SOME STUDIES ON EXISTENCE OF FIXED POINTS AND COINCIDENCE POINTS. A Summary SUBMITTED TO THE H.N.B. GARHWAL UNIVERSITY SRINAGAR (GARHWAL)

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1 SOME STUDIES ON EXISTENCE OF FIXED POINTS AND COINCIDENCE POINTS A Summary SUBMITTED TO THE H.N.B. GARHWAL UNIVERSITY SRINAGAR (GARHWAL) FOR THE AWARD OF THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS UNDER THE SUPERVISION OF: Dr.U.C. Gairola DEPARTMENT OF MATHEMATICS H.N.B.GARHWAL UNIVERSITY CAMPUS PAURI PAURI SUBMITTED BY: Amit Mewari DEPARTMENT OF MATHEMATICS H.N.B.GARHWAL UNIVERSITY CAMPUS PAURI PAURI REGISTRATION NO.: HNBGU/Res/21450 SEPTEMBER, 2007

2 SUMMARY The Thesis entitled SOME STUDIES ON EXISTENCE OF FIXED POINTS AND COINCIDENCE POINTS is divided into the following five chapters: CHAPTER I CHAPTER II : INTRODUCTION : FIXED POINT THEOREM FOR TWO SYSTEMS OF TRANSFORMATION FOR INTEGRAL TYPE INEQUILITY CHAPTER III : FIXED AND COINCIDENCE POINTS FOR MULTIVALUED MAPPINGS CHAPTER IV : COMMON FIXED POINTS FOR MAPPINGS IN FUZZY METRIC SPACE CHAPTER V : COINCIDENCE POINTS IN SMMETRIC SPACE

3 The first chapter of the thesis is purely introductory in nature. A brief introduction of fixed points, Banach contraction principle, Jungck contraction principle, fuzzy sets, fuzzy metric spaces and symmetric spaces has been given. A detailed review of literature on existence of fixed points and coincidence points for mappings in different spaces viz. metric space, Banach space, fuzzy metric space, symmetric spaces has been also incorporated in this chapter. Besides, this introductory chapter the thesis contains four more chapters and Bibliography. In the second chapter some results for existence of common fixed points for two systems of transformations satisfying an integral inequality have been established. These results generalize the results of Matkowski [10], Singh- Kulsherestha [12] and Gairola [5]. The third chapter of the thesis is divided into two parts. Part A contains some new results on existence of fixed points for a general multivalued contractive condition known as p- contraction [4]. This result extends several well known results on fixed points for single valued mappings and generalizes several known results for the existence of fixed points for multivalued mappings. Part B of this chapter contains some results on coincidence points for non expansive type multivalued and singlevalued mappings. These results are good generalizations of the results of Torkoglu-Fisher [13] and Ciric [1-2].

4 Fourth chapter is devoted to the study of common fixed points for mappings satisfying compatibility condition in the fuzzy metric space. In this chapter we follow the definition of fuzzy metric space defined by Kramosil- Michalek [9], George-Veeramani [6-7] and notions and definitions used in Grabiec [8]. The main result of this chapter contains the common fixed point theorem for the mappings satisfying most general contractive condition given in Rhoades [11]. The last chapter of the thesis contains the study of fixed points and coincidence points for the mappings in symmetric settings of the space. Some results for the existence of fixed points and coincidence points for multivalued mappings have been established for rational type contractive conditions on the mappings in symmetric space. The contractive condition used is analogous of the contractive condition given in Das and Gupta [3]. Also it has been proved that the results established are generalizations and extensions of many well known results for metric spaces in the symmetric settings.

5 REFERENCES 1. Ciric, Lj.B.; On some nonexpansive type mappings and fixed points, Ind. J. Pure. Appl. Math., 24(3) (1993), Ciric, Lj. B.; Fixed points for generalized multivalued contractions. Math. Vesnik. (24)(1972), Das, K. and Gupta, S.; An extension of Banach contraction principle through rational expression, Ind. J. Pure App. Math, 6(1975), Delbosco, D.; A unified approach for all contractive mappings, Jnanabha, 16(1986), Gairola, U.C.; A fixed point theorem for a system of transformations satisfying a general contractive condition of integral type on product spaces, (Communicate). 6. George, A.; Veeramani, P.; On some results of Analysis for Fuzzy Metric Space, Fuzzy Sets and Systems, 64(1994), George, A.; Veeramani, P.; On some results of analysis for the Fuzzy metric spaces, Fuzzy Sets and Systems, 90(1997), Grabiec, M.; Fixed points in fuzzy metric spaces, Fuzzy sets and Systems, 27(1988), Kramosil, I. and Michalek, J.; Fuzzy metric and statistical metric spaces, Kybernetic, 11(1975),

6 10. Matkowski, J.; Integrable solutions of Functional Equation, Dissertation Math., Vol. CXXVII (Rozprawy) Warszawa, (1975). 11. Rhoades, B.E.; Comparison of various defines of contractive mapping, Trans. of Amer. Math. Soc., 266(1977), Singh, S.L.; Kulshrestha, C.; A common fixed point theorem for two systems of transformation, Pusan. Kyo. Math. J., 2 (1986), Torkoglu, D.; Fisher, B.; On some non-expansive type multivalued mappings and fixed points, Ind. J. Math., 43(3) (2001),

7 PUBLICATIONS 1. A Coincidence Point Theorems in Metric Space (Communicated). 2. Fixed Point Theorem for Generalized Multivalued Contraction (Communicated).