On the singular impulsive functional integral equations involving nonlocal conditions
The authors consider a singular impulsive functional integral equation of the form \[ u(t)= T(t) H(u)+{1\over \Gamma(\alpha)} \int^t_0 (t-s)^{\alpha-1} T(t-s) F(s,u(s))\,ds \] for \(t\in[0, T]\), but \(t\) not equal to \(t_i\), and \(\Delta u(t_i)= I_i(u(t_i))\), \(i= 1,\dots,n\). Here \(T(t)\) is a compact semigroup on the Banach space \(X\), \(0<\alpha <1\), \(H\) is a nonlocal operator mapping \(\text{PC}([0,T]; X)\) into \(X\). (PC = piecewise continuous) and \(I_i: X\to X\), \(\Delta u(t_i)= u(t_i^+)- u(t_i^-)\), \(i= 1,\dots,n\). Under certain growth conditions on \(H\), \(I_i\) and \(F\) (including a Lipschitz condition on \(H\) but not on \(I_i\)), the authors prove the existence of a solution. The proof is based on Schauders fixed point theorem.
singular impulsive functional integral equation
scientific article; zbMATH DE number 6110921
scientific article