Volume 10, Issue 1

In this paper, we study varieties and characterize convolution operators on the algebras related to the new structures of KPC-hypergroups, which are a generalization of the classical ones.

In this paper, by employing a critical point theorem, we establish the existence of infinitely many solutions for fourth-order impulsive differential equations depending on two real parameters.

In this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following p⁢(x){p(x)}-Kirchhoff system: {-M⁢(I0⁢(u))⁢△p⁢(x)⁢u=λp⁢(x)⁢[λ1⁢f⁢(v)+μ1⁢h⁢(u)]in ⁢Ω,-M⁢(I0⁢(v))⁢△p⁢(x)⁢v=λp⁢(x)⁢[λ2⁢g⁢(u)+μ2⁢τ⁢(v)]in ⁢Ω,u=v=0on ⁢∂⁡Ω,\left\{\begin{aligned} &\displaystyle{-}M(I_{0}(u))\triangle_{p(x)}u=\lambda^{% p(x)}[\lambda_{1}f(v)+\mu_{1}h(u)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M(I_{0}(v))\triangle_{p(x)}v=\lambda^{p(x)}[\lambda_{2}g(u)+% \mu_{2}\tau(v)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain with C2{C^{2}} boundary ∂⁡Ω{\partial\Omega}, △p⁢(x)⁢u=div⁡(|∇⁡u|p⁢(x)-2⁢∇⁡u){\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)}, p⁢(x)∈C1⁢(Ω¯){p(x)\in C^{1}(\overline{\Omega})}, with 1<p⁢(x){1<p(x)}, is a function satisfying 1<p-=infΩ⁡p⁢(x)≤p+=supΩ⁡p⁢(x)<∞{1<p^{-}=\inf_{\Omega}p(x)\leq p^{+}=\sup_{\Omega}p(x)<\infty}, λ, λ1{\lambda_{1}}, λ2{\lambda_{2}}, μ1{\mu_{1}} and μ2{\mu_{2}} are positive parameters, I0⁢(u)=∫Ω1p⁢(x)⁢|∇⁡u|p⁢(x)⁢𝑑x{I_{0}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx}, and M⁢(t){M(t)} is a continuous function.

To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in ℚ⁢[X]\mathbb{Q}[X]. We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results, where polynomials are replaced by packed words. We obtain, in particular, a non-commutative duality principle.

In this article, we consider the Schrödinger semigroup for the Laplacian Δ on ℝn{\mathbb{R}^{n}}, and characterize the image of a Sobolev space in L2⁢(ℝn,eu2⁢d⁢u){L^{2}(\mathbb{R}^{n},e^{u^{2}}du)} under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on ℝn{\mathbb{R}^{n}}.