Best proximity problems for new types of $\mathcal{Z}$-proximal contractions with an application

Yıl 2020, Cilt: 69 Sayı: 2, 1405 - 1417, 31.12.2020

Hüseyin Işık Hassen Aydi

https://doi.org/10.31801/cfsuasmas.727181

Cited By: 2

Öz

In this study, we establish existence and uniqueness theorems of best proximity points for new types of $\mathcal{Z}$-proximal contractions defined on a complete metric space. The presented results improve and generalize some recent results in the literature. Several examples are constructed to demonstrate the generality of our results. As applications of the obtained results, we discuss sufficient conditions to ensure the existence of a unique solution for a variational inequality problem.

Anahtar Kelimeler

Best proximity point, $ (\alpha\eta) $-admissible mapping, $\mathcal{Z}$-contraction, simulation function, variational inequality

Kaynakça

• Al-Thagafi, M. A., Shahzad, N., Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear Anal., 70 (2009), 1209-1216.
• Altun, I., Aslantas, M., Sahin, H., Best proximity point results for p-proximal contractions, Acta Math. Hungar., (2020), https://doi.org/10.1007/s10474-020-01036-3.
• Argoubi, H., Samet, B., Vetro, C., Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl., 8 (6) (2015), 1082-1094.
• Aydi, H., Felhi, A., On best proximity points for various α-proximal contractions on metric-like spaces, J. Nonlinear Sci. Appl., 9 (2016), 5202--5218.
• Aydi, H., Felhi, A., Best proximity points for cyclic Kannan-Chatterjea-Ćirić type contractions on metric-like spaces, J. Nonlinear Sci. Appl., 9 (2016), 2458--2466.
• Caballero, J., Harjani, J., Sadarangani, K., A best proximity point theorem for Geraghty-contractions, Fixed Point Theory Appl., 2012:231 (2012).
• Eldred, A. A., Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2) (2006), 1001-1006.
• Fang, S. C., Petersen, E. L., Generalized variational inequalities, J. Optim. Theory Appl., 38 (1982), 363-383.
• Hussain, N., Kutbi, M. A., Salimi, P., Best proximity point results for modified α-ψ-proximal rational contractions, Abstr. Appl. Anal., 2013, Article ID 927457 (2013).
• Işık, H., Aydi, H., Mlaiki, N., Radenović, S., Best proximity point results for Geraghty type Z-proximal contractions with an application, Axioms, 8 (3) (2019), 81.
• Işık, H., Sezen, M. S., Vetro, C., ϕ-Best proximity point theorems and applications to variational inequality problems, J. Fixed Point Theory Appl., 19 (4) (2017), 3177-3189.
• Jleli, M., Samet, B.: Best proximity points for α-ψ-proximal contractive type mappings and application, Bull. Sci. Math., 137 (2013), 977-995.
• Karapınar, E., Khojasteh, F., An approach to best proximity points results via simulation functions, J. Fixed Point Theory Appl., 19 (2017), 1983-1995.
• Khojasteh, F., Shukla, S., Radenović, S., A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (6) (2015), 1189-1194.
• Kinderlehrer, D., Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
• Kumam, P., Aydi, H., Karapınar, E., Sintunavarat, W., Best proximity points and extension of Mizoguchi-Takahashi's fixed point theorems, Fixed Point Theory Appl., 2013:242 (2013).
• Roldan-Lopez-de-Hierro, A. F., Karapınar, E., Roldan-Lopez-de-Hierro, C., Martinez-Moreno, J., Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), 345-355.
• Sadiq Basha, S., Veeramani, P., Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103 (2000), 119-129.
• Sahin, H., Aslantas, M., Altun, I., Feng-Liu type approach to best proximity point results, for multivalued mappings, J. Fixed Point Theory Appl., 22 (2020), 11.
• Samet, B., Best proximity point results in partially ordered metric spaces via simulation functions, Fixed Point Theory Appl., 2015:232 (2015).
• Sankar Raj, V., A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (14) (2011), 4804-4808.
• Tchier, F., Vetro, C., Vetro, F., Best approximation and variational inequality problems involving a simulation function, Fixed Point Theory Appl., 2016:26 (2016).
• Todd, M. J., The Computations of Fixed Points and Applications, Springer, Berlin, 1976.