Multivalued Operators Research Articles

A proximal bundle approach for solving the generalized variational inequalities with inexact data

This paper introduces a proximal bundle scheme to solve generalized variational inequalities with inexact data. Under optimality conditions, the problem can be equivalently represented as seeking out the zero point of the sum of two multi-valued operators whose domains are the real Hilbert space. The two operators denoted by T and f, respectively, are the monotone operator and the subdifferential of a lower semi-continuous, non-differentiable, convex function. Our approach is based on the principles of the proximal point strategy, which involves incorporating inexact information into the subproblems and approximating them using a series of piecewise linear convex functions. Moreover, we put forward a novel stopping criterion to identify the adequacy of the current approximation. This approach serves to make the subproblems more manageable, and it has been proven that obtaining inexact information can ensure that the linearization error during the iteration process remains non-negative, thus avoiding triggering noise attenuation. Subsequently, we verify the convergence of the algorithm under relatively mild assumptions (the operator T is para-monotone and may be multi-valued). Ultimately, we present the findings of elementary numerical experiments to declare the method's efficacy.

Semigroups Generated by Multivalued Operators and Domain Convergence for Parabolic Problems

Semigroups Generated by Multivalued Operators and Domain Convergence for Parabolic Problems

Fractional elliptic obstacle systems with multivalued terms and nonlocal operators

In this paper, we introduce and study a fractional elliptic obstacle system, which is composed of two elliptic inclusions with fractional (pi, qi)-Laplace operators, nonlocal functions, and multivalued terms. The weak solution of fractional elliptic obstacle system is formulated by a fully nonlinear coupled system driven by two nonlinear and nonmonotone variational inequalities with constraints. The nonemptiness and compactness of solution set in the weak sense are proved via employing a surjectivity theorem to the multivalued operators formulated by the sum of a multivalued pseudomonotone operator and a maximal monotone operator.

Generalized Yosida inclusion problem involving multi-valued operator with XOR operation

Abstract In this article, we study a generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation. It is shown that the generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation is equivalent to a fixed point equation. We have proved that the generalized Yosida approximation operator is Lipschitz continuous. Finally, we prove an existence and convergence result for our problem.

A new generalization of Ćirić's multi-valued operators

A new generalization of Ćirić's multi-valued operators

Fixed points and best proximity points for some general classes of multi-valued operators

Fixed points and best proximity points for some general classes of multi-valued operators

On generalized logistic equations with non-local term of feedback control type

We consider a problem of finding functions λ(x),u(x) such that −Δpu=λf(x,u,∇u)−g(x,u) in Ω and u=0 on ∂Ω, λ(x)∈F(x,u(x)) a.e. in Ω. Assume that, the single-valued functions f,g and the multi-valued function F satisfy certain growth conditions. Using the fixed point index theory for multi-valued operators with non-necessary convex values, we prove the existence of one or two nontrivial non-negative solutions of the problem.

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Application of the Mittag-Leffler kernel in stochastic differential systems for approximating the controllability of nonlocal fractional derivatives

This research focuses on utilizing the Mittag-Leffler kernel within stochastic differential systems to estimate the controllability of nonlocal Atangana–Baleanu fractional derivatives. By assuming the automatic control of the corresponding linear system, a novel set of necessary and sufficient conditions for the approximate controllability of the fractional stochastic differential inclusions of Atangana–Baleanu is established. Furthermore, the study explores the approximate controllability of the proposed system with infinite delay. The investigation relies on the fixed-point theorem for multivalued operators and fractional calculus to derive these outcomes. Lastly, an illustrative example is provided to highlight the practical implications of the research findings.

Study of Nonlinear Second-Order Differential Inclusion Driven by a Φ−Laplacian Operator Using the Lower and Upper Solutions Method

In this paper, we study a second-order differential inclusion under boundary conditions governed by maximal monotone multivalued operators. These boundary conditions incorporate the classical Dirichlet, Neumann, and Sturm–Liouville problems. Our method of study combines the method of lower and upper solutions, the analysis of multivalued functions, and the theory of monotone operators. We show the existence of solutions when the lower solution σ and the upper solution γ are well ordered. Next, we show how our arguments of proof can be easily exploited to establish the existence of extremal solutions in the functional interval σ,γ. We also show that our method can be applied to the periodic case.

Fixed point methods with multi-valued operators for control problems

Fixed point methods with multi-valued operators for control problems

On dominated multivalued operators involving nonlinear contractions and applications

<abstract> <p>The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended <italic>b</italic>-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended <italic>b</italic>-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature.</p> </abstract>

A monotone iterative method for second order nonlinear problems with boundary conditions driven by maximal monotone multivalued operators

A monotone iterative method for second order nonlinear problems with boundary conditions driven by maximal monotone multivalued operators

On the existence of solutions and optimal control for set-valued quasi-variational–hemivariational inequalities with applications

In this paper, we study the existence and optimal control of quasi-hemivariational inequalities by a method different from the one based on Minty's technique. We use an approach similar to the Galerkin method based on a minimax inequality formulation associated with the Brézis pseudomonotonicity notion of multi-valued operators, an implicit Browder–Tikhonov regularization method and a fixed point theorem. This leads us to avoid any kind of monotonicity-type conditions used in recent papers to obtain the convexity of the solution set of the variational selections. We provide applications to the optimal control of implicit obstacle problems of fractional Laplacian type involving a generalized gradient operator, and to the optimal control of contact problems for elastic locking materials. Our approach improves some recent results in the literature.

Stabilization of the heat equation with disturbance at the flux boundary condition

In this paper, we address the problem of boundary stabilization of the heat equation subjected to an unknown disturbance, which is assumed to be acting at the flux boundary condition. By means of Lyapunov techniques and the use of the sign multivalued operator, which is responsible of rejecting the effects of the disturbance, we design a multivalued feedback law to obtain the exponential stability of the closed‐loop system. The well‐posedness of the closed‐loop system, which is a differential inclusion, is shown with the maximal monotone operator theory.

Existence results for nonexpansive multi-valued operators and nonlinear integral inclusions

Existence results for nonexpansive multi-valued operators and nonlinear integral inclusions

Validating the effect of fuel moisture content by a multivalued operator in a simplified physical fire spread model

Validating the effect of fuel moisture content by a multivalued operator in a simplified physical fire spread model

Convergence To Approximate Solutions of Multivalued Operators

The goal of this study is to provide a new explicit iterative process method approach for solving maximal monotone(M.M )operators in Hilbert spaces utilizing a finite family of different types of mappings as( nonexpansive mappings,resolvent mappings and projection mappings. The findings given in this research strengthen and extend key previous findings in the literature. Then, utilizing various structural conditions in Hilbert space and variational inequality problems, we examine the strong convergence to nearest point projection for these explicit iterative process methods Under the presence of two important conditions for convergence, namely closure and convexity. The findings reported in this research strengthen and extend key previous findings from the literature

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.

Double phase implicit obstacle problems with convection term and multivalued operator

This paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke’s generalized gradient. We develop a general framework to deliver an existence result for the double phase implicit obstacle problem under consideration. Our proof is based on the Kakutani–Ky Fan fixed point theorem together with the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded pseudomonotone mapping.