Nonlinear impulsive differential inclusions with integral boundary conditions
The paper is concerned with the existence of solutions for the boundary value problem \[ \begin{cases} y''(t)\in F(t,y(t)),&\text{a.e.}\quad t\in [0,1], t\neq t_i, i=1,2,\ldots, m,\\ \Delta y|_{t=t_{i}}=I_{i}(y(t_i^{-})), &i=1,2,\dots,m, \\ \Delta y'|_{t=t_{i}}=\overline{I_{i}}(y(t_{i}^{-})), &i=1,2,\dots,m,\\ y(0)-k_{1}y'(0)=\int_{0}^{1}h_{1}(s,y(s))ds,\\ y(1)+k_{2}y'(1)=\int_{0}^{1}h_{2}(s,y(s))ds, \end{cases} \] where \(F:[0,1]\times {\mathbb R}\rightarrow {\mathcal P}({\mathbb R})\) is a multivalued map, \({\mathcal P}({\mathbb R})\) is the family of all subsets of \({\mathbb R}\), \(h_{1},h_{2}: [0,1]\times {\mathbb R}\rightarrow {\mathbb R}\) are continuous functions, \(I_i,\overline{I_i}: {\mathbb R}\rightarrow {\mathbb R}\) are given functions, \(0=t_{0}<t_{1}<\cdots<t_m<t_{m+1}=1\), \(k_1,k_{2}\) are nonnegative constants, \(\Delta y|_{t=t_{i}}=y(t_{i}^{+})-y(t_{i}^{-})\), \(\Delta y'|_{t=t_i}=y'(t_{i}^{+})-y'(t_{i}^{-})\), \(y(t_{i}^+)=\lim_{h\rightarrow0^+}y(t_i+h)\) and \(y(t_{i}^-)=\lim_{h\rightarrow0^+}y(t_i-h)\) are the right- and left-hand limits of \(y(t)\) at \(t=t_{i}\), respectively. Both the cases of convex and nonconvex valued right-hand sides are studied, via fixed point theorems.
scientific article; zbMATH DE number 5547394