Hankel Determinants of a Linear Combination of Three Successive Catalan Numbers

We evaluate the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. This is done by finding a Jacobi linear functional, such that their moments are, up to a multiplicative constant, the Catalan numbers. The values of such determinants are then expressed in terms of Jacobi polynomials.  

Existence Results for Critical Semi-linear Equations on Heisenberg Group Domains

Following the work of G. Citti and F. Uguzzoni who studied Yamabe type problems on Heisenberg group domains, we consider here the following critical semi-linear equation on domains of the Heisenberg group ${{\mathbb{H}^1}}$ : $$(P) \left\{\begin{array}{lll}-{\Delta_{H}}u\quad =\quad K{u^{3}}\quad\,{\rm in}\,\,\Omega,\\ \quad\quad\,{u}\quad > \quad0\qquad\,\,\,\,{\rm in}\,\,\Omega,\\ \quad\quad\,{u}\quad = \quad 0 \quad\quad\,\,\,{\rm on}\,\partial \Omega, \end{array}\right. $$ where Δ _H is the sublaplacian on ${{\mathbb{H}^1}}$ and K is a C ³ positive function defined on Ω. Using a version of the Morse Lemma at infinity, we give necessary conditions on K to insure the existence of solutions for (P).  

Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval

In this paper, we study the existence of mild solutions for initial value problems for semilinear Volterra integrodifferential equations in a Banach space. The arguments are based on the concept of measure of noncompactness in Fréchet space and the Tikhonov fixed point theorem.  

Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.  

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.  

Fixed point theorems for the sum of two weakly sequentially continuous mappings

In the present paper, we prove some Krasnosel’skii-Leray-Schauder type fixed point theorems for weak topology. Some fixed point theorems for the sum of two weakly sequentially continuous mappings are also presented. Our results extend and improve on ones from several earlier works.  

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.  

The symmetric D ω -semi-classical orthogonal polynomials of class one

We give the system of Laguerre–Freud equations associated with the D _ω -semi-classical functionals of class one, where D _ω is the divided difference operator. This system is solved in the symmetric case. There are essentially two canonical cases. The corresponding integral representations are given.  

Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform

In this paper we consider the Dunkl operators T _j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.   

$$L^{p}$$-theory of elliptic differential operators with unbounded coefficients

In this paper, we consider the generation of strongly continuous analytic semigroups on \(L^p((0,\omega ),\mu _{p}\, dx)\) and \(L^p((0,\omega ), dx), 1<p<\infty \), by a family of second order elliptic operators of the form

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As in [24], we shall prove the generation results on \(L^2\)-spaces using the sesquilinear forms. More general results are obtained by using interpolation procedure and Neuberger’s theorem.