$\alpha_\kappa-$Implicit Contraction in non-AMMS with Some Applications

Year 2018, Volume: 1 Issue: 2, 212 - 219, 25.12.2018

Ekber Girgin , Mahpeyker Öztürk

https://doi.org/10.33401/fujma.492561

Cited By: 1

Abstract

In this article, we establish $\alpha_\kappa-$implicit contraction and provide some fixed point results in non-AMMS. Our results progress and generalize some famous consequences in a suitable resource. As an implementation, we study stability in the sense of Ulam-Hyers and a fixed point problem's well-posedness. In addition, some examples are given for new concepts. Also, an application to integral equations is discussed.

Keywords

non-Archimedean modular metric space , Implicit contraction , Ulam-Hyers stability , Well-posedness

References

• [1] V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal., 72 (2010), 1-14.
• [2] V. V. Chistyakov, Modular metric spaces, II: Application to superposition operators, Nonlinear Anal., 72 (2010), 15-30.
• [3] C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2011(93) (2011) ,9 pages.
• [4] M. Paknazar, M. A. Kutbi, M. Demma, P. Salimi, On non-Archimedean Modular metric space and some nonlinear contraction mappings, J. Nonlinear Sci. Appl., (2017), in press.
• [5] V. Popa, Fixed point theorems for implicit contractive mappings, Stud. Cerc. St. Ser. Mat. Univ. Bacau, 7 (1997), 129-133.
• [6] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha-\psi-$contractive type mappings, Nonlinear Analysis, 75 (2012) (2012), 2154-2165.
• [7] H. Aydi, $\alpha-$implicit contractive pair of mappings on quasi b-metric spaces and application to integral equations, J. Nonlinear Convex Anal., 17(12) (2015), 2417-2433.
• [8] A. Hussain, T. Kanwal, Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point, Transections of A. Razmadze Mathematical Enstitute, 172(3) (2018), 48-490.
• [9] M. Abbas, A. Hussain, B. Popovic, S. Radenovic, Istratescu-Suzuki-Ciric type fixed point results in thee framework of G-metric spaces, J. Nonlinear Sci. Appl., 9 (2016), 6077-6095.
• [10] N. Hussain, C. Vetro, F. Vetro, Fixed point results for $\alpha-$implicit contractions with application to integral equations, Nonlinear Anal. Model. Control, 21(3) (2016), 362-378.
• [11] V. Berinde, Approximating fixed points of implicit almost contractions, Hacet. J. Math. Stat., 41 (2012), 93-102.