Existence and uniqueness of solutions of a fourth-order boundary value problem with non-homogeneous boundary conditions

Abstract

Let $m \ge 2$ and $a, b, c > 0$. We consider the existence and uniqueness of solutions for the fourth order iterative boundary value problem, \begin{equation*} x^{(4)}(t) = -f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)), \qquad -a \le t \le a \end{equation*} \noindent where $x^{[2]}(t)$ =$x \big(x(t) \big)$ and for $j = 3,\dots, m$, $x^{[j]}(t)$ = $x \big( x^{[j-1]}(t) \big)$, with solutions satisfying one of the following sets of conjugate boundary conditions: \begin{equation*} \begin{aligned} x(-a) &= -a, &\quad x'(-a)& = b,& \quad x''(-a) &= c,& \quad x(a) &= a,\\ x(-a) &= -a, &\quad x(a) &= a,& \quad x'(a) &= b,& \quad x''(a)& = c. \end{aligned} \end{equation*} The main tool used is the Schauder fixed point theorem.

Keywords

• differential equations
• iterative differential equations
• boundary value problems