History, Development And Application of
Pseudocontractive Mapping With Fixed Point
Theory

Chetan Kumar Sahu; Dr. S. C. Srivastava; Dr. S. Biswas

doi:https://doi.org/10.14445/22315373/IJMTT-V66I4P503

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 4 | Year 2020 | Article Id. IJMTT-V66I4P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I4P503

History, Development And Application of Pseudocontractive Mapping With Fixed Point Theory

Chetan Kumar Sahu, Dr. S. C. Srivastava, Dr. S. Biswas

Citation :

Chetan Kumar Sahu, Dr. S. C. Srivastava, Dr. S. Biswas, "History, Development And Application of Pseudocontractive Mapping With Fixed Point Theory," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 4, pp. 13-16, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I4P503

Abstract

In this Paper, Brief history, origin and relevant role of Pseudocontractive mapping and Fixed point theory in Mathematics with the help of some definitions, theorems and types of this theory is presented. This article is to make accessible material that might be of interest to students and research scholars. Some important results from beginning up to now are incorporated in this paper. The theory of fixed point with Pseudo contractive mapping is one of the most powerful tool of modern mathematical analysis. Theorem concerning the existence and properties of fixed points are known as fixed point theorem. Fixed point theory with Pseudocontractive is a beautiful mixture of analysis, topology & geometry which has many applications in various fields such as mathematics engineering, physics, economics, game theory, biology, chemistry, optimization theory and approximation theory etc. Fixed point theory has its own importance and developed tremendously for the last one and half century. The purpose of the present paper is to study the history, development and application of Pseudocontractive mapping with fixed point theory.

Keywords

Fixed Point, Banach Space, Contraction mapping, Pseudocontractive mapping.

References

[1] F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci, U.S.A., 53 (1965), 1272-1276.
[2] Z.D. Mitrovic An application of a generalized KKM principle on the existence of an equilibrium point, Ser. Mat. 12 (2001), 64 - 67.
[3] S. Park, Generalizations of Ky Fan’s matching theorems and their applications, J. Math. Anal. Appl. 41 (1989), 164 - 176.
[4] R. E. Bruck, Jr. A common fixed point theorm for a commuting family of nonexpansive mappings, Pacific J. Math., 53 (1974), 59-71.
[5] Gohde, D: Zum Prinzip der kontraktive Abbidung. Math.nachr 30,251-258(1965)
[6] Fan, Ky, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234 - 240.
[7] M. A. Krasnoselskii, Two observations about the method of successive approximations,Usp.Mat.Nauk 10, 123-127(1955)
[8] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc Japan, 19 (1967), 508-520.
[9] W. A. Kirk and R. Schonberg, Zeros of m-accretive operators in Banach spaces.Israel J. Mulh. 35 (1980). l-8.
[10] R. I.Kachurovsky. On monotone operators and convex functionals. Uspekhi Mat. 15 (1960), 213-215.
[11] E. H.Zarantonello. The closure of the numerical range contains the spectrum. Bull. Amer. Math. Sac. 70 (1964), 771-778.
[12] G. J.Minty. On nonlinear integral equations of the Hammerstein type. In “Nonlinear Integral Equations,” pp. 99 -154. Univ. of Wise. Press, Madison Wise., 1964.
[13] WA Kirk, Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions. Proceedings of the American Mathematical Society 1975, 50: 143–149. 10.1090/S0002-9939-1975-0380527-7
[14] F.E Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967), 875– 882.
[15] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc Japan, 19 (1967), 508-520.
[16] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004-1006.
[17] F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967) 197–228