There are several ways to generalize the notion of the Sobolev space to the setting of metric spaces equipped with a Borel measure. We describe next two definitions of the Sobolev space on a metric space (S, d) equipped with a Borel masure μ that is finite on every ball. Following [11], for 1 ≤ p < ∞, we define the Sobolev space M(S, d, μ) as the set of all u ∈ L(S) for which there exists 0 ≤ g...

‎in this paper we study the existence of fixed points for mappings defined on complete metric spaces‎, ‎satisfying a general contractive inequality depending on two additional mappings‎.

In this paper, we prove a general fixed point theorem in $textrm{S}$-metric spaces for maps satisfying an implicit relation on complete metric spaces. As applications, we get many analogues of fixed point theorems in metric spaces for $textrm{S}$-metric spaces.

a projective parameter of a geodesic as solution of certain ode is defined to be a parameter which is invariant under projective change of metric. using projective parameter and poincaré metric, an intrinsic projectively invariant pseudo-distance can be constructed. in the present work, solutions of the above ode are characterized with respect to the sign of parallel ricci tensor on a finsler s...

We prove that a space is quotient π-image of a locally separable metric space if and only if it has a πand double cs∗cover. We also investigate quotient π-s-images of locally separable metric spaces.

in this paper, tripled coincidence points of mappings satisfying  -contractive conditions in the framework of partially ordered gb-metric spaces are obtained. our results extend the results of aydi et al. [h. aydi, e. karapnar and w. shatanawi, tripled xed point results in generalized metric space, j. applied math., volume 2012, article id 314279, 10 pages]. moreover, some examples of the mai...

The time-optimal trajectory for an airplane from some starting point to some final point is studied by many authors. Here, we consider the extension of robot planer motion of Dubins model in three dimensional spaces. In this model, the system has independent bounded control over both the altitude velocity and the turning rate of airplane movement in a non-obstacle space. Here, in this paper a g...

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the -thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log.gCp /. The same result also holds for the -thick part of the moduli space of metric graphs of rank n equipped with the ...

Providing each simplex in C(S) with the standard euclidean metric of side-length 1 equips the complex of curves with the structure of a geodesic metric space whose isometry group is just M̃g,m (except for the twice punctured torus). However, this metric space is not locally compact. Masur and Minsky [MM1] showed that nevertheless the geometry of C(S) can be understood quite explicitly. Namely, C...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.