Volume 2, Issue 1

In this paper, we consider a positive real number p ≥ 1, a weight function μ (·) and give different existence results in the weighted L p (ℝ + , dμ )-space of some nonlinear integral equations of Hammerstein and Urysohn's types. It is well known that solving the existence problems of such equations on unbounded domains is more challenging than the case of a bounded domain. The main ingredient of our existence results is the Schauder's fixed point theorem. Hence, a special interest is devoted to the compactness as well as the continuity of the integral operators associated with the above integral equations. Moreover, some examples are provided to illustrate the different results of this work.

By using the trigonometric Dunkl intertwining operator and its dual introduced by the author in [Trimèche, Adv. Pure Appl. Math.], we define and study the hypergeometric translation operators associated with the Cherednik operators T j , j = 1, 2, . . . , d . Next with the help of these translation operators, we define and study the hypergeometric convolution product of functions and distributions associated with the operators T j , j = 1, 2, . . . , d .

We present a constructive method for the robust approximation to solutions of some elliptic equations in a plane domain from incomplete and corrupted boundary data. We state this inverse problem in generalized Hardy spaces of functions satisfying the conjugate Beltrami equation, of which we give some properties, in the Hilbertian framework. The issue is then reworded as a constrained approximation (bounded extremal) problem which is shown to be well-posed. A practical motivation comes from modelling plasma confinement in a tokamak reactor. There, the particular form of the conductivity coefficient leads to Bessel-exponential type families of solutions of which we establish density properties.

This paper contains new results on L 1 symbols of bounded (resp. compact) Toeplitz operators on Bergman spaces in the unit disk of the complex plane, ℂ. Our results improve and extend earlier results of N. Zorboska, J. Miao and D. Zheng. We also extend these results to the unit ball of C n . Part of our study can be done for more general linear operators on Bergman spaces.

In this paper we consider pseudo-differential operators associated with the Dunkl transform on the real line. We establish the Calderón–Vaillancourt theorem for such operators. We also obtain the L p -boundedness for the Hörmander's class (0 ≤ δ < 1).

We give the definitions of n -self-cotilting comodules and n -cotilting comodules in comodule categories. We also obtain that n -self-cotilting comodules and n -cotilting comodules can induce equivalences between categories of comodules.