Extremal solutions to a system of <i>n</i>
 nonlinear differential equations and regularly varying functions

Řehák, P.; Matucci, Serena

doi:10.1002/mana.201400252

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…Finally, we note that there is a burgeoning literature regarding the application of the theory of regular variation to the asymptotic behaviour of ordinary and functional differential equations (see for example the monographs of Marić [25], and Řehák [32] and recent representative papers such as those of Chatzarakis et al, [10], Matucci and Řehák [27,28], and Takasi and Manojlović [34]).…”

Section: Results With Regular Variationmentioning

confidence: 99%

Growth Rates of Sublinear Functional and Volterra Differential Equations

Appleby¹,

Patterson²

2018

SIAM J. Math. Anal.

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This paper considers the growth rates of positive solutions of scalar nonlinear functional and Volterra differential equations. The equations are assumed to be autonomous (or asymptotically so), and the nonlinear dependence grows less rapidly than any linear function. We impose extra regularity properties on a function asymptotic to this nonlinear function, rather than on the nonlinearity itself. The main result of the paper demonstrates that the growth rate of the solution can be found by determining the rate of growth of a trivial functional differential equation (FDE) with the same nonlinearity and all its associated measure concentrated at zero; the trivial FDE is nothing other than an autonomous nonlinear ODE. We also supply direct asymptotic information about the solution of the FDE under additional conditions on the nonlinearity, and exploit the theory of regular variation to sharpen and extend the results. µ 2 (ds). (1.2)

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Section: Results With Regular Variationmentioning

confidence: 99%

Growth Rates of Sublinear Functional and Volterra Differential Equations

Appleby¹,

Patterson²

2018

SIAM J. Math. Anal.

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“…However, it seems that extracting good asymptotic information along the lines needed to prove a converse of Theorem 6 may make greater requests on this theory. The literature regarding the application of the theory of regular variation to the asymptotic behaviour of ordinary and functional differential equations is extensive and growing (see for example the monographs of Marić [15] and Řehák [21] and recent representative papers such as [8], [16], [17] and [22]).…”

Section: Resultsmentioning

confidence: 99%

Hartman-Wintner growth results for sublinear functional differential equations

Appleby,

Patterson

2016

Preprint

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In this paper, we determine rates of growth to infinity of scalar autonomous nonlinear functional and Volterra differential equations. In these equations, the right-hand side is a positive continuous linear functional of a nonlinear function of the state. We assume the nonlinearity grows sublinearly at infinity, leading to subexponential growth in the solutions. Our main results show that the solutions of the functional differential equations are asymptotic to those of an auxiliary autonomous ordinary differential equation when the nonlinearity grows more slowly than a critical rate. If the nonlinearity grows more rapidly than this rate, the ODE dominates the FDE. If the nonlinearity tends to infinity at exactly this rate, the FDE and ODE grow at the same rate, modulo a constant non-unit factor. Finally, if the nonlinearity grows more slowly than the critical rate, then the ODE and FDE grow at the same rate asymptotically. We also prove a partial converse of the last result. In the case when the growth rate is slower than that of the ODE, we calculate sharp bounds on the solutions.

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“…The recent development of asymptotic analysis of ordinary differential equations by means of regularly varying functions (see [17]- [19], [23]- [26], [29], [33]- [35] and monograph [28] for results up to 2000. ), suggests to investigate the discrete problem of the existence of strongly decreasing solutions in the framework of regularly varying sequences.…”

Section: Consider the Nonlinear Difference Equation Of Second Order (E)mentioning

confidence: 99%

Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

Kapesic

Manojlović

2019

Filomat

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Positive decreasing solutions of the nonlinear difference equation ∆(p n |∆x n | α−1 ∆x n) = q n |x n+1 | β−1 x n+1 , n ≥ 1, α > β > 0, are studied under the assumption that p, q are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.

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Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

Cited by 12 publications

References 17 publications

Growth Rates of Sublinear Functional and Volterra Differential Equations

Growth Rates of Sublinear Functional and Volterra Differential Equations

Hartman-Wintner growth results for sublinear functional differential equations

Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

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