Best Proximity point for Z-contraction and Suzuki type Z-contraction mappings with an application to fractional calculus

Somayya Komal, Poom Kumam, Dhananjay Gopal


In this article, we introduced the best proximity point theorems for $\mathcal{Z}$-contraction and Suzuki type $\mathcal{Z}$-contraction in the setting of complete metric spaces. Also by the help of weak $P$-property and $P$-property, we proved existence and uniqueness of best proximity point. There is a simple example to show the validity of our results. Our results extended and unify many existing results in the literature. Moreover, an application to fractional order functional differential equation is discussed.


best proximity point; weak P-property; Suzuki type Z-contraction; functional differential equation

Subject classification

54H25; 34A08

Full Text:



M. Abbas, A. Hussain and P. Kumam, A coincidence best proximity point problem in $G$-metric spaces, Abst. and Appl. Anal. 2015 (2015), Article ID 243753, 12 pages.

A. Akbar and M. Gabeleh, Generalized cyclic contractions in partially ordered metric spaces, Optim. Lett. 6 (2012), 1819-1830.

A. Akbar and M. Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl. 153 (2012), 298-305.

H. Argoubi, B. Samet and C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl. 8 (2015), 1082-1094.

S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math. 3 (1922), 133-181.

S. S. Basha, Extensions of Banach's contraction principle, Numer. Funct. Anal. Optim. 31 (2010), 569-576.

S. S. Basha, Best proximity points: Global optimal approximate solution, J. Glob. Optim. 49 (2010), 15-21.

S. S. Basha, Best proximity point theorems generalizing the contraction principle, Nonlinear Anal. 74 (2011), 5844-5850.

S. S. Basha, Common best proximity points: Global minimization of multi-objective functions, J. Glob. Optim. 54 (2012), 367-373.

S. S. Basha, N. Shahzad and R. Jeyaraj, Best proximity points: Approximation and optimization, Optim. Lett. 7 (2011), 145-155.

D. Delboso and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625.

M. Edestein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79.

K. Fan, Extensions of two fixed point theorems of F. E. Browder, Mathematische Zeitschrift 112 (1969), 234-240.

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), 11-41

F. Khojasteh, S. Shukla and S. Redenovi, A new approach to the study fixed point theorems via simulation functions, Filomat 29 (2015), 1188-1194.

P. Kumam, D. Gopal and L. Budhiyi, A new fixed point theorem under Suzuki type $Z$-contraction mappings, preprint.

A. Nastasi and P. Vetro, Fixed point results on metric and partial metric spaces via simulation functions, J. Nonlinear Sci. Appl. 8 (2015), 1059-1069.

M. Olgun, O. Bier and T. Alyldz, A new aspect to Picard operators with simulation functions, Turk. J. Math. 40 (2016), 832-837.

V. S. Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal. 74 (2011), 4804-4808.

A. Roldán-López-de-Hierro, E. Karapinar, C. Roldán-López-de-Hierro, J. Martínez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math. 275 (2015), 345-355.

A. Roldán-López-de-Hierro and N. Shahzad, Common fixed point theorems under (R,S)-contractivity conditions, Fixed Point Theory Appl. 2016 (2016), 55.

A. Roldán-López-de-Hierro and N. Shahzad, New fixed point theorem under R-contractions, Fixed Point Theory Appl. 2015 (2015), 98.

N. Shahzad, A. Roldán-López-de-Hierro, F. Khojasteh, Some new fixed point theorems under (A,S)-contractivity conditions, RACSAM, to appear.

W. Sintunavarat and P. Kumam, Coupled best proximity point theorem in metric spaces, Fixed Point Theory Appl. 2012 (2012), 93.

Y. Sun,Y. Su and J. Zhang, A new method for research of best proximity point theorems for non linear mappings, Fixed Point Theory Appl. 2014 (2014), 116.

T. Suzuki, A generalized Banach contraction principle which characterizes metric completeness, Proc. Am. Math. Soc. 136, no. 5 (2008), 1861-1869.

J. Zhang, Y. Su and Q. Chang, A note on á best proximity point theorem for Geraghty-contractions', Fixed Point Theory Appl. 2013 (2013), 99.

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