spaces. Our proposed method converges strongly, does not require the estimation of operator norm and it is without imposing the strict condition of compactness; these make our method to be potentially more applicable than most existing methods in the literature. Under standard and mild assumption of monotonicity of the SHMVIP associated mappings, we establish the strong convergence of the iterative algorithm.We present some applications of our main result to approximate the solution of Split Hierarchical Convex Minimization Problem (SHCMP) and Split Hierarchical Variational Inequality Problem (SHVIP). Some numerical experiments are presented to illustrate the performance and behavior of our method. The result presented in this paper extends and complements some related results in literature.

An inertial iterative method for solving split equality problem in Banach spaces

<abstract><p>In this paper, a new self-adaptive algorithm with the inertial technique is proposed for solving the split equality problem in $ p $-uniformly convex and uniformly smooth Banach spaces. Under some mild control conditions, a strong convergence theorem for the proposed algorithm is established. Furthermore, the results are applied to split equality fixed point problem and split equality variational inclusion problem. Finally, numerical examples are provided to illustrate the convergence behaviour of the algorithm. The main results in this paper improve and generalize some existing results in the literature.</p></abstract>

An efficient iterative method for solving split variational inclusion problem with applications

<p style='text-indent:20px;'>A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.</p>

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.

Two self-adaptive inertial projection algorithms for solving split variational inclusion problems

<abstract><p>This paper is to analyze the approximation solution of a split variational inclusion problem in the framework of Hilbert spaces. For this purpose, inertial hybrid and shrinking projection algorithms are proposed under the effect of a self-adaptive stepsize which does not require information of the norms of the given operators. The strong convergence properties of the proposed algorithms are obtained under mild constraints. Finally, a numerical experiment is given to illustrate the performance of proposed methods and to compare our algorithms with an existing algorithm.</p></abstract>

A general iterative algorithm for split variational inclusion problems and fixed point problems of a pseudocontractive mapping

A general iterative algorithm for split variational inclusion problems and fixed point problems of a pseudocontractive mapping

PMiCA – Parallel multi-step inertial contracting algorithm for solving common variational inclusions

PMiCA – Parallel multi-step inertial contracting algorithm for solving common variational inclusions

New Self-Adaptive Methods with Double Inertialsteps for Solving Splitting Monotone Variational Inclusion Problems with Applications

New Self-Adaptive Methods with Double Inertialsteps for Solving Splitting Monotone Variational Inclusion Problems with Applications

An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.

Dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds

In this paper, we investigate a new class of dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds. Then we prove that the dynamical system has a unique solution under some suitable assumptions. Moreover, the global exponential stability and invariance property of the dynamical systems are also established. Our main results in this work are new and extend the existing ones in the literature.

Inertial proximal point algorithm for variational inclusion in Hadamard manifolds

In this paper, we consider the inertial proximal point algorithm for finding a zero point of variational inclusions on Hadamard manifolds. Under suitable conditions, it is proved that the sequence generated by the algorithm converges to an element of the set of zero points of variational inclusion problem. As applications, we utilize our results to study the minimization problem and saddle point problem in the setting of Hadamard manifolds.

A strongly convergent algorithm for solving common variational inclusion with application to image recovery problems

In this paper, we present an accelerated approach for addressing the common variational inclusion problem in real Hilbert spaces that incorporates several techniques. Under conventional and appropriate assumptions, the algorithm's strong convergence theorem is established, and the applicability and benefits of the innovative approach for image recovery problems are demonstrated.

A Modified Tseng’s Method for Solving the Modified Variational Inclusion Problems and Its Applications

The goal of this study was to show how a modified variational inclusion problem can be solved based on Tseng’s method. In this study, we propose a modified Tseng’s method and increase the reliability of the proposed method. This method is to modify the relaxed inertial Tseng’s method by using certain conditions and the parallel technique. We also prove a weak convergence theorem under appropriate assumptions and some symmetry properties and then provide numerical experiments to demonstrate the convergence behavior of the proposed method. Moreover, the proposed method is used for image restoration technology, which takes a corrupt/noisy image and estimates the clean, original image. Finally, we show the signal-to-noise ratio (SNR) to guarantee image quality.

The Impact of a 3‐D Earth Structure on Glacial Isostatic Adjustment in Southeast Alaska Following the Little Ice Age

AbstractIn Southeast Alaska, extreme uplift rates are primarily caused by glacial isostatic adjustment (GIA), as a result of ice thickness changes from the Little Ice Age to the present combined with a low‐viscosity asthenosphere. Previous GIA models adopted a 1‐D Earth structure. However, the actual Earth structure is likely more complex due to the long history of subduction and tectonism and the transition from a continental to an oceanic plate. Seismic evidence indeed shows a laterally heterogenous Earth structure. In this study, a numeral model is constructed for Southeast Alaska, which allows for the inclusion of lateral viscosity variations. The viscosity follows from scaling relationships between seismic velocity anomalies and viscosity variations. We use this scaling relationship to constrain the thermal effect on seismic variations and investigate the importance of lateral viscosity variations. We find that a thermal contribution to seismic anomalies of 10% is required to explain the GIA observations. This implies that non‐thermal effects control seismic anomaly variations in the shallow upper mantle. Due to the regional geologic history, it is likely that hydration of the mantle impact both viscosity and seismic velocity. The best‐fit model has a background viscosity of 5.0 × 1019 Pa‐s, and viscosities at ∼80 km depth range from 1.8 × 1019 to 4.5 × 1019 Pa‐s. A 1‐D averaged version of the 3‐D model performed slightly better, however, the two models were statistically equivalent within a 2σ measurement uncertainty. Thus, lateral viscosity variations do not contribute significantly to the uplift rates measured with the current accuracy and distribution of sites.