COMMON COUPLED FIXED POINT RESULTS FOR HYBRID PAIR OF MAPPING UNDER GENERALIZED (𝜓, 𝜃, 𝜑)-CONTRACTION WITH APPLICATION

• Handa, Amrish (Department of Mathematics, Govt. P. G. Arts and Science College)
• Accepted : 2019.07.18
• Published : 2019.08.31

Abstract

We introduce (CLRg) property for hybrid pair $F:X{\times}X{\rightarrow}2^X$ and $g:X{\rightarrow}X$. We also introduce joint common limit range (JCLR) property for two hybrid pairs $F,G:X{\times}X{\rightarrow}2^X$ and $f,g:X{\rightarrow}X$. We also establish some common coupled fixed point theorems for hybrid pair of mappings under generalized (${\psi},{\theta},{\varphi}$)-contraction on a noncomplete metric space, which is not partially ordered. It is to be noted that to find coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. As an application, we study the existence and uniqueness of the solution to an integral equation. We also give an example to demonstrate the degree of validity of our hypothesis. The results we obtain generalize, extend and improve several recent results in the existing literature.

References

1. M. Abbas, L. Ciric, B. Damjanovic & M. A. Khan: Coupled coincidence point and common fixed point theorems for hybrid pair of mappings. Fixed Point Theory Appl. 2012, 4.
2. M.A. Ahmed & H.A. Nafadi: Common fixed point theorems for hybrid pairs of maps in fuzzy metric spaces. J. Egyptian Math. Soc. 2013, Article in press.
3. A. Alotaibi & S.M. Alsulami: Coupled coincidence points for monotone operators in partially ordered metric spaces. Fixed Point Theory Appl. 2011, 44.
4. S.M. Alsulami: Some coupled coincidence point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. Fixed Point Theory Appl. 2013, 194.
5. T.G. Bhaskar & V. Lakshmikantham: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65 (2006), no. 7, 1379-1393. https://doi.org/10.1016/j.na.2005.10.017
6. S. Chauhan, W. Sintunavarat & P. Kumam: Common Fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property. Applied Mathematics 3 (2012), no. 9, 976-982. https://doi.org/10.4236/am.2012.39145
7. B. Deshpande & A. Handa: Nonlinear mixed monotone-generalized contractions on partially ordered modified intuitionistic fuzzy metric spaces with application to integral equations. Afr. Mat. 26 (2015), no. 3-4, 317-343. https://doi.org/10.1007/s13370-013-0204-0
8. B. Deshpande & A. Handa: Application of coupled fixed point technique in solving integral equations on modified intuitionistic fuzzy metric spaces, Adv. Fuzzy Syst. 2014, Article ID 348069.
9. B. Deshpande & A. Handa: Common coupled fixed point theorems for hybrid pair of mappings satisfying an implicit relation with application. Afr. Mat. 27 (2016), no. 1-2, 149-167. https://doi.org/10.1007/s13370-015-0326-7
10. B. Deshpande & A. Handa: Common coupled fixed point theorems for two hybrid pairs of mappings under ${\varphi}-{\psi}$ contraction. ISRN 2014, Article ID 608725.
11. B. Deshpande & A. Handa: Common coupled fixed point for hybrid pair of mappings under generalized nonlinear contraction. East Asian Math. J. 31 (2015), no. 1, 77-89. https://doi.org/10.7858/eamj.2015.008
12. B. Deshpande & A. Handa: Common coupled fixed point theorems for hybrid pair of mappings under some weaker conditions satisfying an implicit relation. Nonlinear Analysis Forum 20 (2015), 79-93.
13. B. Deshpande & A. Handa: Common coupled fixed point theorems for two hybrid pairs of mappings satisfying an implicit relation. Sarajevo J. Math. 11 (2015), no. 23, 85-100. https://doi.org/10.5644/SJM.11.1.07
14. B. Deshpande & A. Handa: Common coupled fixed point theorem under generalized Mizoguchi-Takahashi contraction for hybrid pair of mappings. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 22 (2015), no. 3, 199-214.
15. D. Guo & V. Lakshmikantham: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11 (1987), no. 5, 623-632. https://doi.org/10.1016/0362-546X(87)90077-0
16. J. Harjani, B. Lopez & K. Sadarangani: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. 74 (2011), 1749-1760. https://doi.org/10.1016/j.na.2010.10.047
17. J. Harjani & K. Sadarangani: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72 (2010), no. 3-4, 1188-1197. https://doi.org/10.1016/j.na.2009.08.003
18. M.A. Khan & Sumitra: CLRg property for coupled fixed point theorems in fuzzy metric spaces. Int. J. Appl. Phy. Math. 2 (2012), no. 5, 355-358.
19. V. Lakshmikantham & L. Ciric: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70 (2009), no. 12, 4341-4349. https://doi.org/10.1016/j.na.2008.09.020
20. W. Long, S. Shukla & S. Radenovic: Some coupled coincidence and common fixed point results for hybrid pair of mappings in 0-complete partial metric spaces. Fixed Point Theory Appl. 2013, 145.
21. N.V. Luong & N.X. Thuan: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 74 (2011), 983-992. https://doi.org/10.1016/j.na.2010.09.055
22. J.T. Markin: Continuous dependence of fixed point sets. Proc. Amer. Math. Soc. 38 (1947), 545-547.
23. S.B. Nadler: Multivalued contraction mappings. Pacific J. Math. 30 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475
24. J.J. Nieto & R. Rodriguez-Lopez: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22 (2005), 223-239. https://doi.org/10.1007/s11083-005-9018-5
25. A.C.M. & M.C.B. Reurings: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132 (2004), 1435-1443. https://doi.org/10.1090/S0002-9939-03-07220-4
26. A. Razani & V. Parvaneh: Coupled coincidence point results for (${\psi}$, ${\alpha}$, ${\beta}$)-weak contractions in partially ordered metric spaces. J. Appl. Math. 2012, Article ID 496103.
27. J. Rodriguez-Lopez & S. Romaguera: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets Syst. 147 (2004), 273-283. https://doi.org/10.1016/j.fss.2003.09.007
28. B. Samet, E. Karapinar, H. Aydi & V.C. Rajic: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013, 50.
29. F. Shaddad, M.S.M. Noorani, S.M. Alsulami & H. Akhadkulov: Coupled point results in partially ordered metric spaces without compatibility. Fixed Point Theory Appl. 2014, 204.
30. N. Singh & R. Jain: Coupled coincidence and common fixed point theorems for set-valued and single-valued mappings in fuzzy metric space. Journal of Fuzzy Set Valued Analysis 2012, Article ID jfsva-00129.
31. W. Sintunavarat & P. Kumam: Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. Journal of Applied Mathematics 2011, Article ID 637958.
32. W. Sintunavarat, P. Kumam & Y J. Cho: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012, 170.