Multivalued Monotone Operators Research Articles

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.

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

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.

New iterative regularization methods for solving split variational inclusion problems

<p style='text-indent:20px;'>The paper proposes some new iterative algorithms for solving a split variational inclusion problem involving maximally monotone multi-valued operators in a Hilbert space. The algorithms are constructed around the resolvent of operator and the regularization technique to get the strong convergence. Some stepsize rules are incorporated to allow the algorithms to work easily. An application of the proposed algorithms to split feasibility problems is also studied. The computational performance of the new algorithms in comparison with others is shown by some numerical experiments.</p>

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.

Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

In this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.

The sum theorem for maximal monotone operators in reflexive Banach spaces revisited

The goal of this note is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.

Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian

In this paper, we study an elliptic obstacle problem with a generalized fractional Laplacian and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, we employ a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping to prove that the set of weak solutions to the problem is nonempty, bounded and closed. Then, we introduce a sequence of penalized problems without obstacle constraints. Finally, we prove that the Kuratowski upper limit of the sets of solutions to penalized problems is nonempty and is contained in the set of solutions to original elliptic obstacle problem, i.e., \(\emptyset \ne w\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n=s\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n\subset \mathcal S\).

Existence of solutions for double phase obstacle problems with multivalued convection term

The main goal of this paper is the study of an elliptic obstacle problem with a double phase phenomena and a multivalued reaction term which also depends on the gradient of the solution. Such term is called multivalued convection term. Under quite general assumptions on the data, we prove that the set of weak solutions to our problem is nonempty, bounded and closed. Our proof is based on a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.

Homogenization of multivalued monotone operators with variable growth exponent

<p style='text-indent:20px;'>We consider the Dirichlet problem for an elliptic multivalued maximal monotone operator <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal A}_\varepsilon $\end{document}</tex-math></inline-formula> satisfying growth estimates of power type with a variable exponent. This exponent <inline-formula><tex-math id="M2">\begin{document}$ p_\varepsilon(x) $\end{document}</tex-math></inline-formula> and also the symbol of the operator <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal A}_\varepsilon $\end{document}</tex-math></inline-formula> oscillate with a small period <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> with respect to the space variable <inline-formula><tex-math id="M5">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We prove a homogenization result for this problem.

Fixed points of multi-valued monotone operators and the solvability of a fractional integral inclusion

AbstractBased on the characterizations of reproducing cones, some fixed point theorems for multi-valued increasing, decreasing, and mixed monotone operators are established. As an application, the solvability of a fractional integral inclusion is discussed.

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Abstract Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X⊃D(T) → 2X* is a maximal monotone multi-valued operator and C: X⊃D(C) → X* is a generalized pseudomonotone quasibounded operator with L ⊂ D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x, with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.

Implicit eigenvalue problems for maximal monotone operators

We study the implicit eigenvalue problem of the form where T is a maximal monotone multi-valued operator and the operator C satisfies condition ( ) or ( ). In a regularization method by the duality operator, we use the degree theories of Kartsatos and Skrypnik upon conditions of C as well as Browder’s degree. There are two cases to consider: One is that C is demicontinuous and bounded with condition ( ); and the other is that C is quasibounded and densely defined with condition ( ). Moreover, the eigenvalue problem is also discussed.

Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments

We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays.

Some eigenvalue results for maximal monotone operators

We study the eigenvalue problem of the form 0 ∈ T x − λ C x , where X is a real reflexive Banach space with its dual X ∗ and T : X ⊃ D ( T ) → 2 X ∗ is a maximal monotone multi-valued operator and C : D ( T ) → X ∗ is a not necessarily continuous single-valued operator. Using the index theory for countably condensing operators, we extend some related results of Kartsatos to the countably condensing case instead of compactness of the approximant J μ . Moreover, the solvability of the perturbed problem 0 ∈ T x + C x is discussed in an analogous method to the above problem.

Duality and subdifferential for convex functions on complete [formula omitted] metric spaces

Thanks to the recent concept of quasilinearization of Berg and Nikolaev, we have introduced the notion of duality and subdifferential on complete C A T ( 0 ) (Hadamard) spaces. For a Hadamard space X , its dual is a metric space X ∗ which strictly separates non-empty, disjoint, convex closed subsets of X , provided that one of them is compact. If f : X → ( − ∞ , + ∞ ] is a proper, lower semicontinuous, convex function, then the subdifferential ∂ f : X ⇉ X ∗ is defined as a multivalued monotone operator such that, for any y ∈ X there exists some x ∈ X with x y ⃗ ∈ ∂ f ( x ) . When X is a Hilbert space, it is a classical fact that R ( I + ∂ f ) = X . Using a Fenchel conjugacy-like concept, we show that the approximate subdifferential ∂ ϵ f ( x ) is non-empty, for any ϵ > 0 and any x in efficient domain of f . Our results generalize duality and subdifferential of convex functions in Hilbert spaces.

Existence and uniqueness of fixed points for mixed monotone multivalued operators in Banach spaces

In this paper, the existence and approximation of fixed points for two classes of systems of mixed monotone (downward and upward) multivalued operators are discussed. We present some new fixed point theorems of mixed monotone(downward and upward)operators which need not be continuous and compact. We also indicate the condition to ensure the uniqueness of the fixed point. At last we get some applications ofour theorems.

Fixed points for mixed monotone multivalued operators in Banach spaces with applications

In this paper, the existence and iterative approximation of fixed points for a class of systems of mixed monotone multivalued operator are discussed. We present some new fixed point theorems of mixed monotone operators and increasing operators which need not be continuous or satisfy a compactness condition. We also give some applications to differential inclusions with discontinuous right hand side in Banach spaces and to Hammerstein integral inclusions on R N .

Existence of solutions to initial value problems for the second order mixed monotone type of impulsive differential inclusions

In this paper we consider the initial value problems for the second order mixed monotone type of impulsive differential inclusions. By introducing a special partial order, we present the existence of maximal and minimal fixed points for mixed monotone multivalued operators in Banach spaces. Applying the results, we establish the existence and uniqueness of above impulsive differential inclusions.

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.