## Mikhail Kamenskii, Paul Raynaud de Fitte, Ngai-Ching Wong, Margarita Zvereva, A model of deformations of a discontinuous Stieltjes string with a nonlinear boundary condition

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DOI: 10.23952/jnva.5.2021.5.08

Volume 5, Issue 5, 1 October 2021, Pages 737-759

Abstract. 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 $-\left(pu^\prime_\mu\right)(x)+\left(pu^\prime_\mu\right)(0)+ \int_0^xud[Q]=F(x)-F(0)$ with derivatives with respect to the measure $\mu$ generated by a given strictly increasing function $\mu(x)$ 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 $\int_0 ^xud [Q]$ is understood in the generalized sense according to Stieltjes. We are looking for solutions $u(x)$ in the class of $\mu$-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 $-p(l-0)u'_\mu (l-0) -\gamma u(l)$ $\in N _ {[- k, k]}u (l),$ 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 $\gamma$) so that $|u(l)|\leq k$. 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.