Existence of solutions for nonlinear singular $q$-Sturm–Liouville problems

In this paper, we study a nonlinear $q$-Sturm–Liouville problem on the semi-infinite interval, in which the limit-circle case holds at infinity for the $q$-Sturm–Liouville expression. This problem is considered in the Hilbert space $L_{q}^{2}\left( 0,\infty\right)$. We study this problem by using a special way of imposing boundary conditions at infinity. In the work, we recall some necessary fundamental concepts of quantum calculus such as $q$-derivative, the Jackson $q$-integration, the $q$-Wronskian, the maximal operator, etc. We construct the Green function associated with the problem and reduce it to a fixed point problem. Applying the classical Banach fixed point theorem, we prove the existence and uniqueness of the solutions for this problem. We obtain an existence theorem without the uniqueness of the solution. In order to get this result, we use the well-known Schauder fixed point theorem.