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In this paper, we have provided some fixed point results for self-mappings fulfilling generalized contractive conditions on altered metric spaces. In addition, some applications of the main results to continuous data dependence of the fixed points of operators defined on these spaces were shown.

One recent and prolific direction in the development of fixed point theory is to consider an operator T:X→X defined on a metric space (X,d) which is an F—contraction, i.e., T verifies a condition of type τ+F(d(T(x),T(y))≤F(d(x,y)), for all x,y∈X, T(x)≠T(y), where τ>0 and F:(0,∞)→R satisfies some suitable conditions which ensure the existence and uniqueness for the fixed point of operator T. Moreover, the notion of F-contraction over a metric space (X,d) was generalized by considering the notion of (G,H)—contraction, i.e., a condition of type G(d(Tx,Ty))≤H(d(x,y)), for all x,y∈X, Tx≠Ty for some appropriate G,H:(0,∞)→R functions. Recently, the abovementioned F-contraction theory was extended to the setup of cone metric space over the topological left modules. The principal objective of this paper is to introduce the concept of vectorial dislocated metric space over a topological left module and the notion of A-Cauchy sequence, as a generalization of the classical Cauchy sequence concept. Furthermore, based on the introduced concept, a fixed point result is provided for an operator T:X→X, which satisfies the condition (G,H)—contraction, where G,H are defined on the interior of a solid cone.

In this paper, we introduce a new contraction-type mapping and provide a fixed-point theorem which generalizes and improves some existing results in the literature. Thus, we prove that the Boyd and Wong theorem (1969) and, more recently, the fixed-point results due to Wardowski (2012), Turinici (2012), Piri and Kumam (2016), Secelean (2016), Proinov (2020), and others are consequences of our main result. An application in integral equations and some illustrative examples are indicated.

In this paper, we establish some conditions for the existence and uniqueness of the monotonic solutions for nonhomogeneous systems of first-order linear differential equations, by using a result of the fixed points theory for sequentially complete gauge spaces.

In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as well as substantial generalizations and improvements of several well known results in the recent literature. In addition, the paper contains an example which shows that our main results are applicable on a non-metrizable cone metric space over a topological left module. The article proves that fixed point theorems in the framework of cone metric spaces over a topological left module are more effective and more fertile than standard results presented in cone metric spaces over a Banach algebra.