We introduce the notion of an ls-π-Ponomarev-system to give necessary and sufficient conditions for f : (M,M0) → X to be a strong wcmapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give c...

In this paper, we introduce the M-cone metric space over Banach algebra as a generalization of both M-metric and cone investigate some fixed point results in new settings. Some examples are presented illustrations. Finally, supported by an application to examine existence uniqueness solution for Fredholm integral equation.

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a classical Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become l...

Let (X, d) be a metric space and m ∈ X. Suppose that φ : X×X → R is a nonnegative symmetric function. We define a metric d on X which is equivalent to d. If d is totally bounded, its completion is a compactification of (X, d). As examples, we construct two compactifications of (R, dE), where dE is the Euclidean metric and s ≥ 2. key words. equivalent metric; completion; compactification Mathema...

This paper focuses on the study of open curves in a Riemannian manifold M , and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [13] to define a Riemannian metric on the space of immersions M = Imm([0, 1],M) by pullback of a natural metric on the tangent bundle TM. This induces a first-o...

Let (A,LA) be a quantum metric space. Then clearly S(A) with the metric ρLA is a compact balanced convex metric space, i.e. the metric ρLA is convex and balanced. Another important property of S(A) is that the R−valued affine continuous functions Af(S(A)) separate the points of S(A). Conversly, let (X, ρ) be a compact balanced convex metric space on which Af(X) separate the points of X. Then Af...

In this paper, we obtain a characterization of linear spaces which may be normed so as to become complete, linear, normed metric spaces. In this connection, K. Kunugui and M. Fréchet have shown that every metric space S is isometric with a subset of a complete, linear, normed metric space. I t follows from our result that if the cardinal number of 5 is the limit of a denumerable sequence of car...

Let us consider analogous problem for the metric space (X, d) with at least four points. Let A,B,C ∈ X be three fixed points. Then, for the point M ∈ X we suppose that a triangle can be formed from the distances d1 = d(M,A), d2 = d(M,B) and d3 = d(M,C) iff the following conjunction of inequalities is true: d1 + d2 − d3 ≥ 0 and d2 + d3 − d1 ≥ 0 and d3 + d1 − d2 ≥ 0. (1.1) If in conjunction (1.1)...

A stationary space-time (M,g) is a 4-manifold M with a smooth Lorentzian metric g, of signature (−,+,+,+), which has a smooth 1-parameter group G ≈ R of isometries whose orbits are time-like curves in M . We assume throughout the paper that M is a chronological space-time, i.e. M admits no closed time-like curves, c.f. §1.1 for further discussion. Let S be the orbit space of the action G. Then ...