Document Type : Research Paper
Authors
- Ebru Altiparmak ^{} ^{}
- Ibrahim Karahan
Department of Mathematics, Faculty of Science, Erzurum Technical University, P.O.Box 25050, Erzurum, Turkey.
Abstract
In this paper, necessary and sufficient conditions for the existence and uniqueness of fixed points of generalized Geraghty type contraction mappings are given in complete partial $b_{v}(s) $-metric spaces. The results are more general than several results that exist in the literature because of the considered space. A numerical example is given to support the obtained results. Also, the existence and uniqueness of the solutions of an integral equation has been verified considered as an application.
Keywords
[2] O. Acar and I. Altun, A fixed point theorem for $ F$-Geraghty contraction on metric-like spaces, Fasc. Math., 59 (2017), pp. 5-12.
[3] H. Afshari, H. Alsulami and E. Karapinar, On the extended multivalued Geraghty type contractions, J. Nonlinear Sci. Appl., 9 (2016), pp. 4695-4706.
[4] S. Aleksic, Z.D. Mitrovic and S. Radenovic, A fixed point theorem of Jungck in $b_{v(s)}$-metric spaces, Period. Math. Hungar., 77 (2018), pp. 224-231.
[5] B. Alqahtani, A. Fulga and E. Karapinar, A short note on the common fixed points of the Geraghty contraction of type $E_{S,T}$, Demonstr. Math., 51 (2018), pp. 233-240.
[6] I. Altun and K. Sadarangani, Generalized Geraghty type mappings on partial metric spaces and fixed point results, Arab J. Math., 2 (2013), pp. 247-253.
[7] M. Arshad and A. Hussain, Fixed point results for generalized rational $alpha $-Geragty contraction, Miskolc Math. Notes, 18 (2017), pp. 611-621.
[8] H. Aydi, A. Felhi and H. Afshari, New Geraghty type contractions on metric-like spaces, J. Nonlinear Sci. Appl., 10 (2017), pp. 780-788.
[9] S. Chandok, Some fixed point theorems for $( alpha ,beta ) $-admissible Geraghty type contractive mappings and related results, Math. Sci., 9 (2015), pp. 127-135.
[10] A.K. Dubey, U. Mishra and W.H. Lim, Some new fixed point theorems for generalized contractions involving rational expressions in complex valued $b$-metric spaces, Nonlinear Funct. Anal. Appl., 24 (2019), pp. 477-483.
[11] D. Dukic, Z. Kadelburg and S. Radenovic, Fixed Points of Geraghty-Type Mappings in Various Generalized Metric Spaces, Abstr. Appl. Anal., 2011 (2011), 13 pages.
[12] O. Ege, Complex valued rectangular b-metric spaces and an application to linear equations, J. Nonlinear Sci. Appl., 8 (2015), pp. 1014-1021 .
[13] I.M. Erhan, Geraghty type contraction mappings on Branciari $b$-metric spaces, Adv. Theory Analysis Appl., 1 (2017), pp. 147-160.
[14] H. Faraji, D. Savic and S. Radenovic, Fixed point theorems for Geraghty type mappings in $b$-metric spaces and Applications, Axioms, 8 (2019), 12 pages.
[15] M.A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), pp. 604-608.
[16] M.E. Gordji, H. Baghani, H. Khodaei and M. Ramezani, Geraghty's fixed point theorem for special Multi-valued mappings, Thai J. Math., 10 (2010), pp. 225-231.
[17] H. Huang, L. Paunovic and S. Radenovic, On some new fixed point results for rational Geraghty contractive mappings in ordered b-metric spaces, J. Inequal. Appl., 8 (2015), pp. 800-807.
[18] A. Hussain, Modified Geraghty contraction involving fixed point theorems, Jordan J. Math. Stat., 10 (2017), pp. 95-112.
[19] M. Jovanovic, Z. Kadelburg and S. Radenovic, Common fixed point results in metric-type spaces, Fixed Point Theory Appl., 2010 (2010), 15 pages.
[20] Z. Kadelburg and P. Kumam, S. Radenovic and W. Sintunavarat, Common coupled fixed point theorems for Geraghty-type contraction mappings using monotone property, Fixed Point Theory Appl., 2015 (2015), 14 pages.
[21] I. Karahan and I. Isik, Partial $b_{v( s)} $ , Partial $v$-generalized and $b_{v( theta )} $ metric spaces and related fixed point theorems, Facta Univ. Ser. Math. Inform., (In Press).
[22] E. Karapinar, On best proximity point of $psi $ -Geraghty contractions, Fixed Point Theory Appl., 2013 (2013), 9 pages.
[23] E. Karapinar, $alpha $-$psi $-Geraghty contraction type mappings and some related fixed point results, Filomat, 28 (2014), pp. 37-48.
[24] E. Karapinar and B. Samet, A note on '$psi $-Geraghty type contractions', Fixed Point Theory Appl., 2014 (2014), 5 pages.
[25] Z.D. Mitrovic, H. Aydi, Z. Kadelburg and G.S. Rad, On some rational contractions in $b_{v( s)} $-metric spaces, Rend. Circ. Mat. Palermo 2 (2019), 11 pages.
[26] Z.D. Mitrovic, H. Aydi and S. Radenovic, On Banach and Kannan type results in cone $b_{v( s)} $-metric spaces over Banach algebra, Acta Math. Univ. Comenian., LXXXIX (2020), pp. 143-152.
[27] Z.D. Mitrovic and S. Radenovic, The Banach and Reich contractions in $b_{v( s)} $-metric spaces, J. Fixed Point Theory Appl., 19 (2017), pp. 3087-3095.
[28] Z. Mostefaoui, M. Bousselsal and J.K. Kim, Some results in fixed point theory concerning rectangular b-metric spaces, Nonlinear Funct. Anal. Appl., 24 (2019), pp. 49-59.
[29] Y.J. Piao, Fixed point theorems for contractive and expansive mappings of Geraghty type on $2$-metric spaces, Adv. Fixed Point Theory, 6 (2016), pp. 123-135.
[30] K.N.V.V. Vara Prasad and A.K. Singh, Fixed point results for rational $alpha $-Geraghty contractive mappings, Adv. Inequal. Appl., 2019 (2019), 15 pages.
[31] V.L. Rosa and P. Vetro, Fixed points for Geraghty contractions in partial metric spaces, J. Nonlinear Sci. Appl., 7 (2014), pp. 1-10.
[32] R.J. Shahkoohi and A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered $b$-metric spaces, J. Inequal. Appl., 2014 (2014), 23 pages.
[33] A.Wiriyapongsanon and N. Phudolsitthiphat, Coincidence point theorems for Geraghty-type contraction mappings in generalized metric spaces, Thai J. Math., (2018), pp. 145-158.