New differentiability criteria for complex-valued functions

We prove a theorem which states that the existence of an asymptotic limit off _¯z aszz ₀ implies -differentiability of a complex-valued functionf(z) atz ₀.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 329–337, April, 1994.     

On some criteria for the univalence of analytic functions

Criteria for the univalence of functions f(z) which are regular in the domain ¦z¦ < 1 are obtained in the form of restrictions on the modulus of the quotient f (z) [f '(z)]^–1. Analogous results are obtained also for functions which are analytic in the interior of the unit disk except for a simple pole at infinity.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 359–366, March, 1973.In conclusion the author wishes to thank L. A. Aksent'ev and F. G. Avkhadiev for discussion of the results and valuable remarks.     

Evans functions for periodic waves on infinite cylindrical domains

Using Galerkin approximations, an Evans function for spatially periodic waves on infinite cylindrical domains is constructed. It is also shown that the Evans function can be used to define a parity index for periodic waves that detects whether the wave admits an odd number of real unstable eigenvalues. This parity index depends only on local information for the existence problem of the wave: in particular, it uses information about the linear dispersion relation near zero and the orientability of the unstable and stable manifolds along the nonlinear wave. The results are applied to small-amplitude wave trains for a scalar equation on an infinite strip.     

On preponderant differentiability of typical continuous functions

In the literature, several definitions of a preponderant derivative exist. An old result of Jarník implies that a typical continuous function on has a (strong) preponderant derivative at no point. We show that a typical continuous function on has an infinite (weak) preponderant derivative at each point from a -dense subset of .

     

Inclusion and differentiability criteria forL p -classes of infinitely differentiable functions

Summary In this paper, we give necessary and sufficient conditions for a Carleman class in metricL ^p to be included in another or to be differentiable.     

On approximate differentiability of functions with bounded deformation

We prove that functions with bounded deformationu: Ω → ℝⁿ, Ω ⊂ ℝⁿ, i.e., such mappings that the symmetric part of the gradient is a measure, are approximately differentiable a.e. Then we generalize the result to a more general class of functions. This research was carried out while the author stayed in the ICTP in Trieste in 1995. He wishes to thank the ICTP for the hospitality. The author was partially supported by KBN grant no. 2-PO3A-034-08.     

Certain imbedding theorems for spaces of periodic functions of infinite order

Translated from Matematicheskie Zametki, Vol. 43, No. 4, pp. 509–517, April, 1988.     

Almost periodic solutions for some integrodifferential equations with infinite delay

By using a Liapunov functional, the conditions of existence and uniqueness of almost periodic solutions for some integrodifferential equations are obtained.     

On differentiability of ruin functions under Markov-modulated models

This paper analyzes the continuity and differentiability of several classes of ruin functions under Markov-modulated insurance risk models with a barrier and threshold dividend strategy, respectively. Many ruin related functions in the literature, such as the expectation and the Laplace transform of the Gerber–Shiu discounted penalty function at ruin, of the total discounted dividends until ruin, and of the time-integrated discounted penalty and/or reward function of the risk process, etc, are special cases of the functions considered in this paper. Continuity and differentiability of these functions in the corresponding dual models are also studied.     

On differentiability and analyticity of positive definite functions

We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.     

On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.     

On some definite integrals and infinite sums containing products of special functions

A method is proposed for evaluation of some definite integrals and infinite sums containing products of Bessel, Struve and other special functions.     

Pseudo almost periodic solutions of infinite class for some functional differential equations

The aim of this work is to investigate the existence and uniqueness of pseudo almost periodic solutions for some neutral partial functional differential equations in a Banach space when the delay is distributed using the variation of constants formula and the spectral decomposition of the phase space developed in Adimy et al. [M. Adimy, K. Ezzinbi, and A. Ouhinou, Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 317(2) (2006), pp. 668–689]. Here, we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part is assumed to be pseudo almost periodic with respect to the first argument and Lipschitz continuous with respect to the second argument.