A generalization of Caristi’s theorem with applications to nonlinear mapping theory
David J. Downing, W. A. Kirk · Pacific Journal of Mathematics · 1977
Suppose X and Y are complete metric spaces, g: X-+X an arbitrary mapping:, and f:X->Y a closed mapping (thus, for {xJczX the conditions x n -+x and f(x n )->y imply f(x) = y).It is shown that if there exists a lower semicontinuous function φ mapping f(X) into the nonnegative real numbers and a constant c > 0 such that for all x in X, max {d(x, g(x)), cd(f(x), f(g(x))}^φ(f(x))-φ(f(9(x))), then g has a fixed point in X.This theorem is then used to prove surjectivity theorems for nonlinear closed mappings /: X -» F, where X and Y are Banach spaces.