COMMON COUPLED FIXED POINT THEOREM UNDER WEAK ψ − ϕ CONTRACTION FOR HYBRID PAIR OF MAPPINGS WITH APPLICATION

Yıl 2017, Cilt: 7 Sayı: 1, 7 - 24, 01.06.2017

Bhavana Deshpande A. Handa L. Narayan Mishra

Öz

We establish a common coupled fixed point theorem for hybrid pair of mappings under weak ψ − ϕ contraction on a non-complete metric space, which is not partially ordered. It is to be noted that to nd coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. Moreover, an example and an application to integral equations are given here to illustrate the usability of the obtained results. We improve, extend, and generalize several known results.

Anahtar Kelimeler

Coupled fixed point, coupled coincidence point, weak ψ − ϕ contraction, w-compatibility, F-weakly commuting mappings.

Kaynakça

• Abbas,M., Ali,B., and Amini-Harandi,A., (2012), Common fixed point theorem for hybrid pair of mappings in Hausdorff fuzzy metric spaces, Fixed Point Theory Appl., 225.
• Abbas,M., Ciric,L., Damjanovic,B., and Khan,M.A., Coupled coincidence point and common fixed point theorems for hybrid pair of mappings, Fixed Point Theory Appl.1687-1812-2012-4.
• Berinde,V., (2012), Coupled fixed point theorems for ϕ−contractive mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal., 75, pp.3218-3228.
• Bhaskar,T.G. and Lakshmikantham,V., (2006), Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65(7), pp.1379-1393.
• Ciric,L., Damjanovic,B., Jleli,M., and Samet,B., (2012), Coupled fixed point theorems for generalized Mizoguchi-Takahashi contractions with applications, Fixed Point Theory Appl., 51.
• Deepmala, (2014), A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur (Chhatisgarh) India – 492 010.
• Deepmala and Pathak,H.K., (2013), Common fixed points for hybrid strict contractions in symmetric spaces under relaxed conditions, Antarct. J. Math., 10(6), pp.579-588.
• Deepmala and Pathak,H.K., (2014), Remarks on occasionally weakly compatible mappings versus occasionally weakly compatible mappings, Demonstr. Math., 47(3), pp.695-703.
• Deepmala and Pathak,H.K., (2013), Some common fixed point theorems for D-operator pair with applications to nonlinear integral equations, Nonlinear Funct. Anal. Appl., 18(2), pp.205-218.
• Deshpande,B. and Handa,A., (2015), Nonlinear mixed monotone-generalized contractions on partially ordered modified intuitionistic fuzzy metric spaces with application to integral equations, Afr. Mat., 26(3-4), pp.317-343.
• Deshpande,B. and Handa,A., (2014), Application of coupled fixed point technique in solving integral equations on modified intuitionistic fuzzy metric spaces, Adv. Fuzzy Syst., Volume 2014, Article ID 348069.
• Deshpande,B. and Handa,A., (2014), Common coupled fixed point theorems for two hybrid pairs of mappings under ϕ − ψ contraction, ISRN, Volume 2014, Article ID 608725.
• Deshpande,B. and Handa,A., (2015), Common coupled fixed point for hybrid pair of mappings under generalized nonlinear contraction, East Asian Math.J., 31(1), pp.77-89.
• Debnath,P., (2014), Fixed points of contractive set valued mappings with set valued domains on a metric space with graph, TWMS J. App. Eng. Math., 4(2), pp.169-174.
• Gangopadhyay,M., Saha,M., and Baisnab,A.P., (2013), Some fixed point theorem in partial metric spaces, TWMS J. App. Eng. Math., 3(2), pp.206-213.
• Gordji,M.E., Baghani,H., and Kim,G.H., (2012), Common fixed point theorems for (ψ, ϕ)−weak nonlinear contraction in partially ordered sets, Fixed Point Theory Appl., 62.
• Guo,D. and Lakshmikantham,V., (1987), Coupled fixed points of nonlinear operators with applica- tions, Nonlinear Anal., 11(5), pp.623-632.
• Lakshmikantham,V. and Ciric,L., (2009), Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70(12), pp.4341-4349.
• Long,W., Shukla,S., and Radenovic,S., (2013), Some coupled coincidence and common fixed point results for hybrid pair of mappings in 0-complete partial metric spaces, Fixed Point Theory Appl., 145.
• Luong,N.V. and Thuan,N.X., (2011), Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal., 74, pp.983-992.
• Jain,M., Tas,K., Kumar,S., and Gupta,N., (2012), Coupled common fixed point results involving a ϕ − ψ contractive condition for mixed g-monotone operators in partially ordered metric spaces, J. Inequal. Appl., 285.
• Markin,J.T., (1947), Continuous dependence of fixed point sets, Proc. Amer. Math. Soc., 38(1947), pp.545-547.
• Mishra,L.N., Tiwari,S.K., Mishra,V.N., and Khan, I.A., (2015), Unique Fixed Point Theorems for Generalized Contractive Mappings in Partial Metric Spaces, Journal of Function Spaces, Volume 2015, Article ID 960827, 8 pages.
• Mishra,L.N., Tiwari,S.K., and Mishra,V.N., (2015), Fixed point theorems for generalized weakly S- contractive mappings in partial metric spaces, J. Appl. Anal. Comput., 5(4), pp.600-612.
• Mursaleen,M., Mohiuddine,S.A., and Agarwal,R.P., (2012), Coupled fixed point theorems for alpha- psi contractive type mappings in partially ordered metric spaces, Fixed Point Theory Appl. 2012, 228.
• Rodriguez-Lopez,J. and Romaguera,S., (2004), The Hausdorff fuzzy metric on compact sets, Fuzzy Sets Syst., 147, pp.273-283.
• Samet,B., Karapinar,E., Aydi,H., and Rajic,V.C., (2013), Discussion on some coupled fixed point theorems, Fixed Point Theory Appl. 2013, 50.
• Singh,N. and Jain,R., (2012), Coupled coincidence and common fixed point theorems for set-valued and single-valued mappings in fuzzy metric space, Journal of Fuzzy Set Valued Analysis, Volume 2012, Article ID jfsva-00129.
• Sintunavarat,W., Kumam,P., and Cho,Y.J., (2012), Coupled fixed point theorems for nonlinear con- tractions without mixed monotone property, Fixed Point Theory Appl. 2012, 170.
• Bhavana Deshpande, for the photograph and short biography, see TWMS J. Appl. and Eng. Math., V.5, No.1, 2015.
• Amrish Handa, for the photograph and short biography, see TWMS J. Appl. and Eng. Math., V.6, No.1, 2016.