Document Type : Original Article

**Author**

Department of Pure Mathematics, Payame Noor University (PNU), P. O. Box: 19395-3697, Tehran, Iran

**Abstract**

In the paper, we introduce a new class of noncyclic $\varphi$-contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp´ınola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic $\varphi$-contractions as a consequence of our main theorems.

**Keywords**

**Main Subjects**

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Funct. Anal. Optim., 34 (2013), pp. 845–860.

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and metric spaces, Fixed Point Theory, 17 (2016), pp. 63–84.

Theory Appl., 153 (2012), pp. 298–305.

[2] , Proximal quasi-normal structure and a best proximity point theorem, J. Nonlinear Convex Anal., 14

(2013), pp. 653–659.

[3] A. A.-H. Akram Safari-Hafshejani and M. Fakhar, Best proximity points and fixed points results for

noncyclic and cyclic fisher quasi-contractions, Numer. Funct. Anal. Optim., 40 (2019), pp. 603–619.

[4] M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear

Anal., 70 (2009), pp. 3665–3671.

[5] A. A. Eldred, W. A. Kirk, and P. Veeramani, Proximal normal structure and relatively nonexpansive

mappings, Studia Math., 171 (2005), pp. 283–293.

[6] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl.,

323 (2006), pp. 1001–1006.

[7] R. Esp´ınola and A. Fernandez-Le ´ on´ , On best proximity points in metric and Banach spaces, Canad. J.

Math., 63 (2011), pp. 533–550.

[8] R. Esp´ınola and M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numer.

Funct. Anal. Optim., 34 (2013), pp. 845–860.

[9] A. Fernandez-Le ´ on and M. Gabeleh ´ , Best proximity pair theorems for noncyclic mappings in Banach

and metric spaces, Fixed Point Theory, 17 (2016), pp. 63–84.

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Mathematics, Cambridge University Press, Cambridge, 1990.

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p-cyclic orbital ϕ-contraction map, Nonlinear Anal. Model. Control, 27 (2022), pp. 91–101.

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metric spaces with the UC and ultrametric properties, Fixed Point Theory, 23 (2022), pp. 507–518.

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property UC, Nonlinear Anal., 71 (2009), pp. 2918–2926.

[14] Q. Zhang and Y. Song, Fixed point theory for generalized ϕ-weak contractions, Appl. Math. Lett., 22 (2009),

pp. 75–78.

Mathematics, Cambridge University Press, Cambridge, 1990.

[11] P. Magadevan, S. Karpagam, and E. Karapı nar, Existence of fixed point and best proximity point of

p-cyclic orbital ϕ-contraction map, Nonlinear Anal. Model. Control, 27 (2022), pp. 91–101.

[12] A. Safari-Hafshejani, The existence of best proximity points for generalized cyclic quasi-contractions in

metric spaces with the UC and ultrametric properties, Fixed Point Theory, 23 (2022), pp. 507–518.

[13] T. Suzuki, M. Kikkawa, and C. Vetro, The existence of best proximity points in metric spaces with the

property UC, Nonlinear Anal., 71 (2009), pp. 2918–2926.

[14] Q. Zhang and Y. Song, Fixed point theory for generalized ϕ-weak contractions, Appl. Math. Lett., 22 (2009),

pp. 75–78.

July 2024

Pages 267-274