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A
generalization of Caristi’s theorem with applications to
nonlinear mapping theory
David Downing and William A. Kirk |
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Vol. 69 (1977), No. 2, 339–346
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Abstract |
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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
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Milestones
Received: 16 July 1976
Published: 1 April 1977
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Authors |
| David Downing | |
| William A. Kirk | |
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