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

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:

$$\{\begin{array}{l}-(\rho (t){\mathrm{\Phi }}_p(u^{\prime }(t)){)}^{\prime }+g(t){\mathrm{\Phi }}_p(u(t))=\lambda f_j(t,u(t)),t\in (s_j,t_{j+1}],j=0,1,...,m,\\ \mathrm{\Delta }(\rho (t_j){\mathrm{\Phi }}_p(u^{\prime }(t_j)))=\mu I_j(u(t_j)),j=1,2,...,m,\\ \rho (t){\mathrm{\Phi }}_p(u^{\prime }(t))=\rho (t_j^{+}){\mathrm{\Phi }}_p(u^{\prime }(t_j^{+})),t\in (t_j,s_j],j=1,2,...,m,\\ \rho (s_j^{+}){\mathrm{\Phi }}_p(u^{\prime }(s_j^{+}))=\rho (s_j^{-}){\mathrm{\Phi }}_p(u^{\prime }(s_j^{-})),j=1,2,...,m,\\ u(0)=0,u(1)=\zeta u(\eta ),\end{array}$$

where ${\mathrm{\Phi }}_p(u):=|u{|}^{p-2}u,p>1,0=s_0<t_1<s_1<t_2<...<s_{m_1}<t_{m_1+1}=\eta <...<s_m<t_{m+1}=1,\zeta >0,0<\eta <1$, $\mathrm{\Delta }(\rho (t_j){\mathrm{\Phi }}_p(u^{\prime }(t_j)))=\rho (t_j^{+}){\mathrm{\Phi }}_p(u^{\prime }(t_j^{+}))-\rho (t_j^{-}){\mathrm{\Phi }}_p(u^{\prime }(t_j^{-}))$ for $u^{\prime }(t_j^{\pm })=\underset{t\to t_j^{\pm }}{lim}{u}^{\prime }(t)$, $j=1,2,...,m$, and $f_j\in C((s_j,t_{j+1}]\times \mathbb{R},\mathbb{R})$, $I_j\in C(\mathbb{R},\mathbb{R})$. $\lambda \in (0,+\mathrm{\infty })$, $\mu \in \mathbb{R}$ are two parameters. $\rho (t)\ge 1$, $1\le g(t)\le c$ 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.