Multivalued Differential Equations Research Articles

Conformable evolution inclusions with causal operators

In this paper, we study local and nonlocal problems of conformable semilinear evolution inclusions with causal operators, which include time lag system and a large classes of parabolic multivalued differential equations. We first investigate the problems with the help of measure of noncompactness and afterwards when the right-hand side is Lipschitz.

Differential equations with maximal monotone operators

The paper deals with multivalued differential equations in abstract spaces. Nonlocal conditions are assumed. The model includes an m-dissipative multioperator which generates an equicontinuous, not necessarily compact, semigroup. The regularity of the nonlinear term also depends on the Hausdorff measure of noncompactness. The existence of integral solutions is discussed, with a topological index argument. A transversality condition is required. The results are applied to a partial differential inclusion in a bounded domain in Rn with nonlocal integral conditions. The model also includes an m-dissipative but not necessarily compact semigroup generated by a suitable subdifferential operator.

Conditions for the Existence of Basic Solutions of Linear Multivalued Differential Equations

Conditions for the Existence of Basic Solutions of Linear Multivalued Differential Equations

Controllability of Semilinear Multi-Valued Differential Inclusions with Non-Instantaneous Impulses of Order α ∈ (1,2) without Compactness

Herein, we investigated the controllability of a semilinear multi-valued differential equation with non-instantaneous impulses of order α∈(1,2), where the linear part is a strongly continuous cosine family without compactness. We did not assume any compactness conditions on either the semi-group, the multi-valued function, or the inverse of the controllability operator, which is different from the previous literature.

State dependent nonconvex sweeping processes in smooth Banach spaces

In the setting of 2-uniformly smooth and q -uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation: -\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \text{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)} This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here N^C(C(t,u(t)); u(t)) stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on V -uniformly generalized prox-regular sets in Banach spaces.

On a Multivalued Differential Equation with Nonlocality in Time

The initial value problem for a multivalued differential equation is studied, which is governed by the sum of a monotone, hemicontinuous, coercive operator fulfilling a certain growth condition and a Volterra integral operator in time of convolution type with exponential decay. The two operators act on different Banach spaces where one is not embedded in the other. The set-valued right-hand side is measurable and satisfies certain continuity and growth conditions. Existence of a solution is shown via a generalisation of the Kakutani fixed-point theorem.

Stationary Oberbeck–Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations

A generalization of the Oberbeck–Boussinesq model consisting of a system of steady state multivalued partial differential equations for incompressible, generalized Newtonian of the p-power type, viscous flow coupled with the nonlinear heat equation is studied in a bounded domain. The existence of a weak solution is proved by combining the surjectivity method for operator inclusions and a fixed point technique.

A non-convex setup for multivalued differential equations driven by oblique subgradients

The paper proves the existence and uniqueness of the solution for the following multivalued deterministic variational inequality with oblique subgradients, considered in a non-convex domain: {x′(t)+H(t,x(t))∂−φ(x(t))∋g(t,x(t)),t≥0,x(0)=x0, where ∂−φ stands for the Fréchet subdifferential of the semiconvex function φ and the matrix application x↦H(⋅,x) is a Lipschitz mapping. The presence of the oblique reflection brought by the term H∂−φ leads to the use of different techniques comparing to the cases of standard reflection in non-convex domains or oblique reflection in convex domains. The last section of the article is focused on the qualitative analysis of a non-convex Skorohod problem, with generalized reflection and, as applications, stochastic variational inequalities driven by oblique Fréchet subgradients are addressed.

Multivalued fractional differential equations as a model for an impact of two bodies

Impact models of two colliding bodies are usually generated by fundamental physical and geometrical principles. Initially, the number of unknown variables is not the same as the number of equations expressing the principles, so the models should be complemented with an appropriate set of constitutive equations which contain enough information on the physical properties of the system and therefore allow accurate predictions of its behavior. Along these lines we here discuss a problem when a block, moving along a line on a dry surface, impinges against another block being at rest, through a deformable straight rod of negligible mass. Among the variety of all possible choices that can be used, we suggest the constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and the Coulomb friction law given in the set-valued form. Owing to the presence of dry friction and the proposed fractional model, known as the fractional Zener model, the problem belongs to the class of set-valued fractional differential equations (or multivalued differential equations of arbitrary real order) leading to the equivalent Cauchy problem given in terms of two coupled integro-differential inclusions, for which the existence result ensuring the contractible solution set exists. By use of the combinatorial analysis of the problem, we identify 11 imaginable scripts and separate 10 feasible ones that were confirmed numerically by a procedure that combines nonlocal and nonsmooth modules.

Homogenization of Multivalued Partial Differential Equations via Reflected Backward Stochastic Differential Equations

We study the limit of the solution of multivalued semi-linear Partial Differential Equations (PDEs for short) involving a second order differential operator of parabolic type where the non-linear term is a function of the solution, not of its gradient. Our basic tool is the approach given by Pardoux [Pardoux, E. Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In Stochastic Analysis and Related Topics: The Geilo Workshop, 1996; Decreusefond, L., Gjerde, J., Oksendal, B., Ustüunel, A.S., Eds.; Birkhäuser, 1998; 79–127] and Ouknine [Ouknine, Y. Reflected BSDE with jumps. Stoch. Stoch. Reports 65, 111–125]. In particular, we use the weak convergence of an associated reflected Backward Stochastic Differential Equation (BSDE for short) involving the subdifferential operator of a lower semi-continuous, proper and convex function. An homogenization property for solutions of semi-linear PDEs in Sobolev spaces is also proved.

Averaging of multivalued differential equations

Nonlinear multivalued differential equations with slow and fast subsystems are considered. Under transitivity conditions on the fast subsystem, the slow subsystem can be approximated by an averaged multivalued differential equation. The approximation in the Hausdorff sense is of order O(ϵ1/3) as ϵ → 0.

Initial and boundary value problems for nonconvex valued multivalued functional differential equations

In this note we investigate the existence of solutions of initial and boundary value problems defined on a compact interval to some classes of functional differential inclusions. We shall rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler and Schafer′s theorem combined with lower semicontinuous multivalued operators with decomposable values.

Extremal solutions of classes of multivalued differential equations

This paper is concerned with the existence of generalized extremal solutions for classes of multivalued boundary value problems. The results obtained generalize and improve the recent results obtained by Cichon [Ann. Soc. Math. Polonae 17 (1992) 11], Ibrahim [Int. J. Math. Math. Sci. (USA) 21 (1) (1998) 165], Papageorgiou [J. Math. Anal. Appl. 185 (1) (1994) 146], Tolstonogov [Soviet Math. Dokl. 43 (2) (1991) 481] and Truong [Bollettino U.M.I 7 (8-B) (1994) 349].

Multivalued differential equations in banach spaces and their applications

This paper studies the existence of solutions to a class of multivalued differential equations by using a surjectivity result for multivalued (S+) type mappings. The authors then apply their results to evolution hemivariational inequalities and parabolic equations with nonmonotone discontinuities, which generalize and extend previously known theorems.

Multivalued differential equations in Banach spaces

New existence principles and results are presented for differential inclusions in Banach spaces.

A concept of invariance and attractivity for multivalued differential equations

In this work, we shall introduce as well as study the problem of selective invariance and attractivity of sets given by multi-valued differential equations. Not as usual, from the control theory point of view, but here, we discuss it from the differential games view point from which this problem can be interpreted as a qualitative problem. One of our results can be applied to the game of pursuit and evasion in some formulations.

A Bilinear Control Problem Solved by the Fractional Step Scheme

We consider a control problem for which the cost functional consists of locally Lipschitz functions and is governed by a bilinear multivalued differential equation. We study this problem using the fractional step scheme. Two particular examples are solved using the proposed method.

A note on multivalued differential equations on proximate retracts

This paper discusses viable solutions for differential inclusions in Banach spaces. Existence will be established in two steps. In step 1, a nonlinear alternative of Leray‐Schauder type [8] for maps with closed graphs will be used to establish a variety of existence principles for the Cauchy differential inclusion. Step 2 involves using the results in step 1 together with some tricks involving the Bouligand cone (and sometimes the Urysohn function) so that new existence criteria can be established for multivalued differential equations on proximate retracts.

Existence theorems for nonlinear evolution inclusions

In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u′∈−∂−f(u)+G(u), u(O)=x0, where ∂−f is the Frechet subdifferential of a functionf defined on an open subset Ω of a real separable Hilbert space H, taking its values in R ∪ {+∞} and G is a multifunction from C([0, T], Ω) into the nonempty subsets of L2([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x′∈−∂−f(x)+F(t, x), x(0)=x0, where F:[0, T]×Ω→(H) is a multifunction satisfying some regularity assumptions.

Matrix method for solving multivalued logic differential equations

A method to solve logic Differential equations, i.e. equations containing Logic Derivatives of k-valued logic (MVL) functions is proposed. An initial Differential equation is represented by a system of k logic equations of k variables given as 0-polarity Reed-Muller canonical expansion. This system is solved by means of a truncated orthogonal transform algorithm.