Existence of solutions to the non-self-adjoint Sturm-Liouville problem with discontinuous nonlinearity (Q6588155)

In this paper authors study boundary value problem defined with non-self-adjoint operator \(L\)\N\[\NL(u)=-(p(x)u'(x))'+r(x)u'(x)+q(x)u(x)=\lambda g(x,u(x));a<x<b,\N\]\N\[\Nu(a)=u(b)=0,\N\]\Nwhere spectral parameters \(\lambda\) takes positive values, \(p\in C_{1,\alpha}([a,b]),q\in C_{1,\alpha}([a,b]),r\in C_{1,\alpha}([a,b])\) and \(r\neq0,-\infty<a<b<+\infty,0<\alpha<1.\) They assumed that \(g(x,0)=0\) a.e. on \((a,b)\) and the nonlinearity \(g(x,u)\) is a discontinuous function of the phase variable. Authors defined Strong and Semiregular solution of this BVP and give Theorems which is analog to Theorems in paper [\textit{V. N. Pavlenko} and \textit{D. K. Potapov}, Sb. Math. 206, No. 9, 1281--1298 (2015; Zbl 1333.35144); translation from Mat. Sb. 206, No. 9, 121--138 (2015)]. Also, they gave 4 examples that talk about the application of theoretical results.\N\NResult in this paper are new and non-trivial.