G-cone metric Spaces over Banach Algebras and Some Fixed Point Results
Keywords:
Banach algebras, fixed point,, G-cone metric spaces, Spectral radius.
Abstract
In this paper, the notion of a G-cone metric space over Banach algebras and the generalized contractive mapping defined on G -cone metric space over Banach algebras are introduced. Some new fixed point results for the maps are proved without the assumption of normality. The results are significant extensions of fixed point results of maps on G-cone metric and metric spaces in literature.
References
[1] Gahler, S. 2-Metrische Raume und ihre topologische Struktur. Mathematishe Nachrichten 115-148 (1963).
[2] Gahler, S. Zur geometric 2-metric raume. Apppliquees 40 , 664-669 (1966).
[3] Ha, K. S., White, A. & Cho, Y. J. Strictly convex and strictly 2-convex 2-normed spaces. Mathematica Japonica 33(2) , 375?384 (1988).
[4] Dhage, B. Generalized metric space and mapping with fixed points. Bulletin of the Calcutta Mathematical Society 84 , 329-336 (1992).
[5] Dhage, B. Generalized metric space and topological structure I. Analele Atintifice ale Universitatii Al. I. Cuza din lasi. Serie Noua Mathematica 46 , 3-24 (2000).
[6] Dhage, B. On generalized metric space and topological structure II. Pure and Applied Mathematika Sciences 40(2) , 37-41 (1994).
[7] Dhage, B. On continuity of mapping in D -metric space. Bulletin of Calcutta Mathematical Society 86(6) , 503?508 (1994).
[8] Mustafa, Z.& Sims, B. A new approach to generalized metric spaces. J. Nonlinear Convex Analysis 7(2) , 289-297 (2006).
[9] Mustafa, Z.& Sims, B. Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory and Applications 2009 , 1-10 (2009).
[10] Mustafa, Z.& Sims, B.Some remarks concerning D -metric spaces. Proceedings of the International Conference on Fixed Point Theory and Applications, Valencia (Spain), July 2003 189-198 (2004).
[11] Mustafa, Z. A new structure for generalized metric spaces with applications to fixed point theory. PhD thesis, University of Newcastle, Callaghan, Australia (2005).
[12] Huang, L.& Zhang, X. Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2) , 1468?1476 (2007).
[13] Olaleru, J. O. Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009 , 1-10 (2009).
[14] Olaleru, J. O. Common fixed points of three self-mappings in cone metric spaces. Applied Mathematics, E-Note 11 , 41-49 (2010).
[15] Arandelovic, I.& Keckic, D. J. TVS-cone metric spaces as a special case of metric spaces. (preprint), arXiv:1202.5930 (2012).
[16] Du, W. S. A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 2259-2261 (2010).
[17] Abbas, M., Rajic, V. C., Nazir, T.& Radenovic, S. Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces. Afr. Math. doi:10.1007/s13370-013-0185 (2013).
[18] Hao, L.& Shaoyuan, X. Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory and Applications 2013 , 1-10 (2013).
[19] Rudin W. Functional Analysis. 2nd ed. McGraw-Hill, New York (1991).
[20] Xu, S.& Radenovic, S. Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebra without assumption of normality. Fixed Point Theory Appl. 2014 , 1-12 (2014).
[21] Radenovic, S.& Rhoades, B. E. Fixed point theorem for two non-self mappings in cone metric spaces. Comput. Math. Appl. 57 , 1701?1707 (2009).
[22] Frechet, M. Sur queiques points du calcul fonctionnel. Rendic. Circ. Mat. Palermo (1906).
[23] Banach, S. Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae 133-181 (1922).
[2] Gahler, S. Zur geometric 2-metric raume. Apppliquees 40 , 664-669 (1966).
[3] Ha, K. S., White, A. & Cho, Y. J. Strictly convex and strictly 2-convex 2-normed spaces. Mathematica Japonica 33(2) , 375?384 (1988).
[4] Dhage, B. Generalized metric space and mapping with fixed points. Bulletin of the Calcutta Mathematical Society 84 , 329-336 (1992).
[5] Dhage, B. Generalized metric space and topological structure I. Analele Atintifice ale Universitatii Al. I. Cuza din lasi. Serie Noua Mathematica 46 , 3-24 (2000).
[6] Dhage, B. On generalized metric space and topological structure II. Pure and Applied Mathematika Sciences 40(2) , 37-41 (1994).
[7] Dhage, B. On continuity of mapping in D -metric space. Bulletin of Calcutta Mathematical Society 86(6) , 503?508 (1994).
[8] Mustafa, Z.& Sims, B. A new approach to generalized metric spaces. J. Nonlinear Convex Analysis 7(2) , 289-297 (2006).
[9] Mustafa, Z.& Sims, B. Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory and Applications 2009 , 1-10 (2009).
[10] Mustafa, Z.& Sims, B.Some remarks concerning D -metric spaces. Proceedings of the International Conference on Fixed Point Theory and Applications, Valencia (Spain), July 2003 189-198 (2004).
[11] Mustafa, Z. A new structure for generalized metric spaces with applications to fixed point theory. PhD thesis, University of Newcastle, Callaghan, Australia (2005).
[12] Huang, L.& Zhang, X. Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2) , 1468?1476 (2007).
[13] Olaleru, J. O. Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory and Applications 2009 , 1-10 (2009).
[14] Olaleru, J. O. Common fixed points of three self-mappings in cone metric spaces. Applied Mathematics, E-Note 11 , 41-49 (2010).
[15] Arandelovic, I.& Keckic, D. J. TVS-cone metric spaces as a special case of metric spaces. (preprint), arXiv:1202.5930 (2012).
[16] Du, W. S. A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 2259-2261 (2010).
[17] Abbas, M., Rajic, V. C., Nazir, T.& Radenovic, S. Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces. Afr. Math. doi:10.1007/s13370-013-0185 (2013).
[18] Hao, L.& Shaoyuan, X. Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings. Fixed Point Theory and Applications 2013 , 1-10 (2013).
[19] Rudin W. Functional Analysis. 2nd ed. McGraw-Hill, New York (1991).
[20] Xu, S.& Radenovic, S. Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebra without assumption of normality. Fixed Point Theory Appl. 2014 , 1-12 (2014).
[21] Radenovic, S.& Rhoades, B. E. Fixed point theorem for two non-self mappings in cone metric spaces. Comput. Math. Appl. 57 , 1701?1707 (2009).
[22] Frechet, M. Sur queiques points du calcul fonctionnel. Rendic. Circ. Mat. Palermo (1906).
[23] Banach, S. Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae 133-181 (1922).
Published
2019-10-17
How to Cite
Adewale , O. K., & Osawaru , E. K. (2019). G-cone metric Spaces over Banach Algebras and Some Fixed Point Results. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2019(2), 546 - 557. Retrieved from http://ijmso.unilag.edu.ng/article/view/480
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