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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 $\phi :{ℝ}^{+}\to {ℝ}^{+}$ an increasing continuous function satisfying

$\left(1\right)\phantom{\rule{1.em}{0ex}}\phi \left(t\right)=0\phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}only\phantom{\rule{1.em}{0ex}}if\phantom{\rule{1.em}{0ex}}t=0·$

Furthermore, let b and c be decreasing functions from ${ℝ}^{+}\setminus \left\{0\right\}$ into [0,1) such that $2b\left(t\right)+c\left(t\right)<1$ for every $t>0$. Suppose that T satisfies the condition

$\left(2\right)\phantom{\rule{1.em}{0ex}}\phi \left(d\left(Tx,Ty\right)\right)\le$
$b\left(d\left(x,y\right)\right)·\left\{\phi \left(d\left(x,Tx\right)+\phi \left(d\left(y,Ty\right)\right)\right\}+c\left(d\left(x,y\right)\right)·min\left\{\phi \left(d\left(x,Ty\right)\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}\phi \left(d\left(y,Tx\right)\right)\right\}$

$\forall x\ne y\in X$. 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 $\phi :{ℝ}^{+}\to {ℝ}^{+}$ an increasing continuous function satisfying property (1) of Theorem 3.1. Furthermore for all distinct x,y in K the inequality

$\left(3\right)\phantom{\rule{1.em}{0ex}}\phi \left(d\left(Tx,Ty\right)\right)<\frac{\left(1-c\right)}{2}\left\{\phi \left(d\left(x,Tx\right)\right)+\phi \left(d\left(y,Ty\right)\right)\right\}+c·min\left\{\phantom{\rule{1.em}{0ex}}\phi \left(d\left(x,Ty\right)\right),\phantom{\rule{1.em}{0ex}}\phi \left(d\left(y,Tx\right)\right)\right\}$

holds, where $0\le c\le 1$. 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 for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces
fixed point