Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

“…With regard for (7) and (1), we obtain relation (10), the second condition in (9), and the second representation in (12). Furthermore, by virtue of (13), the following representation is true: (5).…”

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confidence: 87%

“…Thus, all conditions of Theorem 2.1 in [12] are satisfied for the system of differential equations (25). According to this theorem, system (22) has at least one…”

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confidence: 91%

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Asymptotic representations of solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities

Belozerova

2009

Nonlinear Oscill

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UDC 517.925Asymptotic representations are found for solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities.In the present paper, we consider the nonlinear differential equationBy virtue of the first condition in (2), every function ϕ i , i ∈ {0, 1}, is close, in a certain sense, to a power function, namely, it has the form ϕWe say that a function ϕ i (z), i ∈ {0, 1}, satisfies condition S i if, for any continuously differentiable function L :. 2 For Yi = +∞ (Yi = −∞), we suppose that y 0 i > 0 (y 0 i < 0), respectively.

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“…By virtue of Theorem 1.1 in [2], this system has a solution that tends to zero as t → +∞. By virtue of the substitutions used above, it is associated with a solution of Eq.…”

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confidence: 99%

On the asymptotic behavior of solutions of second-order differential equations with nonlinearity close to an exponential

Khar’kov

2008

Nonlinear Oscill

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We obtain asymptotic representations for one class of solutions of a second-order differential equation with nonlinearity close to an exponential. Statement of the Problem and Main ResultsConsider the second-order differential equation(1.2) and the functions L k : (0, +∞) → R \ {0} and F k : I → R are defined by the recurrence relations L r (z) = ln |L r−1 (z)| , L 1 (z) = ln z, z ∈ (0, +∞), F r (y) = L r−1 (ϕ(y)) (F r−1 (y) − 1) , F 1 (y) = ϕ (y)ϕ(y) ϕ 2 (y) , r = 2, . . . , k.(1.3)In the present paper, we investigate the asymptotic behavior of solutions of Eq. (1.1) defined in the left neighborhood of the point ω and such that lim t→ω y(t) = y 0 .The asymptotic behavior of solutions of Eq. (1.1) that belong to the class of so-called P ω (λ)-functions, where the function ϕ(y) satisfies the limit relation lim y→y 0 F 1 (y) = γ, γ ∈ R\{0},

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Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

Cited by 36 publications

References 6 publications

Asymptotic representations of solutions of the nonautonomous ordinary differential $n$-th order equations

Asymptotic representations of solutions of the nonautonomous ordinary differential $n$-th order equations

Asymptotic representations of solutions of nonautonomous second-order differential equations with nonlinearities close to power-type nonlinearities

On the asymptotic behavior of solutions of second-order differential equations with nonlinearity close to an exponential

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