Existence of solution of \(m\)-point boundary value problems of second-order differential systems with impulses
Here, the authors study the existence of solutions to the following \(m\)-point boundary value problem for second-order differential equations with impulses \[ \begin{aligned} (\rho'(t)x'(t))' &=f(t,x(t),x'(t)),\quad t \in [0,1], t \neq t_i,\\ x(t_i^+) - x(t_i^-) &= I_i(x(t_i),x'(t_i)),\quad i =1,\dots,k,\\ x'(t_i^+) - x'(t_i^-) &= J_i(x(t_i),x'(t_i)),\quad i =1,\dots,k,\\ x'(0)&=0,\qquad x(1)=\sum_{j=1}^{m-2}a_jx(\eta_j). \end{aligned} \] where \(\rho:[0,1]\to \mathbb{R}\) is a positive continuous differentiable function, \(f: [0,1]\times \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n,\) \(I_i, J_i: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n\) are continuous, \(k\) is a fixed positive integer, \(t_i\), \(i=1,2,\ldots,k\), are fixed points with \(0<t_1<t_2<\ldots<t_k<1,\) \(\eta_j\in (0,1)\), \(j=1,2,\ldots,m-2\), be given \(0<\eta_1<\eta_2<\ldots<\eta_{m-2}<1,\) \(\eta_j\neq t_i,\) and \(a_j, j=1,2,\ldots, m-2\), are nonnegative real constant numbers. The coincidence degree continuation theorem and the concept of autonomous curvature bound set are used.
scientific article; zbMATH DE number 1860987
scientific article