Best proximity point results for Geraghty p-proximal cyclic quasi-contraction in uniform spaces
Keywords:
Best proximity point, cyclic contraction, Geraghty p-proximal quasi-contraction, Geraghty p-proximal cyclic quasi-contraction, uniform spaces
Abstract
In this work, we develop Geraghty p-proximal cyclic quasi-contraction in uniform spaces. The existence and uniqueness of best proximity points for these contractions are proved. The main results, apart from the fact that they are new in literature, generalize several other similar results in literature. An illustrative example is given to validate the applicability of the results obtained.
References
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Basha, S. S.. Best proximity points: Optimal solution, J. Optim. Theory Appl., 151 (2011), 210–216.
Bilgili, N.; Karapı́nar, E. and Sadarangani, K.. A generalization for the best proximity point of Geraghty-contractions, J. Inequal. Appl., 2013:286, (2013).
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Geraghty, M.;On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608.
Hamzehnejadi, J. and Lashkaripour, R.. Best proximity points for generalized α-φ-Geraghty proximal contraction mappings and its applications, Fixed Point Theory Appl., 2016:72, (2016).
Hussain, N.; Karapı́nar, E.; Sedghi, S.; Shobkolaei, N. and Firouzian, S.; Cyclic (φ)-contractions in uniform spaces and related fixed point results, Abstract and Applied Anal., 2014, article ID 976859, (2014), 7 pages.
Jleli, M. and Samet, B.. An optimization problem involving proximal quasi-contraction mappings, Fixed Point Theory Appl., 2014:141, (2014).
Karapı́nar, E. and Erhan, I. M.. Best proximity point on different type of contractions, Applied Math. Info. Sci., 5 (2011), 558–569.
Kiany, F. and Amini-Harandi, A.. Fixed point theory for generalised Ciric quasi-contraction maps in metric spaces, Fixed Point Theory Appl., (2013), doi:10.1186/1687-1812-2013-26.
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Mongkolkeha, C.; Cho, Y. J. and Kumam, P.. Best proximity point for Geraghty’s proximal contraction mappings, Fixed Point Theory Appl., 2013:180, (2013).
Olaleru, J. O.. Some generalizations of fixed point theorems in cone metric spaces, Fixed Point Theory Appl., 2009:657914, (2010), 10 pages.
Olaleru, J. O.. Common fixed points of three self-mappings in cone metric spaces, Appl. Math. E-Notes 11 (2010), 41–49.
Olaleru, J. O.; Olisama, V. O. and Abbas, M.; Coupled best proximity points for generalised Hardy-Rogers type cyclic (ω)-contraction, Int. J. Math. Anal. Optim., 1 (2015), 33–54.
Olisama, V. O.; Olaleru, J. O. and Akewe, H.. Best proximity points results for some contractive mappings in uniform spaces, Int. J. Anal., (2017), Article I.D. 6173468, 8 pages.
Olisama, V. O.; Olaleru, J. O. and Akewe, H.. Best proximity points results for Hardy-Rogers p-proximal cyclic contraction in uniform spaces, Fixed Point Theory Appl., 2018:18, (2018).
Rodríguez-Montes, J. and Charris, J. A.; Fixed points for W -contractive or W -expansive maps in uniform spaces: toward a unified approach, Southwest J. Pure Appl. Math., 1 (2001), electronic, 93–101.
Umudu, J. C.; Olaleru, J. O. and Mogbademu, A. A.. Fixed points of involution mappings in convex uniform spaces, Commun. Nonlinear Anal., 7(1) (2019), 50–57.
Umudu, J. C.; Olaleru, J. O. and Mogbademu, A. A.. Fixed point results for Geraghty quasi-contraction type mappings in dislocated quasi-metric spaces, Fixed Point Theory Appl., (2020), doi: 10.1186/s13663-020-00683-z.
Weil, A.. Sur les espaces a structure uniforme et sur la topologie generale, Act. Sci. Ind., 551, Paris, 1937.
Amini-Harandi, A.. Best proximity point theorem for cyclic strongly quasi-contraction mappings, J. Glob. Optim., doi:10.1007/s 1089-012-9953-9, (2012).
Banach, S.. Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fundam. Math., 3 (1922), 133–181.
Basha, S. S.. Best proximity points: Optimal solution, J. Optim. Theory Appl., 151 (2011), 210–216.
Bilgili, N.; Karapı́nar, E. and Sadarangani, K.. A generalization for the best proximity point of Geraghty-contractions, J. Inequal. Appl., 2013:286, (2013).
Bourbaki, N.. Topologie generale, Chapitre 1: Structures topologiques, Chapitre 2: Structures uniformes. Quatrieme edition. Actualites Scientifiques et Industrielles, No. 1142. Hermann, Paris, 1965.
Caballero, J.; Harjani, J. and Sadarangani, K.. A best proximity point theorem for Geraghty-contractions, Fixed Point Theory Appl., 2012:231, (2012).
Cho, S.; Bae, J. and Karapı́nar, E.. Fixed point theorem of α- Geraghty contractive maps in metric spaces, Fixed Point Theory Appl., doi:10.1186/1687-1812-2013-329, (2013).
Ciric, L. B.. A generalization of Banach’s contraction principles, Proc. Amer. Math. Soc. 45(2) (1974), 267–273.
Dhagat, V. B.; Singh, V. and Nath, S.. Fixed point theorems in uniform spaces, Int. J. Math. Anal., 3 (2009), 197–202.
Eldred, A. A. and Veeramani, P.. Existence and convergence of best proximity points, J. Math Anal. Appl. 323 (2006), 1001–1006.
Geraghty, M.;On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608.
Hamzehnejadi, J. and Lashkaripour, R.. Best proximity points for generalized α-φ-Geraghty proximal contraction mappings and its applications, Fixed Point Theory Appl., 2016:72, (2016).
Hussain, N.; Karapı́nar, E.; Sedghi, S.; Shobkolaei, N. and Firouzian, S.; Cyclic (φ)-contractions in uniform spaces and related fixed point results, Abstract and Applied Anal., 2014, article ID 976859, (2014), 7 pages.
Jleli, M. and Samet, B.. An optimization problem involving proximal quasi-contraction mappings, Fixed Point Theory Appl., 2014:141, (2014).
Karapı́nar, E. and Erhan, I. M.. Best proximity point on different type of contractions, Applied Math. Info. Sci., 5 (2011), 558–569.
Kiany, F. and Amini-Harandi, A.. Fixed point theory for generalised Ciric quasi-contraction maps in metric spaces, Fixed Point Theory Appl., (2013), doi:10.1186/1687-1812-2013-26.
Kirk, W. A.; Srinavasan, P. S. and Veeramani, P.; Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory Appl., 4 (2003), 79–89.
Mongkolkeha, C.; Cho, Y. J. and Kumam, P.. Best proximity point for Geraghty’s proximal contraction mappings, Fixed Point Theory Appl., 2013:180, (2013).
Olaleru, J. O.. Some generalizations of fixed point theorems in cone metric spaces, Fixed Point Theory Appl., 2009:657914, (2010), 10 pages.
Olaleru, J. O.. Common fixed points of three self-mappings in cone metric spaces, Appl. Math. E-Notes 11 (2010), 41–49.
Olaleru, J. O.; Olisama, V. O. and Abbas, M.; Coupled best proximity points for generalised Hardy-Rogers type cyclic (ω)-contraction, Int. J. Math. Anal. Optim., 1 (2015), 33–54.
Olisama, V. O.; Olaleru, J. O. and Akewe, H.. Best proximity points results for some contractive mappings in uniform spaces, Int. J. Anal., (2017), Article I.D. 6173468, 8 pages.
Olisama, V. O.; Olaleru, J. O. and Akewe, H.. Best proximity points results for Hardy-Rogers p-proximal cyclic contraction in uniform spaces, Fixed Point Theory Appl., 2018:18, (2018).
Rodríguez-Montes, J. and Charris, J. A.; Fixed points for W -contractive or W -expansive maps in uniform spaces: toward a unified approach, Southwest J. Pure Appl. Math., 1 (2001), electronic, 93–101.
Umudu, J. C.; Olaleru, J. O. and Mogbademu, A. A.. Fixed points of involution mappings in convex uniform spaces, Commun. Nonlinear Anal., 7(1) (2019), 50–57.
Umudu, J. C.; Olaleru, J. O. and Mogbademu, A. A.. Fixed point results for Geraghty quasi-contraction type mappings in dislocated quasi-metric spaces, Fixed Point Theory Appl., (2020), doi: 10.1186/s13663-020-00683-z.
Weil, A.. Sur les espaces a structure uniforme et sur la topologie generale, Act. Sci. Ind., 551, Paris, 1937.
Published
2020-12-27
How to Cite
Chinyere Umudu, J., Olajire Olaleru, J., & Alao Mogbademu, A. (2020). Best proximity point results for Geraghty p-proximal cyclic quasi-contraction in uniform spaces. Divulgaciones Matemáticas, 21(1-2), 21-32. Retrieved from https://produccioncientificaluz.org/index.php/divulgaciones/article/view/36601
Section
Research papers
