A Proper Generalization of Banach Principle for Nadler Type Mappings in Cone b-Metric Spaces Over Banach Algebras

Yıl 2019, Cilt: 4 Sayı: 1, 11 - 18, 15.04.2019

Faruk Develi , Muttalip Özavşar Stojan Radenovic

https://doi.org/10.30931/jetas.510813

https://izlik.org/JA84UL84HN

Öz

In this paper, we first consider Nadler type contractions with the
generalized Lipschitz constant k holding r(k)<1 instead of r(sk)<1
where r(k) is the spectral radius of k and s≥1 is the coefficient
of the underlying cone b-metric spaces over Banach algebras. Then, we
prove the corresponding fixed point theorem for such mappings. Finally, we
compare our result with one obtained by the case r(sk)<1 by introducing
some proper examples.

Anahtar Kelimeler

Nadler Mappings; Banach Algebra; Cone b-Metric

Kaynakça

• Nadler, S.B., “Multi-valued contraction mappings”, Pac. J. Math. 30(2) (1969) : 475–488.
• Czerwik, S., “Contraction mappings in b-metric spaces”, Acta Math Inf Univ Ostraviensis 1(1) (1993) : 5–11.
• Huang, L.G., Zhang, X., “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 332(2) (2007) : 1468–1476.
• Du, W.-S., “A note on cone metric fixed point theory and its equivalence”, Nonlinear Anal. 72(5) (2010) : 2259–2261.
• Liu, H., Xu, S., “Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings”, Fixed Point Theory Appl. 2013(1) (2013) : 320.
• Wardowski, D., “On set-valued contractions of Nadler type in cone metric spaces”, Appl. Math. Lett. 24(3) (2011) : 275–278.
• Ozavsar, M., “Nadler mappings in cone b-metric spaces over Banach algebras”, Rendiconti del Seminario Matematico (2018).
• Suzuki, T., “Basic inequality on a b-metric space and its applications”, Journal of inequalities and applications 2017(1) 2017 : 256.
• Huang, H., Radenovic, S., “Common fixed point theorems of generalized Lipschitz mappings in cone b-metric spaces over Banach algebras and applications”, J. Non. Sci. Appl. 8(5) (2015) : 787–799.
• Rezapour, S., Hamlbarani, R., “Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings””, J. Math. Anal. Appl. 345(2) (2008) : 719-724.
• Radenovic, S., Rhoades, B.E., “Fixed point theorem for two non-self mappings in cone metric spaces”, Comput. Math. Appl. 57(10) (2009) : 1701-1707.
• Xu, S., Radenovic, S., “Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality”, Fixed Point Theory Appl. 2014(1) (2014) : 102.
• Rudin, W., Functional Analysis, 2nd edn. McGraw-Hill, New York (1991).
• Kadelburg, Z., Radenovic, S., “A note on various types of cones and fixed point results in cone metric spaces”, Asian J. Math. Appl. (2013).