Fixed point theory on spaces with vector-valued metrics and application

Abstract

The purpose of this work is to prove some common fixed point theorems for two operators on a set endowed with one or two vector-valued metrics. The use of vector-valued metrics makes it possible for each equation of a system to have its own Lipschitz property, while the use of two such metrics makes it possible for the Lipschitz condition to be expressed with respect to an incomplete metric. An application is presented for a system of operatorial equations.

Keywords

• Fixed point,vector-valued metric,matrix convergent to zero

References

1. G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16, 201-206, 1973.
2. R. Kannan, Some remarks on fixed points, Bull. Calcutta Math. Soc. 60, 71-76, 1960.
3. D. O'Regan, R. Precup, Continuation theory for contractions on spaces with two vector- valued metrics, Applicable Analysis. 82, 131-144, 2003.
4. A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Dier. Uvavn. 2, 115-134, 1964.
5. R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Mathematical and Computer Modelling, 49, no. 3-4, 703-708, 2009.
6. S. Reich, Some results concerning contraction mappings, Canad. Math. Bull. 14, 121-124, 1971.
7. I. A. Rus, Principles and applications of the Fixed Point Theory, Dacia, Cluj-Napoca, Romania, 1979.
8. M. Turinici, Finite-dimensional vector contractions and their fixed points, Studia Universitatis Babeş-Bolyai. Mathematica, 35, no. 1, 30-42, 1990.

9. R. S. Varga, Matrix Iterative Analysis, vol. 27 of Springer Series in Computational Mathematics Springer, Berlin, Germany, 2000.