A nonlocal boundary value problem with singularities in phase variables
Sufficient conditions for the solvability are obtained for the second-order nonlinear boundary value problem \[ (g(x'(t)))'=f(t,x(t)),\;x'(t)),\quad x(0)=x(T)=-\gamma\min\{x(t): t\in[0,T]\}. \] Here, \(g\) is a continuous increasing odd function. The right-hand side \(f(t,x,x')>0\) may be singular both at \(x=0\) and \(x'=0\). The number \(\gamma\) is positive. The operation of minimum makes the boundary value problem nonlocal. Boundary conditions that fix the minimal and maximal values of a solution to a second-order differential equation were considered by \textit{S. A. Brykalov} [Differ. Equations 29, No.~6, 802--805 (1993; Zbl 0826.34020)].
singular nonlinear second-order differential equation
scientific article; zbMATH DE number 2137062
scientific article