In this paper we discuss the asymptotic behavior as t→∞ of solutions of a second order differential equation in y depending on two real parameters α and β and two nonzero real constants b and c. The analysis shows that there is a set of rays, each one starting at the origin of the (α,β) plane, which divide this plane into sectors, with different asymptotic representations for y and y′ holding in different sectors. These rays are the negative alpha axis, the positive and negative beta axes, and a countable number of rays in the first quadrant, beginning with the ray β=α/2 and ending with the ray β=2α, which resemble a fan. The results on asymptotic behavior are obtained by use of methods due to Harris and D. A. Lutz [J. Math. Anal. Appl. 57 (1977), 571-586] which construct matrices P and Λ such that the two-dimensional system associated with the original equation, say Y′=A(t)Y, with A(t)=(0−ct2β−21−2btα−1), has the asymptotic integration Y(t)=P(t)[I+o(1)]exp(∫tΛ(s)ds), t→∞. Depending on whether the solution y under consideration is oscillatory or nonoscillatory, the form the representations take is as follows: if y is a nonoscillatory solution and Γ is an open sector in the α,β plane bounded by a pair of consecutive rays there is a positive (real analytic) function φΓ(t)=φΓ(t,α,β) (α,β)∈Γ, t>0, having the property that y(t)/φΓ(t)→finite limit, t→∞. A similar result holds for oscillatory solutions, but with a pair of functions φΓ and ψΓ such that y(t)/φΓ(t)=Acos(ψΓ(t)+θ)⋅[1+o(1)], t→∞, with A and θ constants. In all cases explicit formulae are given for the functions φΓ and ψΓ, along with a discussion of the transitional behavior and the continuity of these functions as the sector rays are traversed. As an illustration of the results, the asymptotic behavior of solutions with β=1, c=1 and b>0 is given, by computing explicitly the various necessary φ and ψ functions. Finally, we state that, in a forthcoming paper, the asymptotic behavior of higher derivatives of solutions will be given.

Asymptotic behavior of solutions of a nonstandard second order differential equation

PUCCI, Patrizia;
1993

Abstract

In this paper we discuss the asymptotic behavior as t→∞ of solutions of a second order differential equation in y depending on two real parameters α and β and two nonzero real constants b and c. The analysis shows that there is a set of rays, each one starting at the origin of the (α,β) plane, which divide this plane into sectors, with different asymptotic representations for y and y′ holding in different sectors. These rays are the negative alpha axis, the positive and negative beta axes, and a countable number of rays in the first quadrant, beginning with the ray β=α/2 and ending with the ray β=2α, which resemble a fan. The results on asymptotic behavior are obtained by use of methods due to Harris and D. A. Lutz [J. Math. Anal. Appl. 57 (1977), 571-586] which construct matrices P and Λ such that the two-dimensional system associated with the original equation, say Y′=A(t)Y, with A(t)=(0−ct2β−21−2btα−1), has the asymptotic integration Y(t)=P(t)[I+o(1)]exp(∫tΛ(s)ds), t→∞. Depending on whether the solution y under consideration is oscillatory or nonoscillatory, the form the representations take is as follows: if y is a nonoscillatory solution and Γ is an open sector in the α,β plane bounded by a pair of consecutive rays there is a positive (real analytic) function φΓ(t)=φΓ(t,α,β) (α,β)∈Γ, t>0, having the property that y(t)/φΓ(t)→finite limit, t→∞. A similar result holds for oscillatory solutions, but with a pair of functions φΓ and ψΓ such that y(t)/φΓ(t)=Acos(ψΓ(t)+θ)⋅[1+o(1)], t→∞, with A and θ constants. In all cases explicit formulae are given for the functions φΓ and ψΓ, along with a discussion of the transitional behavior and the continuity of these functions as the sector rays are traversed. As an illustration of the results, the asymptotic behavior of solutions with β=1, c=1 and b>0 is given, by computing explicitly the various necessary φ and ψ functions. Finally, we state that, in a forthcoming paper, the asymptotic behavior of higher derivatives of solutions will be given.
1993
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/117200
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