Banach Cebirli Koni Modüler Uzaylarda Banach Büzülme Prensibi

Bilinen anlamda büzülme dönüşümü olmayan öyle dönüşümler vardır ki bu dönüşümler bazı yeni metrik ve modüler uzay yapılarında bazı büzülme tipinde koşulları sağlarlar. Biz bu makalede bu durumu göz önünde bulundurarak Banach cebirlerdeki konilerin yardımıyla yeni bir modüler uzay kavramı sunduk. İlk kısımda temel tanım ve notasyonlar verildi. İkinci kısımda Banach Büzülme Prensibinin ? ∗-cebir değerli modüler uzaylardaki sonucuyla klasik modüler uzaylardaki sonucunun denkliği gösterildi. Sonra yukarıda bahsedilen o modüler uzaya giriş yapıldı ve bazı sonuçlar verildi. Son olarak çalışma bir örnekle desteklendi.

Banach Contraction Principle in Cone Modular Spaces with Banach Algebra

There are some mappings, which are not contraction mappings in the usual senses, such that they holdsome contractive type conditions in the settings of some new abstract metric and modular spaces. Inthis paper, taking into account this fact, we introduce such a new type modular space by using thesetting of cones in Banach algebras. In the first section, some basic notions and definitions are given. Inthe second part, it is shown that some result of Banach Contraction Principle in modular space with ?∗-algebra is equal to the result of Banach Contraction Principle of the usual modular space. Then that newmodular space mentioned above is introduced and some results are given. Finally the work is concludedwith an example.

PDF

___

Alsulami, H.H., Agarwal, R.P., Karapınar, E., Khojasteh, F., 2016. A short note on ? ∗ -algebra-valued contraction mappings. Journal of Inequalities and Applications, 50.
Banach, S., 1922. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133-181.
Gerstewitz, Chr., 1983. Nichtkonvexe dualitat in der vektaroptimierung, Wissenschaftlichte Zeitschrift T H Leuna-mersebung, 25.
Huang, H., Radenovic’, S., Deng, G., 2017. A sharp generalization on cone b-metric space over Banach algebra. J. Nonlinear Sci. Appl., 10, 429-435.
Kadelburg Z., Radenovic’, S., 2016. Fixed point results in ? ∗ -algebra-valued metric spaces are direct consequences of their standard metric counterparts. Fixed point theory and applications, 53.
Khamsi, M.A., Kozlowski, W.M., 1990. Fixed point theory in modular function spaces. Nonlinear Anal. 14, no. 11, 935-953.
Khamsi, M.A., Kozlowski, W.M., 2015.Fixed point theory in modular function spaces, Springer/Birkhauser, New York.
Kozlowski, W.M., 1998. Modular Function Spaces, Dekker, New York.
Liu, H., Xu, S., 2013. Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Applications, 320.
Ma Z., Jiang, L., Sun, H., 2014. ? ∗ -algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory Appl. 206, no. 1.
Murphy, G.J., 1990, ? ∗ -algebra and operator theory, Academic press, INC..
Musielak, J., 1983. Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer, Berlin.
Musielak, J., Orlicz, W., 1959. On modular spaces, Studia Math, 18, 49-56.
Nakano, H., 1950. Modulared semi-ordered linear spaces, Tokyo Mathematics Book Series, 1, Maruzen Co., Tokyo.
Rudin, W., 1991. Functional Analysis McGraw-Hill, New York.
Shateri, T.L., 2017, ? ∗ -algebra-valued modular spaces and fixed point theorems, J. Fixed Point Theory Appl. 19, no. 2, 1551-1560.
Xu, S., Radenovic’, S., 2014. Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality, Fixed Point Theory and Applications, 102.