New fixed point theorems in metric space with w-distance

This paper presents some new fixed point results for a class of weakly contractive maps in a metric space. We use the concept of w-distance in the framework of metric spaces. As far as we know, the concept of w-distance is being utilized for the first time to study Suzuki-type antecedent conditions with weakly contractive maps. An illustrative example demonstrates the generality of the main result.

Show more Show less

References:

Ya. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications, 98, I. Gohberg and Yu. Lyubich (eds.), Birkhauser, Basel, 1997, pp. 7-22. https://doi.org/10.1007/978-3-0348-8910-0_2

H. Aydi, T. Wongyat, and W. Sintunavarat, On new evolution of Ri's result via w-distance and the study on the solution for nonlinear integral equations and fractional differential equations, Advances Diff. Equ. 2018 (2018), no. 132. https://doi.org/10.1186/s13662-018-1590-2

D. W. Boyd and J. S. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9

J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251. https://doi.org/10.1090/S0002-9947-1976-0394329-4

K. Darko, E. Karapinar, and V. Rakocevic, On quasi-contraction mappings of Ciric and Fisher type via w-distance, Quaes. Math. 42 (2018), 1-14. https://doi.org/10.2989/16073606.2018.1436614

D. Doric, Common fixed point for generalized (ψ, φ)-weak contractions, Appl. Math. Lett. 22 (2009), 1896-1900. https://doi.org/10.1016/j.aml.2009.08.001

P. N. Dutta and B. S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. 2008 (2008), 406368. https://doi.org/10.1155/2008/406368

I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474. https://doi.org/10.1090/S0273-0979-1979-14595-6

O. Kada, T. Suzuki, and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica 4 (1991), 381-391.

H. Lakzian, H. Aydi, and B. E. Rhoades, Fixed points for (φ, ψ, p)-weakly contractive mappings in metric space with w-distance, Appl. Math. Comp. 219 (2013), 6777-6782. https://doi.org/10.1016/j.amc.2012.11.025

H. Lakzian and I.-J. Lin, The existence of fixed points for nonlinear contractive maps in metric spaces with w-distances, J. Appl. Math. 2012 (2012), 161470. https://doi.org/10.1155/2012/161470

H. Lakzian and B. E. Rhoades, Some fixed point theorems using weaker Meir-Keeler function in metric space with w-distance, Appl. Math. Comp. 342 (2019), 18-25. https://doi.org/10.1016/j.amc.2018.06.048

S. Moradi and Ali Farajzadeh, On fixed points of (ψ-φ)-weak and generalized (ψ-φ)-weak contraction mappings, Appl. Math. Lett. 25 (2012), 1257-1262. https://doi.org/10.1016/j.aml.2011.11.007

O. Popescu, Fixed points for (ψ, φ)-weak contractions, Appl. Math. Lett. 24 (2011), 1-4. https://doi.org/10.1016/j.aml.2010.06.024

A. Razani, Z. Mazlumi and M. Boujary, A fixed point theorem for w-distance, Appl. Sci. 11 (2019), 114-118.

S. Reich, Some fixed point theorems, Atti. Accad. Naz. Lincei 57 (1974), 194-198.

B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1

S. Romaguera and P. Tirado, Presymmetric w-distances on metric spaces, Mathematics 12, no. 2 (2024), 305. https://doi.org/10.3390/math12020305

S. L. Singh, A. Kumar, and S. Stofile, Coincidence and fixed points of weakly contractive maps, J. Adv. Math. Stud. 5, no. 2 (2012), 68-81.

S. L. Singh, R. Kamal, M. De la Sen, and R. Chugh, A fixed point theorem for generalized weak contractions, Filomat 29, no. 7 (2015), 1481-1490. https://doi.org/10.2298/FIL1507481S

T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 1861-1869. https://doi.org/10.1090/S0002-9939-07-09055-7

W. Takahashi, A convexity in metric space and nonexpansive mappings I, Kodai Math. Sem. Rep. 22 (1970), 142-149. https://doi.org/10.2996/kmj/1138846111

J.-S. Ume, Fixed point theorems related to Ciric's contraction principle, J. Math. Anal. Appl. 225 (1998), 630-640. https://doi.org/10.1006/jmaa.1998.6030

T. Wongyat and W. Sintunavarat, The existence and uniqueness of the solution for nonlinear Fredholm and Volterra integral equations together with nonlinear fractional differential equations via w-distance, Advances Diff. Equ. 2017 (2017), 211. https://doi.org/10.1186/s13662-017-1267-2

Show more Show less