New best proximity point results in G-metric space
محورهای موضوعی : History and biographyA. H. Ansari 1 , A. Razani 2 , N. Hussain 3
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
کلید واژه: G-metric space, Best proximity point, Generalized proximal weakly G-contraction,
چکیده مقاله :
Best approximation results provide an approximate solution to the fixedpoint equation $Tx=x$, when the non-self mapping $T$ has no fixed point. Inparticular, a well-known best approximation theorem, due to Fan cite{5},asserts that if $K$ is a nonempty compact convex subset of a Hausdorfflocally convex topological vector space $E$ and $T:K\rightarrow E$ is acontinuous mapping, then there exists an element $x$ satisfying thecondition $d(x,Tx)=\inf \{d(y,Tx):y\in K\}$, where $d$ is a metric on $E$.Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, ArticleID 837943) introduced proximal contractive mappings and established certainbest proximity point results for these mappings in $G$-metric spaces. The aimof this paper is to introduce certain new classes of auxiliary functions andproximal contraction mappings and establish best proximity point theoremsfor such kind of mappings in $G$-metric spaces. As consequences of theseresults, we deduce certain new best proximity and fixed point results in$G$-metric spaces. Moreover, we present certain examples to illustrate theusability of the obtained results.
[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. Radenovic, P. Salimi, S. Pantelic, J. Vujakovic, 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.
