Abstract:
In this paper, we consider a boundary value problem with a nonlinear boundary condition and discontinuous solutions. This problem models the deformation process of a discontinuous Stieltjes string (a chain of Stieltjes strings connected by springs) under the action of an external load. The shape of the string is described by an integro-differential equation with a derivative with respect to the measure and with a generalized Stieltjes integral. This representation allows us to analyze both solutions and relations at each point. We assume that there is an obstacle at the left end of the chain. Depending on the applied external force, the corresponding end of the chain either touches the boundary points of the obstacle or remains free. This creates a nonlinear boundary condition, since it is not known in advance how the solution will behave. The existence and uniqueness theorems of the solution are proved, a formula for the representation of the solution is obtained, loads at which the end of the chain touches the obstacle are found, and the dependence of the solution on the size of the obstacle is studied.
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