Vol 13 - Forthcoming papers

Advances in Pure and Applied Mathematics

List of Articles

In this paper, we reconsider the notion of a Weyl p-almost automorphic function introduced by S. Abbas [1] in 2012 and propose several new ways for introduction of the class of Weyl p-almost automorphic functions (1 ⩽ p < ∞). We first analyze the introduced classes of Weyl p-almost automorphic functions of type 1, jointly Weyl p-almost automorphic functions and Weyl p-almost automorphic functions of type 2 in the one-dimensional setting. After that, we introduce and analyze generalizations of these classes in the multi-dimensional setting, working with general Lebesgue spaces with variable exponents. We provide several illustrative examples and applications to the abstract Volterra integro-differential equations.

We prove that the initial value problem with small data for the asymptotically flat spherically symmetric Einstein-Vlasov-Maxwell system admits the global in time solution in the case of the non zero shift vector. This result extends the one already known for chargeless case.

In this paper, we investigate the existence of at least three weak solutions for a class of nonlocal elliptic equations with Navier boundary value conditions. The proof of our result uses the basic theory and critical point theory of variable exponential Lebesgue Sobolev spaces. Moreover a generalization of Corollary 1.1 in [21] is obtained.

Let $$$1\leq q\leq\alpha < \infty. \left\{(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d):\alpha\leq p\leq\infty \right\}$$$ is a nondecreasing family of Banach spaces such that the Lebesgue space is $$$L^{\alpha}(\mathbb{R}^d)$$$ its minimal element and the classical Morrey space $$$\mathcal{M}_{q}^{\alpha}(\mathbb{R}^d)$$$ is its maximal element. In this note we investigate some closed linear subspaces of $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$. We give a characterization of the closure in $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$ of the set of all its compactly supported elements and study the action of some classical operators on it. We also describe the closure in $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$ of the set $$$\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$$$ of all infinitely differentiable and compactly supported functions on $$$\mathbb{R}^{d}$$$ as an intersection of other linear subspaces of $$$(L^{q}, l^{p})^{\alpha}(\mathbb{R}^d)$$$ and obtain the weak density of $$$\mathcal{C}_{\rm{c}}^{\infty}(\mathbb{R}^d)$$$ in some of these subspaces. We establish a necessary condition on a function $$$f$$$ in order that its Riesz potential $$$I_{\gamma}(|f|) \;(0<\gamma<1)$$$ be in a given Lebesgue space.