A fixed point theorem for (\phi, \shi)-convex contraction in metric spaces

Abstract

In the present paper, we introduce the notion of (\phi, \shi)-convex contraction mapping of order m and establish a fixed point theorem for such mappings in complete metric spaces. The present result extends and generalizes the well known result of Dutta and Choudhary (Fixed Point Theory Appl. 2008 (2008), Art. ID 406368), Rhoades (Nonlinear Anal., 47(2001), 2683-2693), Istratescu (Ann. Mat. Pura Appl., 130(1982), 89-104) and besides many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results.

Keywords

• Fixed Point
• (\phi \shi)-weak contraction
• convex contraction
• metric space

Kaynakça

1. [1] Ya. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, New results in operator theory and its applications, Oper.Theory Adv. Appl., 98 (1997), 7-22.
2. [2] C. D. Alecsa, Some fixed point results regarding convex contractions of presi¢ type, Journal of Fixed Point Theory and Applications 20 (2018), no. 1, Paper No. 19.
3. [3] M. A. Alghamdi, S. H. Alnafei, S. Radenovic, and N. Shahzad, Fixed point theorems for convex contraction mappings on cone metric spaces, Mathematical and Computer Modelling 54 (2011), no. 9-10, 2020-2026.
4. [4] H. Aydi, Common fixed point results for mappings satisfying (ψ, φ)-weak contractions in ordered partial metric spaces, Int. J. Math. Stat. 12 (2012), no. 2, 53-64.
5. [5] D. Doric, Common fixed point for generalized (ψ,φ)-weak contractions, Appl. Math. Lett. 22 (2009), no. 12, 1896-1900.
6. [6] P. N. Dutta and Binayak S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. (2008), Art. ID 406368, 8.
7. [7] A. Fulga, Fixed point theorems in rational form via Suzuki approaches, Results in Nonlinear Analysis 1 (2018), 19-29.
8. [8] A. Fulga and P. Alexandrina, A new generalization of Wardowski fixed point theorem in complete metric spaces, Advances in the Theory of Nonlinear Analysis and its Application, 1 (2017), 57-63.

9. [9] U. C. Gairola and D. Khantwal, Suzuki type fixed point theorems in S-metric space, 5(3-C) (2017), 277-289.
10. [10] U. C. Gairola and R. Krishan, Hybrid contractions with implicit relations, Advances in Fixed Point Theory 5 (2014), no. 1, 32-44.
11. [11] M. Imdad, S. Chauhan, and Z. Kadelburg, Fixed point theorems for mappings with common limit range property satisfying generalized (ψ, ϕ)-weak contractive conditions, Mathematical Sciences 7(2013), no. 1, Article 16.
12. [12] V. I. Istratescu, Some fixed point theorems for convex contraction mappings and convex nonexpansive mappings. I. Libertas Math. 1 (1981), 151-163.
13. [13] V. I. Istratescu, Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. I. Ann. Mat. Pura Appl. (4) 130 (1982), 89-104.
14. [14] V. I. Istratescu, Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters. II. Ann. Mat. Pura Appl. (4) 134 (1983), 327-362.
15. [15] D. Khantwal and U. C. Gairola, An extension of Matkowski's and Wardowski's fixed point theorems with applications to functional equations, Aequationes mathematicae 93 (2019), no. 2, 433-443.
16. [16] R. Miculescu and A. Mihail, A generalization of Matkowski's fixed point theorem and Istratescu's fixed point theorem concerning convex contractions, J. Fixed Point Theory Appl. 19 (2017), no. 2, 1525-1533.
17. [17] S. Moradi and A. Farajzadeh, On the fixed point of (ψ- ϕ)-weak and generalized (ψ- ϕ)-weak contraction mappings, Applied Mathematics Letters 25 (2012), no. 10, 1257-1262.
18. [18] V. Muresan and A. S. Muresan, On the theory of fixed point theorems for convex contraction mappings, Carpathian J. Math. 31 (2015), no. 3, 365-371.
19. [19] H. K. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ, ϕ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Analysis:Theory, Methods & Applications 74 (2011), no. 6, 2201-2209.
20. [20] O. Popescu, Fixed points for (ψ, φ)-weak contractions, Applied Mathematics Letters 24 (2011), no. 1, 1-4.
21. [21] S. Radenovi¢ and Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Computers & Mathe- matics with Applications 60 (2010), no. 6, 1776-1783.
22. [22] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis: Theory, Methods & Applications 47 (2001), no. 4, 2683-2693.
23. [23] S. L. Singh, R. Kamal, M. De la Sen, and R. Chugh, A fixed point theorem for generalized weak contractions, Filomat 29 (2015), no. 7, 1481-1490.
24. [24] T. Suzuki, Fixed point theorem for a kind of Ciric type contractions in complete metric spaces, Advances in the Theory of Nonlinear Analysis and its Application 2 (2018), 33-41.
25. [25] C. Vetro, D. Gopal, and M. Imdad, Common fixed point theorems for (ϕ, ψ)-weak contractions in fuzzy metric spaces, Indian J. Math 52 (2010), no. 3,573-590.