Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses

Abstract

In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order $ p $-Laplacian differential equations with instantaneous and non-instantaneous impulses: $ \begin{equation*} \left\{ {\begin{array}{l} -(\rho(t)\Phi_{p} (u'(t)))'+g(t)\Phi_{p}(u(t))=\lambda f_{j}(t,u(t)),\quad t\in(s_{j},t_{j+1}],\; j=0,1,...,m,\\ \Delta (\rho (t_{j})\Phi_{p}(u'(t_{j})))=\mu I_{j}(u(t_{j})), \quad j=1,2,...,m,\\ \rho (t)\Phi_{p} (u'(t))=\rho(t_{j}^{+}) \Phi_{p} (u'(t_{j}^{+})),\quad t\in(t_{j},s_{j}],\; j=1,2,...,m,\\ \rho(s_{j}^{+})\Phi_{p} (u'(s_{j}^{+}))=\rho(s_{j}^{-})\Phi_{p} (u'(s_{j}^{-})),\quad j=1,2,...,m,\\ u(0)=0, \quad u(1)=\zeta u(\eta), \end{array}} \right. \end{equation*} $ where $ \Phi_{p}(u): = |u|^{p-2}u, \; p > 1, \; 0 = s_{0} < t_{1} < s_{1} < t_{2} < ... < s_{m_{1}} < t_{m_{1}+1} = \eta < ... < s_{m} < t_{m+1} = 1, \; \zeta > 0, \; 0 < \eta < 1 $, $ \Delta (\rho (t_{j})\Phi_{p}(u'(t_{j}))) = \rho (t_{j}^{+})\Phi_{p}(u'(t_{j}^{+}))-\rho (t_{j}^{-})\Phi_{p}(u'(t_{j}^{-})) $ for $ u'(t_{j}^{\pm}) = \lim\limits_{t\to t_{j}^{\pm}}u'(t) $, $ j = 1, 2, ..., m $, and $ f_{j}\in C((s_{j}, t_{j+1}]\times\mathbb{R}, \mathbb{R}) $, $ I_{j}\in C(\mathbb{R}, \mathbb{R}) $. $ \lambda\in (0, +\infty) $, $ \mu\in\mathbb{R} $ are two parameters. $ \rho(t)\geq 1 $, $ 1\leq g(t)\leq c $ for $ t\in (s_{j}, t_{j+1}] $, $ \rho(t), \; g(t)\in L^{p}[0, 1] $, and $ c $ is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results.

Keywords

• three-point boundary value problem
• variational method
• critical points theorem
• instantaneous impulse
• non-instantaneous impulse