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In this paper we consider stochastic evolution equations on UMD-Banach spaces. In a recent paper we proved existence of optimal controls. Here in this paper we develop necessary conditions of optimality whereby one can construct the optimal controls. For illustration we use these results to treat the LQR problem in sufficient details under two sets of alternative and distinct assumptions. [ABSTRACT FROM AUTHOR]

The paper is devoted to properties of generalized set-valued stochastic integrals defined in [10]. These integrals generalize set-valued stochastic in- tegrals defined by E.J. Jung and J.H. Kim in the paper [4]. Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals de- fined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H. Kim, are not in- tegrably bounded. Generalized set-valued stochastic integrals, considered in the paper, are in some non-trivial cases square integrably bounded and can be applied in the theory of stochastic differential equations with set-valued solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the question of optimal control for a class of stochastic evolution equations on infinite dimensional Hilbert spaces with controls appearing in both the drift and the diffusion operators. We con- sider relaxed controls (measure valued random processes) and briefly present some results on the question of existence of mild solutions including their regularity followed by a result on existence of partially observed optimal relaxed controls. Then we develop the necessary conditions of optimality for partially observed relaxed controls. This is the main topic of this paper. Further we present an algorithm for computation of optimal policies followed by a brief discussion on regular versus relaxed controls. The paper is con- cluded by an example of a non-convex problem which is readily solvable by our approach. [ABSTRACT FROM AUTHOR]

This paper is devoted to singular calculus of variations problems with constraint functional not regular at the solution point in the sense that the first derivative is not surjective. In the first part of the paper we pursue an approach based on the constructions of the p-regularity theory. For p-regular calculus of variations problem we formulate and prove necessary and sufficient conditions for optimality in singular case and illustrate our results by classical example of calculus of variations problem. [ABSTRACT FROM AUTHOR]

In this paper, we use the upper and lower solutions method combined with a fixed point theorem for multivalued maps in Banach algebras due to Dhage for investigations of the existence of solutions of a class of discontinuous partial differential inclusions with not instantaneous impulses. Also, we study the existence of extremal solutions under Lipschitz, Carathéodory and certain monotonicity conditions. [ABSTRACT FROM AUTHOR]

In this paper we consider the question of existence of measure valued solutions for neutral differential equations on Banach spaces when there is no mild solutions. We prove the existence of measure solutions and their regularity properties. We consider also control problems of such systems and prove existence of optimal feedback controls for some interesting a-typical control problems. [ABSTRACT FROM AUTHOR]

This paper studies a new class of nonlocal boundary value problems of nonlinear differential equations and inclusions of fractional order with fractional integral boundary conditions. Some new existence results are obtained by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and approximate controllability for impulsive fractional-order stochastic infinite delay integro-differential equations in Hilbert space. By using Krasnoselskii's fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of impulsive fractional stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

This article studies a boundary value problem of nonlinear fractional differential inclusions with anti-periodic type integral boundary conditions. Some existence results are obtained via fixed point theorems. The cases of convex-valued and nonconvex-valued right hand sides are considered. Several new results appear as a special case of the results of this paper. [ABSTRACT FROM AUTHOR]

In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C0-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control problems. We present results on existence of optimal feedback operators. [ABSTRACT FROM AUTHOR]

In this paper we prove existence theorems for integro - differential equations xΔ(t) = f(t, x(t), ∫ 0 t k(t, s, x(s))Δs), x(0) = x0 t ∊ Ia = [0, a] ∩ T, a ∊ R+, where T denotes a time scale (nonempty closed subset of real numbers R), Ia is a time scale interval. Functions f, k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale. [ABSTRACT FROM AUTHOR]

In this paper we consider the random fuzzy differential equations and show their application by an example. Under suitable conditions the Peano type theorem on existence of solutions is proved. For our purposes, a notion of ϵ-solution is exploited. [ABSTRACT FROM AUTHOR]

This paper deals with the formulation of the necessary optimality condition for a topology optimization problem of an elastic body in unilateral contact with a rigid foundation. In the contact problem of Tresca, a given friction is governed by an elliptic variational inequality of the second order. The optimization problem consists in finding such topology of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The topological derivative of the cost functional is calculated and a necessary optimality condition is formulated. The calculated topological derivative is also used in the numerical algorithm to find a descent direction by inserting voids in the domain occupied by the body. Numerical examples are provided. [ABSTRACT FROM AUTHOR]

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ1 > 0 of (-Δp;W1,p 0 (Z)). We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal. [ABSTRACT FROM AUTHOR]

Penalty methods, augmented Lagrangian methods and Nitsche mortaring are well known numerical methods among the specialists in the related areas optimization and finite elements, respectively, but common aspects are rarely available. The aim of the present paper is to describe these methods from a unifying optimization perspective and to highlight some common features of them. [ABSTRACT FROM AUTHOR]

In the paper a class of families F(M) of functions defined on differentiable manifolds M with the following properties: 1F. if M is a linear manifold, then F(M) contains convex functions, 2F. F(·) is invariant under diffeomorphisms, 3F. each f ∈ F(M) is differentiable on a dense Gδ-set, is investigated. [ABSTRACT FROM AUTHOR]

In this paper we establish an estimation for the rate of pointwise convergence of the Chlodovsky-Kantorovich polynomials for functions f locally integrable on the interval [0;∞). In particular, corresponding estimation for functions f measurable and locally bounded on [0;∞) is presented, too. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type (Multiple line equation(s) cannot be represented in ASCII text), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·, ·) is measurable. The corresponding single-valued problem will be considered first. [ABSTRACT FROM AUTHOR]

In this paper we consider the first order difference equation in a Banach space (Multiple line equation(s) cannot be represented in ASCII text). We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation (Multiple line equation(s) cannot be represented in ASCII text) and give conditions when this equation has solutions. In this note we extend the results from [8, 9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness. [ABSTRACT FROM AUTHOR]

In this paper, some fixed point principle is applied to prove the existence of solutions for delay second order differential inclusions with three-point boundary conditions in the context of a separable Banach space. A topological property of the solutions set is also established. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reexive Banach space with dual V*: The system is given by dx + A(dt; x) = f(t; x)(dt) + B(t)u(dt); x(0) = ε, t ϵ I ≡ [0; T] where A is a strongly nonlinear operator valued measure mapping Σ x V to V* with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I x H to H, γ is a countably additive bounded positive measure and the control μ is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point theorems for single valued mappings satisfying an implicit relation on space with two metrics. In addition we give a homotopy result using our theorems. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are given for the existence of solutions for a class of second order stochastic differential inclusions in Hilbert space with the help of Leray-Schauder Nonlinear Alternative. [ABSTRACT FROM AUTHOR]