Multivalued Operators Research Articles

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.

A multifield variational formulation of viscoplasticity suitable to deal with singularities and non-smooth functions

In the present paper a multifield variational formulation of non-smooth viscoplasticity is illustrated which is well suited to deal with multivalued operators and non-smooth functions characterized by singularities. The adopted approach proves to be helpful in describing a geometrical framework of the non-smooth viscoplastic problem and in guiding to enhancements towards other kinds of plastic and viscoplastic material behavior. Alternative equivalent expressions of evolutive laws are discussed. The formulation shows to include Koiter’s theory as a special case. In addition, a multifield variational formulation of non-smooth viscoplasticity is illustrated. A procedure for deriving a parent potential in all the unknown variables is illustrated for non-smooth viscoplasticity problems. From the parent potential a set of associated multifield potentials is furtherly derived in one or more unknown fields and the relevant extremum properties are illustrated so that the non-smooth viscoplasticity structural problem is provided with a complete variational formulation suitable for developments in finite element applications.

On some applications of the controllability principle for fixed point equations

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka–Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for ϕ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

Existence of Urysohn and Atangana-Baleanu Fractional Integral Inclusion Systems Solutions Via Common Fixed Point of Multi-Valued Operators

Existence of Urysohn and Atangana-Baleanu Fractional Integral Inclusion Systems Solutions Via Common Fixed Point of Multi-Valued Operators

Existence of Urysohn and Atangana-Baleanu Fractional Integral Inclusion Systems Solutions Via Common Fixed Point of Multi-Valued Operators

Existence of Urysohn and Atangana-Baleanu Fractional Integral Inclusion Systems Solutions Via Common Fixed Point of Multi-Valued Operators

Second order evolutionary partial differential variational-like inequalities

<abstract><p>This work is dedicated to the study of a second order evolutionary partial differential variational-like inequality in Banach space. We obtain the mild solution of our problem by applying the concept of strongly continuous cosine family of bounded linear operators, fixed point theorem for condensing multi-valued operators and measure of non-compactness. It is proved that the solution set of mixed variational-like inequalities is non-empty, bounded, closed and convex.</p></abstract>

Approximate controllability of neutral delay integro-differential inclusion of order $ \alpha\in (1, 2) $ with non-instantaneous impulses

<p style='text-indent:20px;'>This paper aims to establish the approximate controllability results for fractional neutral integro-differential inclusions with non-instantaneous impulse and infinite delay. Sufficient conditions for approximate controllability have been established for the proposed control problem. The tools for study include the fixed point theorem for discontinuous multi-valued operators with the <inline-formula><tex-math id="M3">\begin{document}$ \alpha- $\end{document}</tex-math></inline-formula>resolvent operator. Finally, the proposed results are illustrated with the help of an example.</p>

Multi-valued variational inequalities in unbounded domains: existence, comparison and extremal solutions

In this paper, we study multi-valued quasilinear elliptic variational inequalities of the form in all of as well as in the exterior domain , where is the p-Laplacian, K is a closed convex subset of the Beppo-Levi space with 1<p<N, or with p = N, and is the indicator functional corresponding to K with its subdifferential . The lower order multi-valued operator is generated by a multi-valued, upper semicontinuous function , and the measurable coefficient a is supposed to decay like: Our main goals are as follows. First, we provide a new approach to an existence theory for the above multi-valued variational inequalities in all for 1<p<N as well as in the exterior domain Ω for the borderline case . Second, we establish an enclosure and comparison principle based on appropriately defined sub-supersolutions, and prove the existence of extremal solutions. Third, by means of the sub-supersolution method provided here, we show that certain rather general classes of variational-hemivariational type inequalities turn out to be only subclasses of the above class of multi-valued elliptic variational inequalities. Finally, we apply the abstract theory to a multi-valued obstacle problem.

Minimization arguments in analysis of variational–hemivariational inequalities

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

On a fractional cantilever beam model in the q-difference inclusion settings via special multi-valued operators

The fundamental goal of the study under consideration is to establish some of the existence criteria needed for a particular fractional inclusion model of cantilever beam in the setting of quantum calculus using new arguments of existence theory. In this way, we investigate a fractional integral equation that corresponds to the aforementioned boundary value problem. In a more concrete sense, we design new multi-valued operators based on this integral equation, which belong to the certain subclasses of functions, called α-admissible and α-ψ-contractive multi-functions, in combination with the AEP-property. Also, we use some inequalities such as Ω-inequality and set-valued version inequalities. Moreover, we add a simulative example for a numerical analysis of our results obtained in this study.

Existence Results for ψ -Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions

In this research work, we study a new class of ψ -Hilfer hybrid fractional integro-differential boundary value problems with nonlocal boundary conditions. Existence results are established for single and multivalued cases, by using suitable fixed-point theorems for the product of two single or multivalued operators. Examples illustrating the main results are also constructed.

Application of some special operators on the analysis of a new generalized fractional Navier problem in the context of q-calculus

The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t. q-RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital α-admissible and α-ψ-contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.

Fixed Point, Data Dependence, and Well-Posed Problems for Multivalued Nonlinear Contractions

The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature.

Formulation and existence of weak solutions for a problem of adhesive contact with elastoplasticity and hardening

This paper presents the weak formulation of a quasi-static evolution model for two deformable bodies with uni-directional adhesive unilateral contact on which external loads act. Small deformations and linearized elastoplasticity with hardening are assumed. The adhesion component is rate-dependent or rate-independent according to the choice of the viscosity coefficient of the glue; elastoplasticity is considered rate-independent. The weak formulation is expressed as a doubly non-linear problem with unbounded multivalued operators, as a function of internal and boundary displacements, plastic and symmetric strain tensors, and the bonding field and its gradient. This paper differs from other formulations by coupling the equations defined inside and on the boundary of the solids in functional form. In addition to this novelty, we verify the existence of solutions by a path other than that displayed in similar articles. The existence of solutions is demonstrated after considering a succession of doubly non-linear problems with an unbounded operator, and verifying that the solution of one of the problems is also a solution to the objective problem. The proof is supported by previous results from non-linear Partial differential equations theory with monotone operators.

Positive Solvability for Conjugate Fractional Differential Inclusion of (k, n − k) Type without Continuity and Compactness

The monotonicity of multi-valued operators serves as a guideline to prove the existence of the results in this article. This theory focuses on the existence of solutions without continuity and compactness conditions. We study these results for the (k,n−k) conjugate fractional differential inclusion type with λ&gt;0,1≤k≤n−1.

Frum-Ketkov type multivalued operators

Let (M,d) be a metric space, X\subset M be a nonempty closed subset and K\subset M be a nonempty compact subset. By definition, an upper semi-continuous multivalued operator F:X\to P(X) is said to be a strong Frum-Ketkov type operator if there exists \alpha\in ]0,1[ such that e_d(F(x),K)\le \alpha D_d(x,K), for every x\in X, where e_d is the excess functional generated by d and D_d is the distance from a point to a set. In this paper, we will study the fixed points of strong Frum-Ketkov type multivalued operators.

Existence of solutions for implicit obstacle problems involving nonhomogeneous partial differential operators and multivalued terms

In this article, we study an implicit obstacle problem with a nonlinear nonhomogeneous partial differential operator and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, and employing Kluge's fixed point principle for multivalued operators, Minty echnique and a surjectivity theorem, we prove that the set of weak solutions to the problem is nonempty, bounded and weakly closed.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/37/abstr.html

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.

Some decomposition for multivalued matrix pencils linear operator

Some decomposition for multivalued matrix pencils linear operator

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., \(\emptyset \ne w\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n=s\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n\subset \mathcal S\).