Class Of Inclusions Research Articles

General variational inclusions involving difference of operators

In this paper, we introduce and consider a new class of general variational inclusions involving the difference of operators in a Hilbert space. We establish the equivalence between the general variational inclusions and the fixed point problems as well as with a new class of resolvent equations using the resolvent operator technique. We use this alternative formulation to discuss the existence of a solution of the general variational inclusions. We again use this alternative equivalent formulation to suggest and analyze a number of iterative methods for finding a zero of the difference of operators. We also discuss the convergence of the iterative method under suitable conditions. Our methods of proofs are very simple as compared with other techniques. Several special cases of these problems are also considered. The results proved in this paper may be viewed as a refinement and an improvement of the known results in this area.

Existence of Mild Solutions for Impulsive Fractional Stochastic Differential Inclusions with State-Dependent Delay

We study the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependent delay. Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan. An example is given to illustrate the theory.

Stabilization and Finite Time Stabilization of Nonlinear Differential Inclusions Based on Control Lyapunov Function

This paper deals with the stabilization and finite time stabilization of a class of nonlinear differential inclusions (NDI). First, we present the definitions of control Lyapunov function, exponentially stable control Lyapunov function (CLF) and finite time stable CLF for NDI. Then, we design controllers to make the systems asymptotically stable, exponentially stable, and finite time stable, respectively. Finally, an example of discontinuous system is given to show the effectiveness of the designed controllers.

Well-posedness of non-autonomous evolutionary inclusions

A class of non-autonomous differential inclusions in a Hilbert space setting is considered. The well-posedness for this class is shown by establishing the mappings involved as maximal monotone relations. Moreover, the causality of the so established solution operator is addressed. The results are exemplified by the equations of thermoplasticity with time dependent coefficients and by a non-autonomous version of the equations of viscoplasticity with internal variables.

Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with infinite delays in Hilbert spaces. The results are obtained by using the fixed-point theorem for multi-valued operators due to Dhage. An example is provided to illustrate the theory.MSC:93B05, 93E03.

Approximate Controllability of Fractional Neutral Stochastic Functional Integro-Differential Inclusions with Infinite Delay

In this paper, we investigate the approximate controllability for a class of fractional neutral stochastic functional integro-differential inclusions involving the Caputo derivative in Hilbert spaces. A new set of sufficient conditions are formulated and proved for the approximate controllability of fractional stochastic integro-differential inclusions under the assumption that the associated linear part of system is approximately controllable. The main techniques rely on the fractional calculus, operator semigroups and Bohnenblust–Karlin’s fixed point theorem. An example is given to illustrate the obtained theory.

Existence of anti-periodic solutions for a class of first order evolution inclusions

The existence of anti-periodic solutions for a class of first order nonlinear evolution inclusions defined in the framework of an evolution triple of spaces is considered. We study the problems under both convexity and nonconvexity conditions on the multivalued right-hand side. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions, the surjectivity result for L-pseudomonotone operators and continuous extreme selection results from multivalued analysis. An example of a nonlinear parabolic problem is given to illustrate our results.

Existence and multiplicity of solutions for second-order impulsive differential inclusions

In this paper we consider a class of a second-order impulsive differential inclusions. Using a variational method based on the non-smooth critical point theory, we prove the existence and multiplicity of anti-periodic solutions.

$\rm BV$ solutions of rate independent differential inclusions

We consider a class of evolution differential inclusions defining the so-called stop operator arising in elastoplasticity, ferromagnetism, and phase transitions. These differential inclusions depend on a constraint which is represented by a convex set that is called the characteristic set. For $\rm BV$ (bounded variation) data we compare different notions of $\rm BV$ solutions and study how the continuity properties of the solution operators are related to the characteristic set. In the finite-dimensional case we also give a geometric characterization of the cases when these kinds of solutions coincide for left continuous inputs.

Periodic Solutions for Semilinear Fourth-Order Differential Inclusions via Nonsmooth Critical Point Theory

Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended.

Control-Sharing and Merging Control Lyapunov Functions

Given two control Lyapunov functions (CLFs), a “merging” is a new CLF whose gradient is a positive combination of the gradients of the two parents CLFs. The merging function is an important trade-off since this new function may, for instance, approximate one of the two parents functions close to the origin, while being close to the other far away. For nonlinear control-affine systems, some equivalence properties are shown between the control-sharing property, i.e., the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given CLFs, and the existence of merging CLFs. It is shown that, even for linear time-invariant systems, the control-sharing property does not always hold, with the remarkable exception of planar systems. The class of linear differential inclusions is also discussed and similar equivalence results are presented. For this class of systems, linear matrix inequalities conditions are provided to guarantee the control-sharing property. Finally, a constructive procedure, based on the recently considered “R-functions,” is defined to merge two smooth positively homogeneous CLFs.

On asymptotics of solutions for a class of functional differential inclusions

On asymptotics of solutions for a class of functional differential inclusions

Existence of solutions for a class of nonconvex differential inclusions

We prove the existence of solutions to the functional differential inclusion of the form where is a Carathéodory function and an upper semicontinuous multifunction with compact values in a Hilbert space such that , with a regular locally Lipschitz function and a linear operator, and .

Convergent Dynamics of Nonreciprocal Differential Variational Inequalities Modeling Neural Networks

The paper addresses convergence of solutions for a class of differential inclusions termed differential variational inequalities (DVIs). Each DVI describes the dynamics of a neural network (NN) evolving in a closed hypercube of \BBR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and defined by a continuously differentiable, cooperative and (possibly) nonreciprocal vector field f. The main result in the paper is that under a new condition on f, which is called strong Kamke-Müller (SKM) condition, the solution semiflow generated by the DVI is strongly order preserving (SOP) and hence it satisfies a Limit Set Dichotomy and enjoys generic convergence properties. A characterization of the SKM condition is given in terms of the interconnection properties of the Jacobian matrix of f. In the case where f is an affine, or a linear, vector field the considered DVIs include two relevant classes of NNs, namely, the linear systems operating on a closed hypercube, also known as linear systems in saturated mode (LSSMs), and the full-range (FR) model of cellular neural networks (CNNs). By applying the results to LSSMs it is obtained that any cooperative LSSM with a (possibly) nonsymmetric and fully interconnected matrix is generically convergent. Analogous results hold for FRCNNs. All the obtained convergence results hold in the general case where the DVIs, and the LSSMs and FRCNNs, possess multiple equilibrium points.

Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces

In ordered Banach spaces, characterizations of ordered -weak-ANODD set-valued mappings are introduced and studied, which is applied to giving an approximate solution for a new class of general nonlinear mixed-order quasi-variational inclusions involving ⊕ operator. By using the resolvent operator associated with an -weak-ANODD set-valued mapping and fixed point theory, an existence theorem of solutions and an approximation algorithm for this kind of inclusions are established and discussed in ordered Banach spaces, and the relation between the first-valued point and the solution of the problems is shown. The results obtained seem to be general in nature. MSC:49J40, 47H06.

Approximate controllability of fractional nonlinear differential inclusions

Fractional differential equations have wide applications in both physical and social sciences. This paper addresses the issue of approximate controllability for a class of fractional nonlinear differential inclusions in Banach spaces. A new set of sufficient conditions are formulated and proved for the approximate controllability of fractional nonlinear differential inclusions. In particular, the results are established with the assumption that the associated linear part of system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of "bang---bang" principle for control systems is given as well by utilizing the relaxation theorem.

Impulsive Problems for Fractional Partial Neutral Functional Integro-Differential Inclusions with Infinite Delay and Analytic Resolvent Operators

In this paper, the existence of mild solutions for a class of impulsive fractional partial neutral functional integro-differential inclusions with infinite delay and analytic α-resolvent operators in Banach spaces is investigated. Sufficient conditions for the existence are derived with the help of the fixed-point theorem for discontinuous multi-valued operators due to Dhage and the fractional power of operators combined with approximation techniques. An example is provided to illustrate the theory.

Infinitely many solutions for a class of differential inclusions involving the $p$-biharmonic

The existence of infinitely many solutions for differential inclusions depending on two positive parameters and involving the $p$-biharmonic operator is established via variational methods.

Total Absorption of Electromagnetic Waves in Ultimately Thin Layers

We consider single-layer arrays of electrically small lossy bi-anisotropic particles that completely absorb electromagnetic waves at normal incidence. Required conditions for electromagnetic properties of bi-anisotropic particles have been identified in the most general case of uniaxial reciprocal and nonreciprocal particles. We consider the design possibilities offered by the particles of all four fundamental classes of bianisotropic inclusions: reciprocal chiral and omega particles and nonreciprocal Tellegen and moving particles. We also study the reflection/transmission properties of asymmetric structures with different properties when illuminated from the opposite sides of the sheet. It has been found that it is possible to realize single-layer grids which exhibit the total absorption property when illuminated from one side but are totally transparent when illuminated from the other side (an ultimately thin isolator). Other possible properties are co-polarized or twist polarized reflection from the side opposite to the absorbing one. Finally, we discuss possible approaches to practical realization of particles with the properties required for single-layer perfect absorbers and other proposed devices.