Existence of solutions for second order impulsive differential equations
The author investigates the existence of solutions of the following periodic boundary value problem for second order differential equations with impulses at fixed moments: \[ x''(t)= f(t,x(t),x'(t)), \] \[ t\neq t_{1}; \quad x(t_{1}^{+})=I(x(t_{1}),x'(t_{1}^{-})); \quad x'(t_{1}^{+})=x'(t_{1}^{-}); \quad x(0)=x(T); \quad x'(0)=x'(T), \] where \(t_{1}\in (0,T)\), and \(f:[0,T]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\), \(I:\mathbb{R}^{2}\rightarrow \mathbb{R}\) are continuous functions. An equivalent abstract equation to this problem is found and then an existence result is proved by using topological degree theory. The author also gives an example to illustrate the main result of the paper.
scientific article; zbMATH DE number 1078265