Solvability of a linear boundary value problem for a Fredholm integro-differential equation with impulsive inputs
For \(T>0\) consider an arbitrary partition \(0=\theta_0<\theta_1<\dots<\theta_m<\theta_{m+1}=T\) by means of so-called impulsive input points. The author studies the boundary value problem for Fredholm integro-differential equations of the form \(\frac{dx(t)}{dt}=A(t)x(t)+\int_0^T K(t,s)x(s)ds+f(t)\) with the boundary conditions \(B_0 x(0)+C_0 x(T)=d_0\in {\mathbb R}^n\) and \(B_j x(\theta_j-0)+C_j x(\theta_j+0)=d_j\in{\mathbb R}^n\), where \(j=1,\dots,m\) and \(t\in(0,T)\setminus\{\theta_j:j=1,\dots,m\}\), \(A\) and \(K\) are continuous \(n\times n\) matrix functions on \([0,T]\) and \([0,T]^2\), respectively, and \(f\) is a piecewise continuous vector function with possible discontinuities at the points \(\theta_j\), \(j=1,\dots,m\). Necessary and sufficient conditions for the solvability and the unique solvability of the above boundary value problem are obtained.
scientific article; zbMATH DE number 6526891