Volume 39 | Number 1 | Year 2016 | Article Id. IJMTT-V39P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V39P503

Fixed point theory off course entails the search for a combination of conditions on a set S and a mapping T : S → S which, in turn, assures that T leaves at least one point of S fixed, i.e. x = T( x ) for some x ϵ S. The theory has several rather well-defined (yet overlapping) branches. The purely topological theory as well as those topics which lie on the borderline of topology and functional analysis (e.g. those related to Leray-Schauder theory) have their roots in the celebrated theorem of L. E. J. Brouwer. This paper presents a review of the available literature on fixed point theorems for various types of maps.

[1] Altun, I. and Turkoglu, D. (2009). Some fixed point theorems for weakly compatible mappings satisfying an implicit relation. Taiwanese J. Math. (13)4: 1291–1304.

[2] Asad, A. J. and Ahmad, Z. (1999). Common fixed point of multi-valued mappings with weak commu-tativity conditions. Radovi, Math. 9: 119–124.

[3] Aydi, H., Jellali, M. and Karapinar E. Common fixed points for generalized _-implicit contractions in partial metric spaces: Consequences and application, RACSAM–Revista de la Real Academia de Ciencias Exactas. Físicas y Naturales. Serie A. Matemáticas. To appear.

[4] Berinde, M and Berinde, V. (2007). On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 326: 772–782.

[5] Browder, F. E. (1976). Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math., (18)2: American Mathematical Society, Providence, RI.

[6] Daffer, P. Z and Kaneko, H. (1995). Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl. 192: 655–666.

[7] Fisher, B. (1983). Common fixed point of four mappings. Bull. Inst. Math. Acad. Sinica, 11: 103–113.

[8] Fisher, B. (1979). Mappings satisfying rational inequality. Nanta Math., 12: 195–199.

[9] Gulyaz. S. and Karapinar, E. (2013). Coupled fixed point result in partially ordered partial metric spaces through implicit function. Hacet. J. Math. Stat. (42)4: 347–357.

[10] Gulyaz, S., Karapinar, E. and Yuce, I. S. (2013). A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation. Fixed Point Theory Appl. 2013, 38: 11.

[11] Hitzler, P. and Seda, A. K. (1999). Multi-valued mappings, fixed point theorems and disjunctive databases. 3rd Irish Workshop on Formal Methods (IWFM-99), EWIC, British Computer Society.

[12] Husain, T. and Latif, A. (1991). Fixed points of multi-valued nonexpansive maps. Internat. J. Math. & Math. Sci. (14)3: 421-430.

[13] Jungck, G. (1976). Commutating mappings and fixed points. Amer. Math. Monthly, 83: 261–263.

[14] Jungck, G. (1986). Compatible mappings and common fixed points. Internat J. Math. Math. Sci., 9(4): 771–779.

[15] Jungck, G. and Rhoades, B. E. (1998). Fixed point for set valued functions without continuity. Ind. J. Pure and Appl. Math., 29(3): 227–238.

[16] Kamran, T. (2004). Coincidence and fixed points for hybrid strict contractions. J. Math. Anal. Appl., 299: 235–241.

[17] Kaneko, H. (1988). A common fixed point of weakly commuting multi-valued mappings. Math. Japon., 33(5): 741–744.

[18] Kaneko, H. and Sessa, S. (1989). Fixed point theorems for compatible multi-valued and single-valued mappings. Internat. J. Math. Math. Sci., 12(2): 257–262.

[19] Karapinar, E. and Erhan, I. M. (2012). Cyclic contractions and fixed point theorems. Filomat. (26)4: 777–782.

[20] Karapinar, E. (2011). Fixed point theory for cyclic weak _- contraction. Appl. Math. Lett. (24)6: 822–825.

[21] Kirk, W. A. and Massa, S. (1990). Remarks on asymptotic and Chebyshev centers. Houston J. Math. (16) 3: 357-364.

[22] Kirk, W.A., Srinivasan, P.S. and Veeramani, P. (2003). Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory. (4)1: 79–89.

[23] Kubiak, T. (1985). Fixed point theorems for contractive type multi-valued mappings. Math. Japon., 30: 89–101.

[24] Kubiaczyk, I. and Mostafa Ali, N. (1996). A multi-valued fixed point theorems in non Archimedean vector spaces. Novi Sad J. Math., 26(2): 111–115.

[25] Kyzyska, S. and Kubiaczyk, I. (1998). Fixed point theorems for upper semicontinuous and weakly upper semicontinuous multi-valued mappings. Math. Japonica, (47)2: 237–240.

[26] Lami Dozo, E. (1973). Multi-valued nonexpansive mappings and Opial’s condition. Proc. Amer. Math. Soc. 38: 286-292.

[27] Lim, T. C. (1974). A fixed point theorem for multi-valued nonexpansive mappings in a uniformly convex Banach space. Bull. Amer. Math. Soc. 80: 1123-1126.

[28] Markin, J. (1968). A fixed point theorem for set valued mappings. Bull. Amer. Math. Soc. 74: 639-640.

[29] Miklaszewski, D. (2001). A fixed point theorem for multivalued mappings with noncyclic values. Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center, 17: 125–131.

[30] Moutawakil, D. E. (2004). A fixed point theorem for multivalued maps in symmetric spaces. Applied Mathematics ENotes, 4: 26-32.

[31] Nadler, S. B. (1969). Multi-valued contraction mappings. Pacific J. Math. (20)2: 457–488.

[32] Nashine, H. K., Kadelburg, Z. and Kumam, P. (2012). Implicit-relation-type cyclic contractive mappings and applications to integral equations. Abstr. Appl. Anal. Art. ID 386253, 15 pp.

[33] Olatinwo, M. O. (2009). A fixed point theorem for multivalued weakly picard operators in b-metric spaces. Demonstratio mathematica, (XLII)3: (2009).

[34] Pacurar, M. (2011). Fixed point theory for cyclic Berinde operators. Fixed Point Theory (12)2: 419–428.

[35] Pacurar, M. and Rus, I. A. (2010). Fixed point theory for cyclic '-contractions. Nonlinear Anal. 72: 1181–1187.

[36] Pant, R. P. (1994). Common fixed points of non-commuting mappings. J. Math. Anal. Appl., 188: 436–440.

[37] Pant, R. P. (1998). Common fixed point theorems for contractive maps. J. Math. Anal. Appl. 236: 251–258.

[38] Pant, R. P. (1999). Common fixed points of Lipschitz type mapping pairs. J. Math. Anal. Appl., 240: 280–283.

[39] Pant, R. P. (1999). Discontinuity and fixed points. J. Math. Anal. Appl., 240: 284–289.

[40] Percup, R. (2002). Fixed point theorems for acyclic multivalued maps and inclusions of Hammerstein type. Seminar on Fixed Point Theory Cluj-Napoca, 3: 327-334.

[41] Petric, M. A. (2010). Some results concerning cyclical contractive mappings. Gen. Math. (18)4: 213–226.

[42] Popa, V. (1997). Some fixed point theorems for implicit contractive mappings. Stud. Cercet. Stiinμ., Ser. Mat., Univ. Bacau 7: 129–133.

[43] Popa,V. (2000). A general coincidence theorem for compatible multi-valued mappings satisfying an implicit relation. Demonstratio Math. (33)1: 159–164.

[44] Popa, V. (1999). A general fixed point theorem for weakly commuting multi-valued mappings. Anal. Univ. Dunarea de Jos, Galaμi, Ser. Mat. Fiz. Mec. Teor., Fasc. II 18 (22): 19– 22.

[45] Popa, V. (2015). A general fixed point theorem for implicit cyclic multi-valued contraction mappings. Annales Mathematicae Silesianae 29: 119–129.

[46] Popa, V. (1999). Some fixed point theorems for compatible mappings satisfying an implicit relation. Demonstratio Math. (32)1: 157–163.

[47] Reich, S. (1983). Some problems and results in fixed point theory. Contemporary Math. 21: 179-187.

[48] Rus, I. A. (2005). Cyclic representations of fixed points. Ann. Tiberiu Popoviciu, Semin. Funct. Equ. Approx. Convexity, 3: 171–178.

[49] Shahzad, N. and Kamran, T. (2001). Coincidence points and R-weakly commuting maps. Arch. Math. (Brno), 37: 179– 183.

[50] Sessa, S. (1982). On weak commutativity condition of mappings in fixed point consideration. Publ. Inst. Math., ( 32)46: 149–155.

[51] Singh, S. L. and Singh, S. P. (1980). A fixed point theorem. Ind. J. Pure Appl. Math., 11: 1584–1586.

[52] Singh, S. L. and Mishra, S. N. (2001). Coincidence and fixed points of non self hybrid contraction. J. Math. Anal. Appl., 256: 486–497.

[53] Sintunavarat, W. and Kumam, P. (2012). Common fixed point theorem for cyclic generalized multi-valued mappings. Appl. Math. Lett. 25: 1849–1855.

[54] Xu, H. K. (2000). Metric Fixed Point Theory for Multivalued Mappings. Dissertations Math. (Rozprawy Mat.) 389.

Masroor Mohammad, Rizwana Jamal, Qazi Aftab Kabir, "Review Article - A study of some fixed point
theorems for various types of maps," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 39, no. 1, pp. 18-21, 2016. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V39P503