Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Positive solutions of the nonlinear second-order differential equation $(p(t)|x'|^{\alpha - 1} x')' + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.     

Asymptotic behavior of solutions of equations of the Emden-Fowler type at infinity

We obtain an asymptotic representation of solutions of equations of the Emden-Fowler type with “supercritical” exponent and prove the existence of solutions with a given asymptotics. The methods used include the construction of supersolutions for deriving a priori estimates and the use of Kondrat’ev’s results for weighted spaces. The existence of solutions is proved by the Leray-Schauder method.     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.     

Existence of positive solutions for systems of nonlinear third-order differential equations

This paper is concerned with boundary value problems for systems of nonlinear third-order three-point differential equations. Using fixed-point theorems, the existence of positive solutions is obtained.     

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations of the form {fx006-01} where α, β are ratios of odd positive integers and {p _n}, {q _n} are positive real sequences defined for all n ∈ ℕ(n ₀). We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of convergence or divergence conditions for the sums {fx006-02}. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 8–27, January, 2008.     

Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter

In previous papers [6-8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver"s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: $y"+a\Lambda^2 y"+b\Lambda^3y=f(x)y"+g(x)y$, with $a,b\in\mathbb{C}$ fixed, $f"$ and $g$ continuous, and $\Lambda$ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver"s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral $P(x,y)$ for large $|x|$.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

On existence of oscillatory solutions of second order Emden-Fowler equations

We study the second order Emden-Fowler equation

(E)     

Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

Sufficient conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the second-order nonlinear differential equation $$x^{\prime\prime}(t)+q(t)|x(t)|^{\gamma}\,{\rm sgn}\, x(t)=0, \quad \quad (A)$$ where γ is a positive constant different from 1 and q : [a, ∞) → (0, ∞) is a continuous integrable function. We show how an application of the theory of regular variation gives the possibility of determining the precise asymptotic behavior of solutions of both superlinear and sublinear equation (A).     

Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients

Sufficient conditions are established for the asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients

     

Asymptotic solutions of nonlinear differential equations

An asymptotic expression for solutions of nonlinear differential equations is obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 676–678, May, 1991.