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On some nonself mappings in Banach spaces. (English) Zbl 0671.47051

The main results of this paper are:
Theorem 3.1. Let X be a Banach space, and K a nonempty, closed subset of X,T:K\(\to X\) a mapping satisfying the condition Tx\(\in K\) for every \(x\in \partial K\) (the boundary of K), and \(\phi:{\mathbb{R}}^+\to {\mathbb{R}}^+\) an increasing continuous function satisfying \[ (1)\quad \phi (t)=0\quad if\quad and\quad only\quad if\quad t=0. \] Furthermore, let b and c be decreasing functions from \({\mathbb{R}}^+\setminus \{0\}\) into [0,1) such that \(2b(t)+c(t)<1\) for every \(t>0\). Suppose that T satisfies the condition \[ (2)\quad \phi (d(Tx,Ty))\leq \]
\[ b(d(x,y))\cdot \{\phi (d(x,Tx)+\phi (d(y,Ty))\}+c(d(x,y)).\min \{\phi (d(x,Ty)\quad,\quad \phi (d(y,Tx))\} \] \(\forall x\neq y\in X\). Then T has a unique fixed point.
Theorem 1 of N. A. Assad, Tamkang J. Math. 7, 91-94 (1976; Zbl 0356.47027) is a special case of Theorem 3.1, Theorem 3.1 generaizes Theorem 2 of M. S. Khan, M. Swaleh and S. Sessa, Bull. Aust. Math. Soc. 30, 1-9 (1984; Zbl 0553.54023). Theorem 4.1. Let X be a Banach space and K a nonempty compact subset of X. Let T:K\(\to X\) be a continuous mapping satisfying the condition that T(x)\(\in K\) for every \(x\in \partial K\) and \(\phi:{\mathbb{R}}^+\to {\mathbb{R}}^+\) an increasing continuous function satisfying property (1) of Theorem 3.1. Furthermore for all distinct x,y in K the inequality \[ (3)\quad \phi (d(Tx,Ty))<\frac{(1-c)}{2}\{\phi (d(x,Tx))+\phi (d(y,Ty))\}+c\cdot \min \{\quad \phi (d(x,Ty)),\quad \phi (d(y,Tx))\} \] holds, where \(0\leq c\leq 1\). Then T has a unique fixed point.
Theorem 4.1 generalizes the fixed point theorem of B. Fisher, Calcutta Math. Soc. 68, 265-266 (1976; Zbl 0378.54035) in case X is a Banach space.
Reviewer: V.Popa

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)

Keywords:

fixed point
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