Vol. 69, No. 2, 1977

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A generalization of Caristi’s theorem with applications to nonlinear mapping theory

David Downing and William A. Kirk

Vol. 69 (1977), No. 2, 339–346

Suppose X and Y are complete metric spaces, g : X X an arbitrary mapping, and f : X Y a closed mapping (thus, for {xn}⊂ X the conditions xn x and f(xn) 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(g(x))), then g has a fixed point in X. This theorem is then used to prove surjectivity theorems for nonlinear closed mappings f : X Y , where X and Y are Banach spaces.

Mathematical Subject Classification 2000
Primary: 47H10
Secondary: 54H25
Received: 16 July 1976
Published: 1 April 1977
David Downing
William A. Kirk