A remark on Rhoades fixed point theorem for non-self mappings (Q1210470)

Summary: Let \(X\) be a Banach space, \(K\) a non-empty closed subset of \(X\) and \(T: K\to X\) a mapping satisfying the contractive definition \[ d(Tx, Ty)\leq h\cdot \max \{d(x, y)/2, d(x, Tx), d(y, Ty), [d(x, Ty)+ d(y, Tx)]/q\},\tag{1.1} \] and the condition \(T(\partial K)\subseteq K\). Then \(T\) has a unique fixed point in \(K\). This result improves Theorem of \textit{B. E. Rhoades} [Math. Jap. 23, 457-459 (1978; Zbl 0396.47038)] and generalizes the corresponding theorem of \textit{N. A. Assad} [Tamkang J. Math. 7, 91-94 (1976; Zbl 0356.47027)].