Existence and properties of semi-bounded global solutions to the functional differential equation with Volterra-type operators on the real line

2017 ◽  
Vol 147 (6) ◽  
pp. 1119-1168
Author(s):  
Maitere Aguerrea ◽  
Robert Hakl
Keyword(s):  
Real Line ◽  
Asymptotic Properties ◽  
Global Solutions ◽  
Continuous Operator ◽  
Volterra Type ◽  
Existence Of A Solution ◽  
Carathéodory Conditions ◽  
Functional Differential ◽  
Image Position ◽  
Continuous Operators

Consider the equationwhere are linear positive continuous operators and f : Cloc(ℝ;ℝ) → Lloc(ℝ;ℝ) is a continuous operator satisfying the local Carathéodory conditions. Efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of –∞, to the equation considered are established provided that ℓ0, ℓ1 and f are Volterra-type operators. The existence of a solution that is positive on the whole real line is discussed as well. Furthermore, the asymptotic properties of such solutions are studied in the neighbourhood of –∞. The results are applied to certain models appearing in the natural sciences.

Oscillatory and asymptotic properties of a class of operator-differential inequalities

1984 ◽  
Vol 96 (1-2) ◽  
pp. 5-13 ◽  
Author(s):  
A. D. Myshkis ◽  
D. D. Bainov ◽  
A. I. Zahariev
Keyword(s):  
Asymptotic Behaviour ◽  
Asymptotic Properties ◽  
Neutral Type ◽  
Differential Inequalities ◽  
Functional Differential ◽  
Image Position ◽  
Oscillatory Properties

SynopsisThe present paper studies some asymptotic (including oscillatory) properties of the solutions of operator-differential inequalities of the formwhere(the latter symbol denotes the space of locally summable functions).As an application of the results obtained, theorems are proved for the asymptotic behaviour of the solutions of certain classes of functional-differential and integro-differential neutral-type equations.

scholarly journals Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Asymptotic Properties ◽  
Distributed Delays ◽  
Asymptotic Convergence ◽  
Time Varying ◽  
Volterra Type ◽  
Linear Dynamics ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Functional Differential System ◽  
Rassias Stability

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear time-varying perturbed functional differential system subject to point, finite-distributed, and Volterra-type distributed delays associated with linear dynamics together with a class of nonlinear delayed dynamics. The boundedness of the error and its asymptotic convergence to zero are investigated with the results being obtained based on the Hyers-Ulam-Rassias analysis.

scholarly journals Oscillating properties of the solutions of a class of neutral type functional differential equations

1980 ◽  
Vol 22 (3) ◽  
pp. 365-372 ◽  
Author(s):  
A.I. Zahariev ◽  
D.D. Bainov
Keyword(s):  
Differential Equations ◽  
Positive Constant ◽  
Asymptotic Properties ◽  
Functional Differential Equations ◽  
Neutral Type ◽  
Constant Delay ◽  
Arbitrary Positive Constant ◽  
Functional Differential ◽  
Image Position

The present paper deals with some oscillating and asymptotic properties of the functional differential equations of the formwhere λ is an arbitrary positive constant and τ > 0 is a constant delay.

scholarly journals Integro-differential equations of Volterra type

1970 ◽  
Vol 3 (1) ◽  
pp. 9-22 ◽  
Author(s):  
M. Rama Mohana Rao ◽  
Chris P. Tsokos
Keyword(s):  
Differential System ◽  
Basic Result ◽  
Sufficient Conditions ◽  
Continuous Operator ◽  
Continuous Functions ◽  
Uniform Asymptotic Stability ◽  
Space Of Continuous Functions ◽  
Volterra Type ◽  
The Stability ◽  
Image Position

The aim of this paper is concerned with studying the stability properties of an integro-differential system by reducing it into a scalar integro-differential equation. A theorem is stated about the existence of a maximal solution of such systems and a basic result on integro-differential inequalities. Utilizing these results we obtain sufficient conditions for uniform asymptotic stability of the trivial solution of the integro-differential system of the form where , with , , C(J) denotes the space of continuous functions, A a continuous operator such that A maps C(J) into C(J). The fruitfulness of the results of the paper are illustrated with two applications.

On the blow-up of solutions of a convective reaction diffusion equation

1993 ◽  
Vol 123 (3) ◽  
pp. 433-460 ◽  
Author(s):  
J. Aguirre ◽  
M. Escobedo
Keyword(s):  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Scalar Function ◽  
Reaction Diffusion Equation ◽  
Semilinear Parabolic Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisWe study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equationwhere u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

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 Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

Asymptotic properties of solutions to discrete Volterra type equations

2021 ◽  
Author(s):  
Janusz Migda ◽  
Magdalena Nockowska‐Rosiak ◽  
Malgorzata Migda
Keyword(s):  
Asymptotic Properties ◽  
Volterra Type

Existence of Positive Global Solutions of Mixed Sublinear-Superlinear Problems

1988 ◽  
Vol 40 (5) ◽  
pp. 1222-1242
Author(s):  
W. Allegretto ◽  
Y. X. Huang
Keyword(s):  
Positive Solutions ◽  
Special Interest ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Linear Type ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Quasilinear Problem

Consider the elliptic quasilinear problem:1in Rn, n ≧ 3, whereWe are interested in establishing sufficient conditions on f for the existence of positive solutions u(x) with specified behaviour at ∞. Of special interest to us are criteria which guarantee that u(x) decays at least as fast as |x|−α for some α ≧ 0, given below, in the case f(x, u, ∇u) contains terms of typeThat is: f is of mixed sublinear-super linear type. Our main result is Theorem 3 below which explicitly states sufficient conditions for the existence of such solutions.

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