$C$-class functions on common fixed point theorems for weak‎ ‎contraction mapping of integral type in modular spaces

Subject Areas : Fixed point theory

H. A. Hammad ¹ , R. A. Rashwan ² , A. H. Ansari ³

1 - Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
2 - Department of Mathematics, Faculty of Science, Assuit University, Assuit 71516, Egypt
3 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Received: 2019-04-14 Accepted : 2019-06-26 Published : 2019-12-01

Keywords: Common fixed point, modular spaces, ρ−compatible maps, comparison function, Lebesgue-Stieltjes integrable mapping, $C$-class functions,

Abstract :

‎In this paper‎, ‎we use the concept of $C$-class functions introduced‎‎by Ansari [4] to prove the existence and uniqueness of‎‎common fixed point for self-mappings in modular spaces of integral‎‎inequality‎. ‎Our results extended and generalized previous known‎‎results in this direction‎.

References:

[1] M. Abbas, B. E. Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type, Fixed Point Theory Appl. (2007), 2007:54101.

[2] A. Ait Taleb, E. Hanebaly, A fixed point theorem and its application to integral equations in modular function spaces, Proc. Amer. Math. Soc. 128 (2000), 419-426.

[3] I. Altun, D. Turkoglu, B. E. Rhoades, Fixed points of weakly compatible maps satisfying a general contractive condition of integral type, Fixed Point Theory and Applications, (2007), 2007:17301.

[4] A. H. Ansari, Note on “φ−ψ-contractive type mappings and related fixed point”, The 2nd Regional Conference on Mathematics and Applications, Payame Noor University, (2014), 377-380.

[5] H. Aydi, A fixed point theorem for a contractive condition of integral type involving altering distances, Int. J. Nonlinear Anal. Appl. 3 (1) (2012), 42-53.

[6] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181.

[7] V. Berinde, Approximating fixed points of weak contractions using picard iteration, Nonlinear Anal. Forum. 9 (1) (2004), 43-53.

[8] M. Beygmohammadi, A. Razani, Two fixed point theorems for mappings satisfying a general contractive condition of integral type in the modular space, Inter. J. Math. & Math. Sci. (2010), 2010:317107.

[9] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002), 531-536.

[10] G. Jungck, Compatible mappings and common fixed point, Int. J. Math. Math. Sci. 9 (1986), 771-779.

[11] M. A. Khamsi, W. M. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal. 14 (1999), 935-953.

[12] S. Kumar, R. Chugh, R. Kumar, Fixed point theorem for compatible mapping satisfying a contractive condition of integral type, Soochow J. Math. 33 (2) (2007), 181-185.

[13] J. Musielak, W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49-65.

[14] H. Nakano, Modulared semi-ordered linear spaces, Tokyo Mathematical Book Series, Maruzen Co. Ltd, Tokyo, Japan, 1950.

[15] H. K. Nashine, H. Aydi, Coupled fixed point theorems for contractions involving altering distances in ordered metric spaces, Math. Sci. (2013), 7:20.

[16] M. O. Olatinwo, A results for approximating fixed points of generaized weak contraction of the integral-type by using Picard iteration, Rev. Colom. de Mate. 42 (2) (2008), 145-151.

[17] M. O. Olatinwo, Some fixed point theorems for weak contraction condition of integral type, Acta. Univ. Apul. 24 (2010), 331-338.

[18] V. Ozturk, A. H. Ansari, Fixed point theorems for (F,ψ,φ)-contractions on ordered S-complete Hausdorff uniform spaces, New Trends. Math. Sci. 5 (2017), 243-249.

[19] V. Ozturk, D. Turkoglu, A. H. Ansari, Fixed points for generalized (F,h,α,µ) − ϕ contractions in b-metric spaces, Commun. Fac. Sci. Univ. Ank. Seri A1: Math. Stat. 67 (2) (2018), 306-316.

[20] H. K. Pathak, R. Tiwari, M. S. Khan, A common fixed point theorem satifying integral type implicit relations, Appl. Math. E-Notes. 7 (2007), 222-228.

[21] R. A. Rashwan, H. A. Hammad, Common fixed point theorems for weak contraction mapping of integral type in modular spaces, Univ. J. Comput. Math. 2 (4) (2014), 69-86.

[22] A. Razani, R. Moradi, Common fixed point theorems of integral type in modular spaces, Bull. Iran. Math. Soc. 35 (2) (2009), 11-24.

[23] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (4) (2001), 2683-2693.

[24] B. E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 63 (2003), 4007-4013.

[25] P. Vijayaraju, B. E. Rhoades, R. Mohanra, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 15 (2005), 2359-2364.