Document Type: Research Paper

Authors

1 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

2 Department of Mathematics, Faculty of Science, Imam Khomeini International University, postal code: 34149-16818, Qazvin, Iran

3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Abstract

Best approximation results provide an approximate solution to the fixed point equation $Tx=x$, when the non-self mapping $T$ has no fixed point. In particular, a well-known best approximation theorem, due to Fan cite{5}, asserts that if $K$ is a nonempty compact convex subset of a Hausdorff locally convex topological vector space $E$ and $T:K\rightarrow E$ is a continuous mapping, then there exists an element $x$ satisfying the condition $d(x,Tx)=\inf \{d(y,Tx):y\in K\}$, where $d$ is a metric on $E$. Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, Article ID 837943) introduced proximal contractive mappings and established certain best proximity point results for these mappings in $G$-metric spaces. The aim of this paper is to introduce certain new classes of auxiliary functions and proximal contraction mappings and establish best proximity point theorems for such kind of mappings in $G$-metric spaces. As consequences of these results, we deduce certain new best proximity and fixed point results in $G$-metric spaces. Moreover, we present certain examples to illustrate the usability of the obtained results.

Keywords

Main Subjects

[1] A. Amini-Harandi, N. Hussain, F. Akbar, Best proximity point results for generalized contractions in metric spaces, Fixed Point Theory and Applications. (2013), 2013:164.

[2] M. Asadi, E. Karapinar, P. Salimi, A new approach to G-metric and related fixed point theorems, Journal of Inequalities and Applications. (2013), 2013:454.

[3] H. Aydi, E. Karapinar, P. Salimi, Some fixed point results in GP-metric spaces, J. Appl. Math. (2012), Article ID 891713.

[4] B. S. Choudhury, P. Maity, Best proximity point results in generalized metric spaces, Vietnam J. Mathematics, 10.1007/s10013-015-0141-3.

[5] C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Analysis 69 (11) (2008), 3790-3794.

[6] K. Fan, Extensions of two fixed point theorems of F.E. Browder, Mathematische Zeitschrift 112 (3) (1969), 234-240.

[7] N. Hussain, E. Karapinar, P. Salimi, P. Vetro, Fixed point results for Gm-Meir-Keeler contractive and G- (α, ψ)-Meir-Keeler contractive mappings, Fixed Point Theory and Applications. (2013), 2013:34.

[8] N. Hussain, M. A. Kutbi, P. Salimi, Best proximity point results for modified α-ψ-proximal rational contractions, Abstract and Applied Analysis. (2013), Article ID 927457, 14 pages.

[9] N. Hussain, A. Latif, P. Salimi, Best proximity point results in G-metric spaces, Abstract and Applied Analysis. (2014), Article ID 837943, 8 pages.

[10] N. Hussain, A Latif, P. Salimi, Best proximity point results for modified Suzuki α-ψ-proximal contractions, Fixed Point Theory and Applications. (2014), 2014:10.

[11] M. Jleli, B. Samet, Remarks on G-metric spaces and fixed point theorems, Fixed Point Theory Appl. (2012), 2012:210.

[12] F. Moradlou, P. Salimi, P.Vetro, Some new extensions of Edelstein-Suzuki-type fixed point theorem to Gmetric and G-cone metric spaces, Acta Mathematica Scientia. 33B (4) (2013), 1049-1058.

[13] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), 289-297.

[14] Z. Mustafa, B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl. (2009), Article ID 917175, 10 pages.

[15] Z. Mustafa, V. Parvaneh, M. Abbas, J.R. Roshan, Some coincidence point results for generalized (ψ, ϕ)- weakly contractive mappings in ordered G-metric spaces, Fixed Point Theory and Applications. (2013),  2013:326.

[16] Z. Mustafa, A new structure for generalized metric spaces with applications to fixed point theory, Ph.D. Thesis, The University of Newcastle, Australia, 2005.

[17] S. Radenovi´c, P. Salimi, S. Panteli´c, J. Vujakovi´c, A note on some tripled coincidence point results in G-metric spaces. Int. J. Math. Sci. Eng. Appl. 6 (2012), 23-38.

[18] A. Razani, V. Parvaneh, On generalized weakly G-contractive mappings in partially ordered G-metric spaces, Abstr. Appl. Anal. (2012), Article ID 701910, 18 pages.

[19] S. Sadiq Basha, Extensions of Banach’s contraction principle. Numer. Funct. Anal. Optim. 31 (2010), 569-576.

[20] P. Salimi, P. Vetro, A result of suzuki type in partial G-metric spaces, Acta Mathematica Scientia 34B (2) (2014), 1-11.

[21] B. Samet, C. Vetro, F. Vetro, Remarks on G-metric spaces, Int. J. Anal. (2013) Article ID 917158, 6 pages.

[22] T. Suzuki, M. Kikkawa, C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal. Theory, Methods & Applications, 71 (7-8) (2009), 2918-2926.