Advances in Pure and Applied Mathematics

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  ABSTRACT: In this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

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  ABSTRACT: We consider the transmutation operators Vk and tVk associated with the Cherednik operators attached to the root system A2, called also in [Adv. Pure Appl. Math. 1 (2010), 293-323; Adv. Pure Appl. Math. 2 (2011), 223-246] the trigonometric Dunkl intertwining operator, and its dual. In this paper, we prove that the operators Vk and tVk are positivity preserving and allows positive integral representations. In particular, we deduce that the Opdam-Cherednik kernel is positive definite.

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  ABSTRACT: So far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardy's theorem for a diamond Lie group, which is a semi-direct product of Rd with the Heisenberg group H2d+1 In this setting, we formulate and prove in this paper some other uncertainty principles (Miyachi, Cowling-Price and Lp-Lq Morgan). This allows us to provide a refined version of Hardy's theorem and to study the sharpness problems.

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  ABSTRACT: In this paper, by defining Lyapunov functionals, we investigate proper sufficient conditions for the uniform stability of the zero solution, and also for the uniform boundedness and uniform ultimate boundedness of all solutions of a certain third-order nonlinear vector delay differential equation of the type

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  ABSTRACT: Let G be an exponential solvable Lie group with Lie algebra defined by a semi-direct product of and . We study a holomorphically induced representation ρ of G from a real linear form f of and a 1-dimensional complex subalgebra of such that the space + generates . Under some assumptions of structure of the Lie algebra, we obtain a sufficient condition for non-triviality of ρ and decompose ρ into a multiplicity-free direct integral of irreducible representations.

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  ABSTRACT: Let G be a locally compact group. We denote by (× G) the hyperspace of closed subgroups of × G endowed with the Chabauty topology. In this paper, some general results about the connectedness of the hyperspace (× G) are established. The main result proved here is that if G is a prosolvable group, then (× G) is connected. We also prove a partial converse to the result just mentioned.

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  ABSTRACT: Let 2n+1r be the reduced Heisenberg Lie group, G = 2n+1r × 2n+1r be the (4n + 2)-dimensional Lie group and ΔG = {(x, x) G : x2n+1r} be the diagonal subgroup of G. Given any discontinuous subgroup Γ G for G/ΔG, we provide a layering of the parameter space (Γ, G, ΔG), which is shown to be endowed with a smooth manifold structure, we also show that the stability property holds. On the other hand, a local (and hence a global) rigidity theorem is obtained. That is, the parameter space (Γ, G, ΔG) admits a rigid point if and only if Γ is finite and this is also equivalent to the fact that the deformation space is Hausdorff.

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ABSTRACT: A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock-Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $\Theta_t(x)=\exp(-x^2/(4t))/\sqrt{4\pi t}$ be the heat kernel. With initial data $f$ that is the distributional derivative of a continuous function, it is shown that $u_t(x):=u(x, t):=f\ast\Theta_t(x)$ is a classical solution of the heat equation $u_{11}=u_2$. The estimate $\|f\ast\Theta_t\|_\infty\leq\|f\|/\sqrt{\pi t}$ holds. The Alexiewicz norm is $\|f\|=\sup_I|\int_If|$, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, $\|u_t-f\|\to 0$ as $t\to 0^+$. The solution of the heat equation is unique under the assumptions that $\|u_t\|$ is bounded and $u_t\to f$ in the Alexiewicz norm for some integrable $f$. The heat equation is also considered with initial data that is the $n$th derivative of a continuous function and in weighted spaces such that $\int_{-\infty}^\infty f(x)\exp(-ax^2)\, dx$ exists for some $a>0$. Similar results are obtained.

Preview · Article · Jan 2015 · Advances in Pure and Applied Mathematics

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  ABSTRACT: In this paper, we obtain the general solution and we study the generalized Hyers–Ulam stability of the mixed type additive-quadratic functional equation

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  ABSTRACT: We prove various versions of uncertainty principles for a certain Fourier transform FA. Here, A is a Chébli function (that is, a Sturm-Liouville function with additional hypotheses). We mainly establish an analogue of Beurling's theorem, and its relatives such as theorems of Gelfand-Shilov type, of Morgan type, of Hardy type, and of Cowling-Price type, for FA and relate them to the characterization of the heat kernel corresponding to FA. Heisenberg's and local uncertainty inequalities are also proved.