Dirichlet Problem with Time Singularities

Abstract

The chapter deals with a singular Dirichlet problem having fixed-time impulses which has the form

$$\begin{aligned} u''(t) + f(t,u(t),u'(t)) = 0 \quad \quad \text {for a.e. }t \in [0,T]\subset \mathbb {R}, \end{aligned}$$

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

$$\begin{aligned} u(0) = A, \quad u(T) = B, \end{aligned}$$

where $p\in \mathbb {N}$, $0 < t_1 < \cdots < t_p < T$, $A,B\in \mathbb {R}$, $f \in \mathrm {Car}_{\mathrm {loc}}((0,T)\times \mathbb {R}^{2})$, f has time singularities at $t = 0$ and $t = T$, $J_i$, $M_i \in \mathbb {C}(\mathbb {R})$, $i=1,\dots , p$. We prove the existence of a solution to this problem under the assumption that there exist lower and upper functions associated with the problem. The solution has continuous first derivative also at the singular points $t =0$ and $t = T$. Our proofs are based on the regularization technique and on the method of a priori estimates.