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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:KX a mapping satisfying the condition TxK for every xK (the boundary of K), and ϕ: + + an increasing continuous function satisfying

(1)ϕ(t)=0ifandonlyift=0·

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

(2)ϕ(d(Tx,Ty))
b(d(x,y))·{ϕ(d(x,Tx)+ϕ(d(y,Ty))}+c(d(x,y))·min{ϕ(d(x,Ty),ϕ(d(y,Tx))}

xyX. 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:KX be a continuous mapping satisfying the condition that T(x)K for every xK and ϕ: + + an increasing continuous function satisfying property (1) of Theorem 3.1. Furthermore for all distinct x,y in K the inequality

(3)ϕ(d(Tx,Ty))<(1-c) 2{ϕ(d(x,Tx))+ϕ(d(y,Ty))}+c·min{ϕ(d(x,Ty)),ϕ(d(y,Tx))}

holds, where 0c1. 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:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
Keywords:
fixed point