Graphical structure of extended b-metric spaces: an application to the transverse oscillations of a homogeneous bar

The purpose of this article is to present the notion of graphical extended b-metric spaces, blending the concepts of graph theory and metric fixed point theory. We discuss the structure of an open ball of the new proposed space and elaborate on the newly introduced ideas in a novel way by portraying suitably directed graphs. We also provide some examples in graph structure to show that our results are sharp as compared to the results in the existing state-of-art. Furthermore, an application to the transverse oscillations of a homogeneous bar is entrusted to affirm the applicability of the established results. Additionally, we evoke some open problems for enthusiastic readers for the future aspects of the study.

On Unique and Nonunique Fixed Points and Fixed Circles in M v b -Metric Space and Application to Cantilever Beam Problem

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

FUNGSI ELEMEN KELAS STUMMEL MODULUS TERBATAS TETAPI BUKAN ELEMEN DARI KELAS STUMMEL

In the paper [1], it was given a function which belongs to the bounded Stummel modulus classes but not in Stummel classes. The given proof of this function properties in that paper was not obvious and very concise. By using the countable linearity property of integral, polar coordinate of integration, other properties of Lebesgue measure and integration, and some observation on the geometric property of the open ball in Euclidean spaces, we prove in detail the properties of this function.

Simply connected topological spaces of weighted composition operators

In this paper, we prove that the topological spaces of nonzero weighted composition operators acting on some Hilbert spaces of analytic functions on the unit open ball are simply connected.

Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

On the domain of implicit functions in a projective limit setting without additional norm estimates

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain D which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse function theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x<R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Generalized contraction involving an open ball and common fixed point of multivalued mappings in ordered dislocated quasi metric spaces

The aim of this work is to obtain fixed point results for multivalued mappings satisfying generalized contractions on the intersection of an open ball and a sequence in left (right) K-sequentially complete ordered dislocated quasi metric space. An example has been built to demonstrate the novelty of results. Our results generalize and extend the results of Altun et al. (J. Funct. Spaces, Article ID 6759320, 2016)

A CLASS OF CRITICAL KIRCHHOFF PROBLEM ON THE HYPERBOLIC SPACE

We investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.