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General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces. (English) Zbl 1199.54209

This article deals with two generalizations of the classical Banach-Caccioppoli fixed point principle for contractions in complete metric spaces. In the first of them, the author considers operators T:XX, where (X,d) is a complete metric space with a metric d, satisfying the following condition

d(Tx,Ty)kM(x,y)+Ld(y,Tx),x,yX,
(M(x,y)=max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}),

where 0(0,1), L0. It is proved that Fix(T), the convergence of the Picard iteration x n+1 =Tx n to some x * Fix(T) for any x 0 X, and the estimate

d(x n ,x * )k n (1-k) 2 d(x 0 ,Tx 0 ),n=0,1,2,;(1)

(note that T can have more than one fixed point). In the second generalization, the author considers operators T:XX satisfying the following condition

d(Tx,Ty)kM(x,y)+Ld(x,Tx),x,yX,

where also 0(0,1), L0 and M(x,y) is defined by the same formula. It is proved that T has a unique fixed point x * , the Picard iteration x n+1 =Tx n converges to x * for any x 0 X and the estimate (1) holds, and, moreover, the following estimate

d(x n+1 ,x * )kd(x n ,x * ),n=0,1,2,,

also holds. In the end of the article, the author gives some illustrative examples and some remarks about relations between new generalizations of the Banach-Caccioppoli fixed point principle and early versions of such generalizations.


MSC:
54H25Fixed-point and coincidence theorems in topological spaces