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The aim of this paper is to prove some existence and uniqueness results of the fixed points for Hardy-Rogers type contraction in cone metric spaces associated with a 𝑐-distance and endowed with a graph. These results prepare a more general statement, since we apply the condition of orbitally 𝐺-continuity of mapping instead of the condition of continuity, and consider cone metric spaces endowed with a graph instead of cone metric spaces.

In this paper, we prove the existence of fixed points for any mapping that is a Banach G-contraction with respect to a c-distance on a cone metric space endowed with a graph.

In this paper, we prove some properties of algebraic cone metric spaces and introduce the notion of algebraic distance in an algebraic cone metric space. As an application, we obtain some famous fixed point results in the framework of this algebraic distance.

The notion of coupled fixed point was introduced in 2006 by Bhaskar and Lakshmikantham. On the other hand, Filipovićet al. [M. Filipovićet al., “Remarks on “Cone metric spaces and fixed-point theorems of T-Kannan and T-Chatterjea contractive mappings”,” Math. Comput. Modelling 54, 1467–1472 (2011)] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper we prove some coupled fixed-point theorems for certain T-contractions and study the existence of solutions of a system of nonlinear integral equations using the results of our work. The results of this paper extend and generalize well-known comparable results in the literature.

We present a generalization of several fixed and common fixed point theorems on c -distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature. Наведено узагальнення деяких теорем про нерухому точку та спільну нерухому точку для с-відстані в упорядкованих конічних метричних просторах. Таким чином, покращено та узагальнено різноманітні результати, що наведені в літературі.