Multivalued Boundary Conditions Research Articles

Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions

Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions

Multiple solutions for a class of [formula omitted]-biharmonic problems with potential boundary conditions

We are concerned with multiplicity of solutions for differential inclusions involving the p-biharmonic operator, submitted to a general potential multivalued boundary condition which, among others, covers many particular boundary conditions that are frequently invoked in the literature. Our approach is a variational one and relies on non-smooth critical point theory.

Anisotropic and isotropic implicit obstacle problems with nonlocal terms and multivalued boundary conditions

In this paper we study an anisotropic implicit obstacle problem driven by the (p(⋅),q(⋅))-Laplacian and an isotropic implicit obstacle problem involving a nonlinear convection term (a reaction term depending on the gradient) which contain several interesting and challenging untreated problems. These two implicit obstacle problems have both highly nonlinear and nonlocal functions and three multivalued terms where two of them are appearing on the boundary and the other one is formulated in the domain. Under very general assumptions on the data, we develop general frameworks to examine the nonemptiness and compactness of the set of weak solutions to the problems under consideration. The proofs of our main results use the theory of nonsmooth analysis, Tychonoff’s fixed point theorem for multivalued operators, the theory of pseudomonotone operators and variational approach.

Identification of discontinuous parameters in double phase obstacle problems

Abstract In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.

Inverse problems for a class of elliptic obstacle problems involving multivalued convection term and weighted $(p,q)$-Laplacian

In this paper, we study the inverse problem of estimating discontinuous parameters and boundary data in a nonlinear elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is given as a sum of a weighted p-Laplacian and a weighted q-Laplacian (called the weighted -Laplacian), a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. Under general hypotheses on the data, we prove the existence of a nontrivial solution to the elliptic obstacle problem, which depends on the first eigenvalue of the Steklov eigenvalue problem for the p-Laplacian. We also establish the boundedness and the closedness of the solution set to the problem. For the parameter-to-solution map, we give a convergence result of the Kuratowski type and prove the solvability of the inverse problem.

Study of a Coupled System with Sub-Strip and Multi-Valued Boundary Conditions via Topological Degree Theory on an Infinite Domain

The existence and uniqueness of solutions for a coupled system of Liouville–Caputo type fractional integro-differential equations with multi-point and sub-strip boundary conditions are investigated in this study. The fractional integro-differential equations contain a finite number of Riemann–Liouville fractional integral and non-integral type nonlinearities, as well as Caputo differential operators of various orders subject to fractional boundary conditions on an infinite interval. At the boundary conditions, we use sub-strip and multi-point contribution. There are various techniques to solve such type of differential equations and one of the most common is known as symmetry analysis. The symmetry analysis has widely been used in problems involving differential equations, although determining the symmetries can be computationally intensive compared to other methods. Therefore, we employ the degree theory due to the Mawhin involving measure of a non-compactness technique to arrive at our desired findings. An interesting pertinent problem has also been provided to demonstrate the applicability of our results.

Coupled Variational Inequalities: Existence, Stability and Optimal Control

In this paper, we introduce and investigate a new kind of coupled systems, called coupled variational inequalities, which consist of two elliptic mixed variational inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, we prove that the set of solutions for the coupled variational inequality (CVI, for short) under consideration is nonempty and weak compact. Then, two uniqueness theorems are delivered via using the monotonicity arguments, and a stability result for the solutions of CVI is proposed, through the perturbations of duality mappings. Furthermore, an optimal control problem governed by CVI is introduced, and a solvability result for the optimal control problem is established. Finally, to illustrate the applicability of the theoretical results, we study a coupled elliptic mixed boundary value system with nonlocal effect and multivalued boundary conditions, and a feedback control problem involving a least energy condition with respect to the control variable, respectively.

Existence and Multiplicity Results for Second-Order Nonlinear Differential Equations with Multivalued Boundary Conditions

In this paper, we consider the following second-order nonlinear differential equations’ problem: a.e on Φ=[0, T] with a discontinuous perturbation and multivalued boundary conditions. By combining lower and upper solutions method, theory of monotone operators and theory of topological degree, we show the existence of solutions of the investigated problem in two cases. At first, α andβ are assumed respectively an ordered pair of lower and upper solutions of the problem, secondly α and β are assumed respectively non ordered pair of lower and upper solutions of the problem. Moreover, we show multiplicity results when the problem admits a pair of lower and strict lower solutions and a pair of upper and strict upper solutions. We also show that our method of proof stays true for a periodic problem.

Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.

Extremal solutions for nonlinear second order differential inclusions

AbstractWe consider a nonlinear second order differential inclusion driven by the scalar p‐Laplacian and with nonlinear multivalued boundary conditions. Assuming the existence of an ordered pair of upper‐lower solutions and using truncation and penalization techniques together with Zorn's lemma, we show that the problem has extremal solutions in the order interval formed by the upper und lower solutions. We present some special cases of interest and show that our method applies also to the periodic problem. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary hemivariational inequality of parabolic type*1

In this paper we consider a parabolic hemivariational inequality with a nonmonotone multivalued boundary condition and with a time-dependent pseudomonotone operator. We establish the existence of solutions for such inequality. Our proof is based on a known surjectivity result for operators of pseudomonotone type. We present an example of semipermeability heat conduction problem to which the result applies.

Boundary hemivariational inequality of parabolic type

In this paper we consider a parabolic hemivariational inequality with a nonmonotone multivalued boundary condition and with a time-dependent pseudomonotone operator. We establish the existence of solutions for such inequality. Our proof is based on a known surjectivity result for operators of pseudomonotone type. We present an example of semipermeability heat conduction problem to which the result applies.

Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.

Strongly nonlinear multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the differential operator depending on both: unknown function x and its derivative x′, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and can be applied to the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the “convex” and “nonconvex” problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.

Nonlinear elliptic problems of Neumann-type

In this paper we study a nonlinear elliptic differential equation driven by thep-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem of the range of the sum of monotone operators, we prove the existence of a (strong) solution.

Nonlinear boundary value problems with maximal monotone terms

In this paper we consider a second order nonlinear differential equation with maximal monotone terms and nonlinear, possibly multivalued boundary conditions. Using the theory of maximal monotone operators and the Leray—Schauder alternative theorem we establish the existence of solutions. Our formulation is very general and includes as special cases the Dirichlet, Neumann and periodic problems.

Nonlinear Elliptic Problems of Neumann-Type

In this paper we study a nonlinear elliptic differential equation driven by the p-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem on the range of the sum of monotone operators, we prove the existence of a (strong) solution.

Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions

In this paper we consider a quasilinear second-order differential inclusion with a convex-valued multivalued term and nonlinear, multivalued boundary conditions. Using the Leray–Schauder fixed-point theorem and techniques from multivalued analysis and from nonlinear analysis, we prove the existence of a solution. Our formulation of the problem is general and includes as special cases the Dirichlet, the Neumann, the periodic problems, as well as certain Sturm–Liouville-type problems.

On a class of noncoercive hemivariational inequalities arising in nonlinear elasticity

The aim of this paper is to extend the results obtained in hyperelasticity by Ball 8,9, Ciarlet and Neĉas 17 and Gugat 30, 31 by considering a large class of nonmonotone possibly multivalued boundary conditions. Thus we are led to noncoercive hemivariational inequalities related to hyperelastic polyconvex materials

Stability of elastic bodies with nonmonotone multivalued boundary conditions of the quasidifferential type

Necessary conditions for the stability of elastic bodies subjected to nonmonotone multivalued boundary conditions are derived. These conditions are assumed to be derived from nonconvex and nonsmooth, quasidifferentiable energy functions. A ‘difference convex’ approximation of the potential energy function is written based on an appropriate quasidifferential formulation. Under appropriate assumptions for the convex and the concave parts we prove the existence of at least one nontrivial solution to the nonlinear eigenvalue problem.