A common fixed point theorem for weakly subsequentially continuous mappings satisfying implicit relation

Said Beloul

doi:10.26637/mjm304/004

Abstract

In this paper, we prove a common fixed point theorem for two weakly subsequentially continuous and compatible of type (E) for two pairs of self mappings, which satisfying implicit relation in metric spaces, an example is given to illustrate our results, also we give an application to solve a partial differential equations, and the study of its generalized Hyers-Ulam stability, our results improve and extend some previous results.

Keywords:

Common fixed point, weakly subsequentially continuous, compatible of type (E), implicit relation

Mathematics Subject Classification:

54H25

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Said Beloul Department of Mathematics and Informatics, College of Sciences and Technology, University of El-Oued, P.O.Box789,El-Oued39000, Algeria. https://orcid.org/0000-0002-2814-2161

Pages: 409-418
Date Published: 01-10-2015
Vol. 3 No. 04 (2015): Malaya Journal of Matematik (MJM)

M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, Math. Anal. Appl, 270, (2002), 181-188 DOI: https://doi.org/10.1016/S0022-247X(02)00059-8

R. Agawal,R.K. Bisht and N. Shahzad, A comparison of various noncommuting conditions in metric fixed point theory and their applications Fixed Point Theory and Applications (1) (2014),1-33. DOI: https://doi.org/10.1186/1687-1812-2014-38

M. Akkouchi,Hyers-ulam-rassias stability of nonlinear volterra integral equations via a fixed point approach,Acta Univ.Apulensis,No. 26(2011), 257-266.

A.Aliouche and A. Djoudi, Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption, Hacet. J. Math. Stat. 36 (1),(2007) 11-18.

A. Aliouche, Common fixed point theorems via an implicit relation and new properties, Soochow J. Math., vol. 33, no. 4,(2007), 593-601.

I. Altun and D. Türkoğlu,Some fixed point theorems for weakly compatible mappings satisfying an implicit relation, Taiwanese.J. Maths.Vol.13, No.4 (2009), 1291-1304. DOI: https://doi.org/10.11650/twjm/1500405509

S.András and A. R. Meszáros,UlamHyers stability of elliptic partial differential equations in Sobolev spaces,Appl. Maths .Comput $229(2014) 131138$. DOI: https://doi.org/10.1016/j.amc.2013.12.021

S. Beloul,Common fixed point theorems for weakly subsequentially continuous generalized contractions with applications,Appl.Maths E-Notes, 15(2015), 173-186.

H. Bouhadjera and C.G.Thobie,Common fixed point theorems for pairs of subcompatible maps,arXiv:0906.3159v1 [math.FA],(2009).

H. Bouhadjera and C.G.Thobie,Common fixed point theorems for pairs of subcompatible maps, New version. arXiv:0906.3159v2 [math.FA],(2011).

S.Chauhan and D.Pant,Fixed point theorems for compatible and subsequentially continuous mappings in Menger spaces,J. Nonlinear Sci. Appl. 7 (2014), 78-89. DOI: https://doi.org/10.22436/jnsa.007.02.02

D. Dorić, Z. Kadelburg and S. Radenović, A note on occasionally weakly compatible mappings and common fixed point, Fixed Point Theory 13,(2012),475-480.

D.H.Hyers, The stability of homomorphism and related topics, in Global Analysis-Analysis on Manifolds (Th. M. Rassias, ed.), Teubner, Leipzig, 1983, 140153.

M. Imdad, J. Ali and M. Tanveer, Remarks on some recent metrical common fixed point theorems, Appl. Math. Lett. 24(2011), 11651169. DOI: https://doi.org/10.1186/1687-1812-2011-28

S.M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl. 320,no2(2006), 549-561. DOI: https://doi.org/10.1016/j.jmaa.2005.07.032

S.M.Jung,Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. DOI: https://doi.org/10.1007/978-1-4419-9637-4

G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83(4)(1976), 261-263. DOI: https://doi.org/10.1080/00029890.1976.11994093

G. Jungck,Compatible mappings and common fixed points, Int. J. Math. Math. Sci. 9 (1986), 771-779. DOI: https://doi.org/10.1155/S0161171286000935

G. Jungck,P.P Murthy and Y.J Cho, Compatible mappings of type (A) and common fixed points, Math. Japon., 38, (1993), 381-390

G. Jungck and B.E Rhoades, Fixed point for set valued functions without continuity, Indian J. Pure Appl. Math. 29 (3),(1998), 227-238.

J.Li, M.Fu,Z.Liu and S.M.Kang, A common fixed point theorem and its application in dynamic programming, App. Math. Sci. 2(17) ( 2008), 829-842

S. Moradi and E. Analoei,Common fixed point of generalized $(psi, varphi)$-weak contraction mappings,Int. J. Nonlinear Anal. Appl. No.1 3(2012), 24-30.

V.L. Lazăr, Ulam-Hyers stability results for partial differential equations, Creative Math.and Inform., No. $1,21(2012), 57-62$ No.3. DOI: https://doi.org/10.14232/ejqtde.2012.1.21

R.P Pant, A common fixed point theorem under a new condition, Indian. J. Pure Appl.Math. 30 (2),(1999), $147-152$.

J.Matkowski,Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.) 127 (1975),68 pp.

R.P.Pant, A common fixed point theorem under a new condition, Indian J. Pure Appl. Math, no. 2, 30 (1999), $147-152$.

R.P Pant,R.K Bisht and D.Arora, Weak reciprocal continuity and fixed point theorems, Ann Univ Ferrara ,57,(2011), 181-190. DOI: https://doi.org/10.1007/s11565-011-0119-3

H. K.Pathak and M.S.Khan, Compatible mappings of type (B) and common fixed point theorems of Gregus type, Czechoslovak Math.J, 45(120)(1995), 685-698. DOI: https://doi.org/10.21136/CMJ.1995.128555

H. K. Pathak, Y. J. Cho, S. M. Kang and B. S. Lee, Fixed point theorems for compatible mappings of type (P) and application to dynamic programming, Le Matematiche (Fasc. I), 50 (1995), 15-33.

H.K. Pathak, Y.J. Cho, S.M. Khan, B. Madharia, Compatible mappings of type (C) and common fixed point theorems of Gregus type, Demonstratio Math. 31 (3)(1998), 499-518.

H.K Pathak,R. Rodriguez-Lopez and R.K Verma A common fixed point theorem using implicit relation and property (E.A.) in metric spaces, Filomat no21:2 (2007), 211-234. DOI: https://doi.org/10.2298/FIL0702211P

V.Popa, A general fixed point theorem for four weakly compatible mappings satisfying an implicit relation, Filomat , 19 (2005), 45-51. DOI: https://doi.org/10.2298/FIL0519045P