scholarly journals Some weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball

2019 ◽  
Vol 11 (1) ◽  
pp. 14-25 ◽  
Author(s):  
A.I. Bandura
Keyword(s):  
Analytic Function ◽  
Differential Equations ◽  
Unit Ball ◽  
Logarithmic Derivative ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Directional Derivatives ◽  
Local Behavior ◽  
Exceptional Set ◽  
Linear Differential

We partially reinforce some criteria of $L$-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products.Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

scholarly journals SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

2010 ◽  
Vol 88 (2) ◽  
pp. 145-167 ◽  
Author(s):  
I. CHYZHYKOV ◽  
J. HEITTOKANGAS ◽  
J. RÄTTYÄ
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Logarithmic Derivative ◽  
Maximum Modulus ◽  
Exceptional Set ◽  
Proximate Order ◽  
Upper Density ◽  
Linear Differential ◽  
Logarithmic Derivatives ◽  
Growth Of Solutions

AbstractNew estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

Linear directional differential equations in the unit ball: solutions of bounded L-index

2019 ◽  
Vol 69 (5) ◽  
pp. 1089-1098 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Unit Ball ◽  
Analytic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Directional Derivatives ◽  
Several Variables

Abstract We study sufficient conditions of boundedness of L-index in a direction b ∈ ℂn ∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of bounded L-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

2020 ◽  
Vol 70 (5) ◽  
pp. 1141-1152
Author(s):  
Vita Baksa ◽  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Unit Ball ◽  
Logarithmic Derivative ◽  
Sufficient Conditions ◽  
Analytic Vector ◽  
Exceptional Set ◽  
Partial Derivatives ◽  
Definition Of ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l1, l2): 𝔹2 → $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function.Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

scholarly journals On non-homogeneous canonical third-order linear differential equations

1994 ◽  
Vol 57 (2) ◽  
pp. 138-148 ◽  
Author(s):  
N. Parhi
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Linear Differential Equations ◽  
Third Order ◽  
Order Linear ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

AbstractIn this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

scholarly journals Boundedness of Solutions for Second Order Differential Equations

2020 ◽  
Vol 12 (4) ◽  
pp. 58
Author(s):  
Daniel C. Biles
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equations ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

We present new theorems which specify sufficient conditions for the boundedness of all solutions for second order non-linear differential equations. Unboundedness of solutions is also considered.

On the Nature of the Spectrum of Singular Second Order Linear Differential Equations

1951 ◽  
Vol 3 ◽  
pp. 335-338 ◽  
Author(s):  
E. A. Coddington ◽  
N. Levinson
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Continuous Functions ◽  
Second Order ◽  
Real Parameter ◽  
Linear Differential Equations ◽  
Order Linear ◽  
The Real ◽  
Linear Differential

Let p(x) > 0, q(x) be two real-valued continuous functions on . Suppose that the differential equation with the real parameter λ

Conjugate points for fractional differential equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Paul Eloe ◽  
Jeffrey Neugebauer
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Conjugate Point ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Existence Of A Solution ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential ◽  
Linear Differential

AbstractLet b > 0. Let 1 < α ≤ 2. The theory of u 0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D 0+α u + λp(t)u = 0, 0 < t < b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation.

Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

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