Fixed Point Theorems for Multi-valued $\alpha$-$F$- contractions in Partial metric spaces with an Application

Year 2021, Volume: 4 Issue: 3, 130 - 148, 30.09.2021

Lucas Wangwe , Santosh Kumar

Abstract

This paper aims to prove a fixed point theorem for multi-valued mapping using $\alpha-F$-contraction in partial metric spaces. Furthermore, a fixed point theorem is proved for F-Hardy-Roger’s multi-valued mappings in ordered partial metric spaces. Specifically, this paper intends to generalize the theorems by Ali and Kamran [3], Sgroi and Vetro
[32] and Kumar [15]. We also provided illustrative examples and an application to integral equations.

Keywords

Fixed point, multi-valued mapping, $\alpha$-F-contraction, Hardy-Rogers contraction, partial metric spaces, integral equation.

Supporting Institution

None

References

• [1] M. Abbas, B. Ali and S. Romaguera, Fixed and periodic points of generalized contraction in metric spaces, Fixed point Theory and Applications, (2013)(1)(2013):1–11.
• [2] O. Acar, G. Durmaz and G. Minak, Generalized multivalued F - contractions on complete metric spaces, Bulleti of the Iranian Mathematical Society, 40(6)(2014):1469-1478.
• [3] M. U. Ali and T. Kamran, Multivalued F-Contractions and related fixed point theorems with an application, Filomat, 30(14)(2016):3779-3793.
• [4] I. Altun, G. Minak and H. Dag, Multivalued F-contractions on complete metric space, J. Nonlinear Convex Anal, 16(4)(2015):659-666.
• [5] H. Aydi, M. Abbas and C. Vetro, Partial Hausdorff and Nadler’s fixed point theorem on partial metric space, Topology appl, 159(14)(2012):3234–3242.
• [6] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3(1922):133–181.
• [7] M. Cosentino and P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat, 28(4)(2014):715-722.
• [8] C. Chifu and G. Petrusel, Fixed point results for multi valued hardyrogers contractions in b-metric spaces, Filomat, 31(8)(2017):2499-2507.
• [9] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl, 1(2012):94.
• [10] G. Durmaz, I. Altun, Fixed points results for -admissible multivalued F-contractions, Miskolc of Mathematics Notes, 17(1)(2016):187-199.
• [11] G. Durmaz, I. Altun and G. Minak, Fixed points of ordered F-contractions, Hacettepe Journal of Mathematics and Statistics, 45(1)(2016):15-21. [12] D. Gopal, M. Abbas, D. K. Patel and C. Vetro, Fixed points of a- type F-contractive mappings with an application to nonlinear fractionaldifferential equation. Acta math. Sci., 36(3)(2016):957-970.
• [13] N. Goswami, N. Haokip and V. N. Mishara, F-contractive type mappings in b-metric spaces and some related fixed points results, Springer open Journal of Fixed point Theory and Applications, 2019(13)(2019):1-17.
• [14] G. E. Hardy and T. D. Rogers, A generalization of a fixed point theorem of Reich, Can. Math. Bull., 16(2)(1973):201–206.
• [15] S. Kumar, Coincidence points for a pair of ordered F-contraction mappings in ordered partial metric space, Malaya Journal of Matematiix, 7(3)(2019): 423-428.
• [16] A. Lukács and S. Kajántó, Fixed point theorems for various types of F-contractions in complete b-metric spaces, Fixed Point Theory, 19(1)(2018):321-334. https://doi.org/10.24193/fpt-ro.2018.1.25
• [17] S. Matthews, Partial metric topology in Papers on General Topology and Applications, Eighth Summer Conference at Queens College, Eds. S. Andima et al., Annals of the New York Academy of Sciences, 728(1994):183–197.
• [18] G. Minak, A. Helvaci and I. Altun, Ciric type generalized F-contractions on complete metric spaces and fixed point results, Filomat, 28(6)(2014): 1143-1151. https://doi.org/10.2298
• [19] B. Mohammadi, S. Rezapour, N Shahzad, Some results on fixed points of $ \alpha-\varphi$ Ciric generalized multifunctions. Fixed Point Theory Appl., (2013) 2013:24 doi:10.1186/1687-1812-2013-24.
• [20] S. B. Nadler, Multi-valued contraction mappings, Amer. Pacific Journal of Mathematics, 30(2)(1969):475–488.
• [21] H. K. Nashine, Z. Kadelburg, S. Radenovic and J. K. Kim, Fixed point theorems under Hardy-Rogers contractive conditions on 0-complete ordered partial metric spaces, Fixed Point Theory and Applications, 2012(1)(2012):180.
• [22] J. J. Nieto and R. Rodrigurz-Lopez, Contractive mappings theorems in partially ordered sets and applications to ordinary differential equations, A journal on the Theory of odered sets and its application, 22(3)(2005):223–239.
• [23] D. O’Regan and A. Petrusel, A fixed point theorems for generalized contraction in ordered metric spaces, J. Math. Anal. Appl, 341(2)(2008):1241-1252.
• [24] S. Oltra and O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Istid Mat univer. Trieste, 36(1-2)(2004):17–26.
• [25] S. Oltra, S. Romaguera and E. A. Sa’nchez-pe’rez, Bicompleting weightable quasi metric spaces and partial metric spaces, Rend. Ciric Mat palermo, 51(1)(2002):151–162.
• [26] D. Paesano and C. Vetro, Multi-valued F-contractions in 0-complete partial metric spaces with application to Volterra type integral equation, Revista de la Real Academia de Ciencias Exactas Fisicas Naturales, 108(2)(2014):1005-1020.
• [27] H. Piri and P. Kumam, Some fixed point theorems concerning F- contraction in complete metric spaces, Fixed Point Theory Appl, (1) (2014):210. https://doi.org/10.1186/1687-1812-2014-210
• [28] S. Radojevic, L. Paunovic, S. Radenovi’c, Abstract metric spaces and Hardy-Rogers type theorems, Applied Math. Letters, 24(2011):553-558.
• [29] A. C. M. Ran and M. C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc, 132(2004):1435–1443.
• [30] S. Reich, Kannarf’s fixed point theorem, Bull. Univ. Mat. Italiana, 4(4) (1971):1-11.
• [31] N. A. Secelean, Weak F-contractions and some fixed point results, Bull. Iran. Math. Soc, 42(3)(2016):779-798.
• [32] M. Sgroi and C. Vetro, Multivalued F-contractions and the solution of certain functional and integral equations, Filomat, 27(7)(2013):1259- 1268.
• [33] N. Shahzard and O. Valero, On-O-complete partial metric spaces and quantitative fixed point technique, Abstr. Appl. Anal, 2013(2013):1–11.
• [34] A. Shoaib, A. Asif, M. Arshad and E. Ameer, Generalized dynamic process for generalized Multivalued Contractions of Hardy-Rogers-type in b-metric spaces, Turkish Journal of Analysis and Number theory, 6(2)(2018): 43-48.
• [35] W. Sintunavarat, A new approach to  􀀀 -contractive mappings and generalized Ulam-Hyers stability, well posedness and limit shadowing results, Carpathian J. Math., 31(2015):395-401.
• [36] D. Wardowski and N. V. Dung, Fixed points of F-weak contractions on complete metric spaces, Demonstr. Math, 47(1)(2014):146-155. https://doi.org/10.2478/dema-2014-0012
• [37] I. Beg and A. R. Butt, Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces, Mathematical Communications. Math, 15(1)(2010):65-76. https://doi.org/10.2478/dema-2014-0012