Existence and uniqueness of solutions for a second-order iterative boundary-value problem

Abstract

We consider the existence and uniqueness of solutions to the second-order iterative boundary-value problem $$ x''(t) = f(t, x(t), x^{[2]}(t)), \quad a \leq t \leq b, $$ where $x^{[2]}(t) = x(x(t))$, with solutions satisfying one of the boundary conditions $x(a) = a$, $x(b) = b$ or $x(a) = b$, $x(b) = a$. The main tool employed to establish our results is the Schauder fixed point theorem.

Keywords

• Iterative differential equation
• Schauder fixed point theorem
• contraction mapping principle