Advances in Analysis

7

Authors: Said R. Grace

We present new oscillation criteria for a class of third order nonlinear differential equations with a nonpositive neutral term. The results obtained improve and complement some related results known in the literature. Two illustrative examples are provided

11

Authors: G. A. ABDULLAYEV, F. G. ABDULLAYEV, A. TAYLAKOVA

We continue the study of estimates of algebraic polynomials in regions bounded by a piecewise asymptotically conformal curve with interior non-zero angles in the weighted Bergman space.

3

Authors: Alp Arslan Kıraç

We extend the classical Ambarzumyan’s theorem to the quasi-periodic boundary value problems by using only a part knowledge of one spectrum. We also weaken slightly the Yurko’s conditions on the first eigenvalue.

6

Authors: Ivan Kupka

In [1] we defined a notion of a generalized derivative for functions defined on a general topological space with values in a linear topological space. Here we develop a theory of Taylor series for this generalized setting.

9

Authors: K. R. Karthikeyan, K. Amarender Reddy, M. Thirucheran

We introduce a new subclass of spiralike biunivalent functions involving q-differential operator. We obtain the coefficient estimates and Fekete-Szegö inequalities for the functions belonging to this class. Relevant connections with various other known classes have been established.

18

Authors: Chenkuan Li, Kyle Clarkson, Vrajna Patel

Let {τn} be a certain sequence of functions in D converging to 1 in D 0 . The commutative neutrix convolution f ♦∗ g of two distributions f and g in D 0 is defined to be the neutrix limit of the sequence 1 2 {(fτn) ∗ g + f ∗ (gτn)} , provided the limit exists. We present relations between this new convolution and other existing distributional convolutions, and demonstrate its strong computational power in evaluating convolutions as well as applications to defining new fractional derivatives and integrals of generalized functions in the new space H which contains D 0 (R +). The neutrix convolutions x λ − ♦∗ x µ + for λ, µ, λ + µ 6= 0, ±1, ±2, · · · and x λ − ♦∗ x s + for λ 6= 0, ±1, ±2, · · · and s = 0, 1, 2, · · · are evaluated, from which other neutrix convolutions are deduced.

5

Authors: Ricardo Estrada

We prove that the product p (x) ∇2mδ (x) of a homogeneous polynomial of degree j in n variables, p, and the iterated Laplacian of the Dirac delta function is of the form p (∇) q (∇) δ (x) for some polynomial q if and only if p (x) = |x| 2l pk (x) for some harmonic homogeneous polynomial of some degree k, j = 2l + k. We also show that the product p (x) ∇2mδ (x), where p is homogeneous of degree j, vanishes for all m > j if and only if p is a harmonic polynomial.

10

Authors: Attou A. Miloua

Let Ω be a region in R 2 and f be a positive C 1 function satisfying lim u→0+ f (u) = ∞. We consider the quasi-linear elliptic equations of the form div (a (u) ∇u) = a 0 (u) 2 |∇u| 2 + f (u) where a is a positive C 1 function. Motivated by the thin film equations, a solution u is said to be a point rupture solution if for some p ∈ Ω, u (p) = 0 and u (p) > 0 in Ω\ {p}. Our main result is a sufficient condition on a and f for the existence of radial point rupture solutions.

14

Authors: Veronika Chrastinová

In the common theory of the inverse problem, a system of differential equations is given and we ask whether this system is identical with the Lagrange system of an appropriate variational integral. In this article, only a small part of the Euler–Lagrange system may be prescribed in advance.The lack of information does not affect the results. The classical Helmholz solvability conditions and the Tonti resolving formula are adapted for this incomplete problem. Elementary and self–contained algorithmical approach is applied.