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This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{'''} + q (t) x^{- γ} = 0$ , by means of regularly varying functions, where $γ$ is a positive constant and $q$ is a positive continuous function on $[a, \infty)$ . It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t \to \infty$ and to acquire...

Decomposition and Moser's lemma.

David E. Edmunds, Miroslav Krbec (2002)

Revista Matemática Complutense

Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

Differences of Decreasing Slowly Varying Functions

Slobodanka Janković, Tatjana Ostrogorski (2000)

Matematički Vesnik

Differentiation of n-convex functions

H. Fejzić, R. E. Svetic, C. E. Weil (2010)

Fundamenta Mathematicae

The main result of this paper is that if f is n-convex on a measurable subset E of ℝ, then f is n-2 times differentiable, n-2 times Peano differentiable and the corresponding derivatives are equal, and $f^{(n - 1)} = f_{(n - 1)}$ except on a countable set. Moreover $f_{(n - 1)}$ is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities

Jaroslav Jaroš, Kusano Takaŝi (2014)

Archivum Mathematicum

The system of nonlinear differential equations $x^{'} + p_{1} (t) x^{α_{1}} + q_{1} (t) y^{β_{1}} = 0, y^{'} + p_{2} (t) x^{α_{2}} + q_{2} (t) y^{β_{2}} = 0, A$ is under consideration, where $α_{i}$ and $β_{i}$ are positive constants and $p_{i} (t)$ and $q_{i} (t)$ are positive continuous functions on $[a, \infty)$ . There are three types of different asymptotic behavior at infinity of positive solutions $(x (t), y (t))$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t \to \infty$ , which can be...

Exponential stability for the wave equation with weak nonmonotone damping.

Martinez, P., Vancostenoble, J. (2000)

Portugaliae Mathematica

F-Normalreihen.

Herbert Möller (1977)

Journal für die reine und angewandte Mathematik

Fonctions A Comportement Régulier Et Convergence Uniforme

D. Aranđelović (1975)

Publications de l'Institut Mathématique

Functions of slow increase and integer sequences.

Jakimczuk, Rafael (2010)

Journal of Integer Sequences [electronic only]

General kernel convolutions with slowly varying functions.

Simić, Slavko (2005)

Publications de l'Institut Mathématique. Nouvelle Série

Generalised regular variation of arbitrary order

Edward Omey, Johan Segers (2010)

Banach Center Publications

Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(t)), each $g_{i}$ being of smaller order than its predecessor $g_{i - 1}$ . The function f is said to be generalised regularly varying of...

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