PPF Dependent Fixed Points of Generalized Weakly Contraction Maps Via $c_g-$simulation Functions

Year 2019, Volume: 1 Issue: 2, 66 - 88, 30.10.2019

G. V. R. Babu , Madugula Vinod Kumar

https://izlik.org/JA27ZU69JC

Abstract

In this paper, we introduce the notion of generalized weakly  $Z_{G,\alpha,\mu,\xi,\eta,\varphi}-$contraction maps with respect to the $C_G-$simulation function and prove the existence of  PPF dependent fixed points of nonself maps in Banach spaces. For such maps, PPF dependent fixed points may not be unique. We provide an example  to illustrate this phenomenon.

Keywords

\alpha-admissible , \mu-subadmissible , C-class function , Razumikhin class , PPF dependent fixed point , simulation function , C_G-simulation function

References

• [1] Ya. I. Alber, S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces New results in Operator theory, Adv. Appl., Vol.98 , Birkhauser Verlag, (1997), 7-22 .
• [2] A. H. Ansari, Note on $\phi-\psi-$ contractive type mappings and related fixed point, The 2nd Regional Conference on Mathematics and Applications, Payame Noor University Tehran, (2014), 377-380.
• [3] A. H. Ansari, J. Kaewcharoen, $C-$ class functions and fixed point theorems for generalized $\alpha-\eta-\psi-\phi-F-$contraction type mappings in $\alpha-\eta$ complete metric spaces, J. Nonlinear Sci. Appl., 9(6)(2016), 4177-4190.
• [4] Antonella Nastasi and P. Vetro, Fixed point results on metric and partial metric spaces via simuation functions, J. Nonlinear Sci. Appl., 8(2015), 1059-1069.
• [5] G.V.R. Babu, G. Satyanarayana and M. Vinod Kumar, Properties of Razumikhin class of functions and PPF dependent fixed points of Weakly contractive type mappings, Bull. Int. Math. Virtual Institute, 9(1)(2019), 65-72.
• [6] G.V.R. Babu and M. Vinod Kumar, PPF dependent coupled fixed points via C􀀀class functions, J. Fixed Point Theory, 2019(2019), Article ID 7.
• [7] G.V.R. Babu and M. Vinod Kumar, PPF dependent fixed points of generalized Suzuki type contractions via simulation functions, Advances in the Theory of Nonlinear Anal. and its Appl., 3(3)(2019), 121-140.
• [8] G.V.R. Babu and M. Vinod Kumar, PPF dependent fixed points of generalized contractions via CG-simulation functions, Communications in Nonlinear Anal., 7(1)(2019), 1-16.
• [9] B. E. Rhoades, Some theorems on weakly contractive mappings, Nonlinear Anal. 47 (2001) 2683-2693.
• [10] Banach S.: Sur les operations dans les ensembles abstraits et leur appliacation aux equations integrales, Fund. math., 3(1922), 133-181.
• [11] Bapurao C. Dhage, On some common fixed point theorems with PPF dependence in Banach spaces, J. Nonlinear Sci. Appl., 5(2012), 220-232.
• [12] S. R. Bernfeld, V. Lakshmikantham, and Y. M. Reddy, Fixed point theorems of operators with PPF dependence in Banach spaces, Appl. Anal., 6(4)(1977), 271-280.
• [13] L. Ciric, S. M. Alsulami, P. Salimi and P. Vetro, PPF dependent fixed point results for triangular $\alpha_{c}-$admissible mappings, Hindawi Publishing corporation, (2014), Article ID 673647, 10 pages.
• [14] S. Cho, Fixed point theorems for generalized weakly contractive mappings in metric spaces with application, Fixed point theory and Appl., 2018(2018).
• [15] S. H. Cho, A fixed point theorem for weakly $\alpha-$contractive mappings with application, Appl. Mathematical Sciences, 7(2013), No. 60, 2953-2965.
• [16] B. S. Choudhury, P. Konar, B. E. Rhoades and N. Metiya, Fixed point theorems for generalized weakly contractive mappings, Nonlinear Anal., 74(2011), 2116-2126.
• [17] Z. Dirci, F. A. McRae and J. Vasundharadevi, Fixed point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear Anal., 67(2007), 641-647.
• [18] D. Doric, Common fixed point for generalized $(\psi,\phi)-$weak contractions, Appl. Mathematics Letters, 22(2009), 1896-1900.
• [19] A. Farajzadeh, A.Kaewcharoen and S.Plubtieng, PPF dependent fixed point theorems for multivalued mappings in Banach spaces, Bull. Iranian Math.Soc., 42(6)(2016), 1583-1595.
• [20] Haitham Quwagneh, Mohd Salmi MD Noorani, Wasfi Shatanawi and Habes Alsamir, Common fixed points for pairs of triangular $ \alpha-$admissible mappings, J. Nonlinear Sci. Appl., 10(2017), 6192 - 6204.
• [21] N. Hussain, S. Khaleghizadeh, P. Salimi and F. Akbar, New Fixed Point Results with PPF dependence in Banach Spaces Endowed with a Graph, Abstr. Appl. Anal., (2013), Article ID 827205.
• [22] E. Karapınar, Fixed points results via simulation functions, Filomat, 30(8)(2016), 2343 - 2350.