Existence and uniqueness of nonlinear impulsive integro-differential equations
The authors study the existence and uniqueness of mild and classical solutions for the following impulsive integro-differential equation \[ u'(t)=Au(t)+f\biggl(t,u(t),\int_{0}^{t}k(t,s)u(s)\,ds\biggr), \quad t\in [0,T_{0}], \;t\not=t_{i}, \] \[ u(0)=u_{0}, \] \[ \Delta u(t_{i})=I_{i}(u(t_{i})), \quad i=1,\ldots,p, \] where \(f:[0,T_{0}]\times X\times X\to X\) is a given function, \(A:D(A)\to X\) is an unbounded operator which generates a strongly continuous semigroup \(T(t), \;t\geq 0, \;k: [0,T_{0}]\times [0,T_{0}]\to R\) a continuous function, \(I_{i}:X\to X\), \(\Delta u(t_{i})=u(t_{i}^{+})-u(t_{i}^{-})\) represents the jump of the state \(u\) at time \(t_{i}\), and \(X\) is a Banach space. The proofs are based upon the semigroup theory and the Banach contraction principle. An example illustrating the abstract theory is missing.
scientific article; zbMATH DE number 5121915