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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 $\phi:{\bbfR}\sp+\to {\bbfR}\sp+$ 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 ${\bbfR}\sp+\setminus \{0\}$ into [0,1) such that $2b(t)+c(t)<1$ for every $t>0$. Suppose that T satisfies the condition $$ (2)\quad \phi (d(Tx,Ty))\le$$ $$ b(d(x,y))\cdot \{\phi (d(x,Tx)+\phi (d(y,Ty))\}+c(d(x,y)).\min \{\phi (d(x,Ty)\quad,\quad \phi (d(y,Tx))\} $$ $\forall x\ne y\in X$. Then T has a unique fixed point. Theorem 1 of {\it 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 {\it M. S. Khan}, {\it M. Swaleh} and {\it 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 $\phi:{\bbfR}\sp+\to {\bbfR}\sp+$ 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 $0\le c\le 1$. Then T has a unique fixed point. Theorem 4.1 generalizes the fixed point theorem of {\it B. Fisher}, Calcutta Math. Soc. 68, 265-266 (1976; Zbl 0378.54035) in case X is a Banach space.
Reviewer: V.Popa

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
fixed point