The authors give results dealing with the following theorem of Krasnoselskii. Theorem: Let \(C\) be a closed bounded convex nonempty subset of a Banach space \(X\). Suppose that \(Ax + By\) lies in \(C\) for all \(x,y\) in \(C\), where \(A\) is continuous and \(A(C)\) is contained in a compact set, \(B\) is a contraction mapping with constant \(k\), \(0 < k < 1\). Then there is an \(x\) in \(C\) with \(x = Ax + Bx\).
In the present paper, two other possibilities are discussed and interesting results with corollaries are proved. In the end, the problem of the existence of solutions for a system of equations is discussed by using Schaefer’s theorem.