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In the present paper, we investigate a problem, describing the deformations process for the Stieltjes strings system located along a geometric star - shaped graph under the influence of an external force. The case when the force can be concentrated at separate points, including a node of the graph, is considered. The non-linear condition arises due to the presence of a limiter on the strings displacement in the node. Using variational methods, the necessary and sufficient conditions for the extremum of an energy functional are established; existence and uniqueness theorems for the solution are proved; an explicit formula for the solution is obtained; and the dependence solution on the length of the limiter is studied. [ABSTRACT FROM AUTHOR]

In this paper, we study a model of deformations of a beam with elastic supports and nonlinear boundary conditions under the influence of an external force. The case when the force can be concentrated at separate points is considered. The minimization problem of the potential energy functional with a constraint on the displacement of the beam end is investigated. The correctness of the model is established, and the necessary and sufficient conditions for the minimum of the potential energy functional are proved. [ABSTRACT FROM AUTHOR]

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]