Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality

Abstract

In this paper, we first present some elementary results concerning cone metric spaces over Banach algebras. Next, by using these results and the related ones about c-sequence on cone metric spaces we obtain some new fixed point theorems for the generalized Lipschitz mappings on cone metric spaces over Banach algebras without the assumption of normality. As a consequence, our main results improve and generalize the corresponding results in the recent paper by Liu and Xu (Fixed Point Theory Appl. 2013:320, 2013). MSC: 54H25; 47H10

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

“…, where α = j + k + l. Since r(α) = r(j + k + l) < 1, by Remark 2.1 in [20], we get ||α n || → 0, which together with Lemma 2.5, Lemma 2.9 and Lemma 2.10 shows that {g(x n )} is a Cauchy sequence. Similarly one can verify that {g(y n )} and {g(z n )} are Cauchy sequences too.…”

“…Noting that r(j +2k +l) < 1 and by Lemma 2.7, Lemma 2.5 and Remark 2.1 in [20], we have lim n→∞ g(r n ) = g(x), lim n→∞ g(s n ) = g(y) and lim n→∞ g(t n ) = g(z). Similarly, we can prove that lim n→∞ g(r n ) = g(u), lim n→∞ g(s n ) = g(v) and lim n→∞ g(t n ) = g(w).…”

“…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.…”

“…In this paper, on the basis of [1], [15] and [20], we introduce the concept of F having the mixed comparable property with respect to g and prove some tripled coincidence point results of F and g provided F has the mixed comparable property with respect to g and some other natural conditions are satisfied. Moreover, we give some support examples of part of our results.…”

“…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.…”