Second Order Problem with Nonlinear Boundary Conditions

Abstract

The chapter is devoted to the impulsive nonlinear boundary value problem

$$\begin{aligned} u''(t) = f(t,u(t),u'(t)) \quad \text {for a.e. }t \in [a,b] \subset \mathbb {R}, \end{aligned}$$

$$\begin{aligned} u(t_i+) = J_i(u(t_i-)),\quad u'(t_i+) = M_i(u'(t_i-)), \quad i = 1,\ldots ,p, \end{aligned}$$

$$\begin{aligned} g_1(u(a),u(b)) = 0, \quad g_2(u'(a),u'(b)) = 0, \end{aligned}$$

where $p\in \mathbb {N}$, $f \in \mathrm {Car}([a,b]\times \mathbb {R}^{2})$, $g_1$, $g_2 \in \mathbb {C}(\mathbb {R}^2)$, $J_i$, $M_i \in \mathbb {C}(\mathbb {R})$, $i=1,\ldots , p$. Impulses are considered at the fixed points $t_1,\ldots , t_p$, $a<t_1<\cdots <t_p<b$. We prove the solvability of the problem under the assumption that there exists a well-ordered pair of lower and upper functions associated with the problem. No growth restrictions are imposed on the functions f, $g_1$, $g_2$, $J_i$, $M_i$, $i=1,\ldots ,p$.