Abstract
We present a condition that guarantees the existence and uniqueness of fixed (or best proximity) points in complete metric space (or uniformly convex Banach spaces) for a wide class of cyclic maps, called p–cyclic summing maps. These results generalize some known results from fixed point theory. We find a priori and a posteriori error estimates of the fixed (or best proximity) point for the Picard iteration associated with the investigated class of maps, provided that the modulus of convexity of the underlying space is of power type. We illustrate the results with some applications and examples.
Highlights
Introduction and PreliminariesBanach contraction principle and its numerous generalizations turn out to be a powerful tool in mathematical research
It is well known that a plentiful number of contractive-type maps that are known to have fixed points can be generalized to ensure the existence of best proximity points
A reason for this may be that the modulus of convexity is greater in the direction of the best proximity point ξ than in the other directions, but, for the estimation of the error, we do not use it
Summary
Banach contraction principle and its numerous generalizations turn out to be a powerful tool in mathematical research. A condition that is completely different from the known ones and which warrants the existence and uniqueness of the best proximity points and for the cases when the distances between them are not equal is considered in [19] These new types of maps were named p–cyclic summing contraction maps, Mathematics 2020, 8, 1060; doi:10.3390/math8071060 www.mdpi.com/journal/mathematics. Error estimates about fixed points for self (or cyclic) maps, starting with the classical Banach contraction principle, some resent results from this year e.g., [21,22] and the approximations of fixed points in [23,24], for example, are one of the greatest advantages in the applications of the fixed points technique. Main Result—We define the notion of p–cyclic summing contraction map and we state and prove that any such map has a unique best proximity point and we obtain error estimates, when a sequence of successive iterations is used. Conclusions—We discuss some open problems and possible future generalizations
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