Generalized Caristi's fixed point theorems
Using the concept of \(w\)-distance [\textit{O. Kada, T. Suzuki} and \textit{W. Takahashi}, Math. Jap. 44, No.~2, 381--391 (1996; Zbl 0897.54029)], the following generalization of Caristi's fixed point theorem is proved: Let \(X\) be a complete metric space, \(\psi: X \rightarrow [0, \infty)\) be a lower semicontinuous function and \(w\) be a \(w\)-distance on \(X\). Let \(g:X\rightarrow (0, \infty)\) be a function such that for some \(r>0\), \[ \sup\{g(x):x\in X, \psi(x)\leq\inf_{z\in X}\psi(z)+r\}<\infty. \] Let \(T: X\rightarrow 2^X\) be a multivalued map such that for each \(x\in X\) there exists \(y\in T(x)\) satisfying \[ w(x,y)\leq g(x)(\psi(x)-\psi(y)). \] Then \(T\) has a fixed point \(x_0\in X\) such that \(w(x_0, x_0)=0.\)
scientific article; zbMATH DE number 5574352