Mild solutions of nonlocal semilinear evolution equations on unbounded intervals via approximation solvability method in reflexive Banach spaces

We study the existence of mild solutions for a nonlocal semilinear evolution equation on unbounded interval by means of an approximation solvability method without assuming compactness on the evolution system and on the nonlinearity. The method is based on the reduction to a finite dimensional problem by means of the projections solved by a fixed point approach that includes a compactness criterion on $BC([0,+infty),X)$. Continuation principle and weak topology are used as well.