Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings

Abstract

In this paper we introduce the concept of cone metric spaces with Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. With this modification, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants. An example shows that our main results concerning the fixed point theorems in the setting of cone metric spaces with Banach algebras are more useful than the standard results in cone metric spaces presented in the literature. MSC: 54H25; 47H10

Tripled coincidence points for mixed comparable mappings in partially ordered cone metric spaces over Banach algebras

“…The following definitions and results come from Huang and Zhang [7] and Liu and Xu [14], which are needed in the sequel.…”

“…But the proofs of the main results in [14] strongly depend on the condition that the underlying solid cone is normal. In 2014, without the assumption of normality of the cone involved, Xu and Stojan [20] proved the conclusions of [14] remain valid by means of some properties of spectral radius.…”

“…The results given by V. Berinde and M. Borcut [1] generalized and extended the works of Bhaskar and Lakshmikantham and Sabetghadam. In 2007, Huang and Zhang [7] introduced the concept of cone metric spaces as a generalization of general metric spaces, in which the distance d(x, y) of x and y is defined to be a vector in an ordered Banach space E and proved that the Banach contraction principle remains true in the setting of cone metric spaces. Since then, many fixed point results of the mappings with certain contractive property on cone metric spaces have been proved on the basis of the work of Huang and Zhang [7] (see [2,3,4,5,8,9,10,11,12,13,14,15,16,17,18,20] and the references therein). Among those works, the results of [15] attract much attention since the authors of [15] introduced the concept of cone metric spaces over Banach algebras by replacing Banach spaces with Banach algebras in order to generalize the Banach contraction principle to a more general form.…”

“…Among those works, the results of [15] attract much attention since the authors of [15] introduced the concept of cone metric spaces over Banach algebras by replacing Banach spaces with Banach algebras in order to generalize the Banach contraction principle to a more general form. In 2013, Liu and Xu [14] proved some fixed point theorems of generalized Lipschitz mappings with weaker and natural conditions on the generalized Lipschitz constant k by means of spectral radius in the setting of cone metric spaces over Banach algebras. But the proofs of the main results in [14] strongly depend on the condition that the underlying solid cone is normal.…”

“…In 2013, Liu and Xu [14] proved some fixed point theorems of generalized Lipschitz mappings with weaker and natural conditions on the generalized Lipschitz constant k by means of spectral radius in the setting of cone metric spaces over Banach algebras. But the proofs of the main results in [14] strongly depend on the condition that the underlying solid cone is normal. In 2014, without the assumption of normality of the cone involved, Xu and Stojan [20] proved the conclusions of [14] remain valid by means of some properties of spectral radius.…”