Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

Continuous first-order methods for monotone inclusions in a Hilbert space

Equations in a Hilbert space that involve multivalued monotone mappings are examined. Solutions to such equations are understood in the inclusion sense. A continuous first-order method and its regularized version are constructed on the basis of the resolvent of the maximal monotone operator, and sufficient conditions for them to converge strongly are obtained.     

Approximate controllability of differential inclusions in Hilbert spaces

In this paper, controllability for the system originating from semilinear functional differential equations in Hilbert spaces is studied. We consider the problem of approximate controllability of semilinear differential inclusion assuming that semigroup, generated by the linear part of the inclusion, is compact and under the assumption that the corresponding linear system is approximately controllable. By using resolvent of controllability Gramian operator and fixed point theorem, sufficient conditions have been formulated and proved. An example is presented to illustrate the utility and applicability of the proposed method.     

Iterative processes of the second order for monotone inclusions in a Hilbert space

We study equations with multiple-valued operators in a Hilbert space. We understand their solutions in the sense of inclusion. We reduce such equations to mixed variational inequalities or to equations with single-valued operators. For constructed problems we propose implicit iterative processes of the second order and establish sufficient conditions for their strong convergence.     

Localization of the method of guiding functions in the problem about periodic solutions of differential inclusions

In this paper we consider the problem on the existence of forced oscillations in nonlinear objects governed by differential inclusions. We propose certain modifications of the methods of generalized and integral guiding functions.     

A zero sum differential game in a Hilbert space

We study a zero sum differential game of fixed duration in a separable Hilbert space. We prove a minimax principle and establish the equivalence between the dynamic programming principle and the existence of a saddle point equilibrium. We also prove sufficient conditions for optimality.     

Existence results for impulsive neutral stochastic evolution inclusions in Hilbert space

This paper is concerned with the existence of mild solutions of a class of impulsive neutral stochastic evolution inclusions in Hilbert space in the case where the right hand side is convex or nonconvex-valued. The results are obtained by using two fixed point theorems for multivalued mappings and evolution system theory.     

Norm continuity and stability for a functional differential equation in Hilbert space

A functional differential equation in Hilbert space with initial data on [−h,0] is considered. An unbounded operator A and a square integrable weight function are acting in the distributed delay term. For a not necessarily continuous weight function the norm continuity of the associated solution semigroup is established at every t>h. In the case that the spectrum of A is real and negative, the asymptotic stability of the solution is obtained.     

Lyapunov functionals for certain linear delay differential equations in a Hilbert space

A theory of stability via Lyapunov functionals is developed for a general class of autonomous delay differential equation whose values lie in a Hilbert space.     

Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space

In this paper the existence of mild solutions of a class of semilinear stochastic evolution inclusions with delays in a Hilbert space is studied. The results are obtained by using a fixed point theorem for condensing map due to Martelli. The result is illustrated with an example.     

Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions

We suggest new methods for the solution of a periodic problem for a nonlinear object described by the differential inclusion x′(t) ∈ F(t, x_t) under the assumption that the multimapping F has convex compact values and satisfies the upper Carathéodory conditions. We also study the case in which this multimapping is not convex-valued but is normal. The class of normal multimappings includes, for example, bounded almost lower semicontinuous multimappings with compact values and mappings satisfying the Carathéodory conditions. In both cases, a generalized integral guiding function is used to study the problem.     

Stability conditions for abstract functional differential equations in Hilbert space

Stability conditions for functional differential equations of the form: du(t)/dt = Au(t) + bAu(t ? h) + (a* Au)(t) are studied, where A is the infinitesimal generator of an analytic semigroup in a Hilbert space, b ≠ 0 and the convolution term contains a square integrable real function a ≠ 0. Norm discontinuity of the solution semigroup of the equation with discrete delay is avoided by studying the inverse of the characteristic operator. Sufficient and necessary conditions for the uniform exponential stability of the solution semigroup are obtained. The results are applied to a retarded partial integrodifferential equation.     

On the existence of solutions for random differential inclusions in a Banach space

We prove two existence theorems for random differential inclusions defined in a separable Banach space. One is about differential inclusions defined on all of the Banach space X and the other for differential inclusion defined on a closed convex subset K. Both theorems are proved through the use of analogous deterministic results, which we also include, and techniques from the theory of measurable multifunctions.