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Variational methods are used to study a model of the deformation of a discontinuous Stieltjes string (a chain of strings held together by springs) located along the segment [0; l]. The model is described by the integro-differential equation pu0 m - (x)+ pu0 m - (0)+ R x 0 ud[Q] = F(x)F(0) with derivatives with respect to the measure m generated by a given strictly increasing function m(x) on the segment [0; l], where the function u(x) determines the deformation of the string, p(x) characterizes the elasticity of the string, the functions Q(x) and F(x) describe the elastic response of the external environment and the external load, respectively. The integral R x 0 ud[Q] is understood in the generalized sense according to Stieltjes. We are looking for solutions u(x) in the class of m-absolutely continuous functions on [0; l], whose derivatives have bounded variation on [0; l]. We assume that one of the boundary conditions is nonlinear and has the form p(l 0)u0 m (l 0)gu(l) 2 N[k;k]u(l); where N[k;k]u(l) denotes the outward normal cone at the point u(l) to the segment [k;k]. This condition arises due to the presence of the limiter [k;k] on the motion of the elastically fixed right end of the string (by a spring with elasticity g) so that ju(l)j = k. In this paper, necessary and sufficient conditions for the minimization of the energy functional of the Stieltjes string system are established, the critical loads at which the contact of the end of the string with the boundary points of the limiter occurs are determined, and the dependence of the solution on the length of the limiter is studied. [ABSTRACT FROM AUTHOR]

We provide a new result of existence of equilibria of a single-valued Lipschitz function on a compact set of which is locally the epigraph of a Lipschitz functions (such a set is called epilipschitz set). Equivalently this provides existence of fixed points of the map . The main point of our result lies in the fact that we do not impose that is an "inward vector" for all point of the boundary of . Some extensions of the existence of equilibria result are also discussed for continuous functions and set-valued maps.

In this paper, a model of deformations of a Stieltjes string with localized interactions with the environment (elastic supports and forces are concentrated at no more than countable set of points) is studied via variational methods. One of the boundary conditions is nonlinear and occurs due to a sleeve, which limits the movement of the right end of the string. The necessary and sufficient conditions for the extremum of the energy functional are established; the critical loads at which the contact of the end of the string with the sleeve occurs are determined; the dependence of the solution on the sleeve length is studied. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the initial boundary value problem describing oscillation process on a geometric graph with hysteresis type boundary conditions. The analogue of the d'Alembert formula is obtained. [ABSTRACT FROM AUTHOR]

In the present paper we investigate the initial - boundary value problems describing oscillation processes with boundary conditions of hysteresis type. Analogues of the d’Alembert formula are obtained. Some control problems are analysed and explicit forms of control functions are presented. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an Rδ-set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented. [ABSTRACT FROM AUTHOR]

We are considering the Cauchy problem for a semilinear fractional differential inclusion in a Banach space. By using the fixed point theory for condensing multivalued maps, we prove the local and global theorems of the existence of mild solutions to this problem. We verify the compactness of the solutions set and its continuous dependence on parameters and initial data. We demonstrate also the application of the averaging principle to the investigation of the continuous dependence of the solutions set on a parameter in the case when the right-hand side of the inclusion is rapidly oscillating. [ABSTRACT FROM AUTHOR]

This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation. [ABSTRACT FROM AUTHOR]