Abstract
The existence and asymptotic behavior at infinity of increasing positive solutions of second order quasilinear ordinary differential equations $(p(t)\varphi (x'(t)) )^{\prime}+q(t)\psi(x(t))=0 $ are studied in the framework of regular variation, where p and q are continuous functions regularly varying at infinity and φ, ψ are both continuous functions regularly varying at zero and regularly varying at infinity, respectively.
Highlights
1 Introduction It is of particular interest in the theory of qualitative analysis of differential equations to determine the exact asymptotic behavior at infinity of the solutions under the appropriate assumptions for the coefficients of an equation
The recent research shows that the problem should be studied in the framework of regularly varying functions
S acting on some closed convex subsets X of C[t, ∞), which should be chosen in such a way that F is a continuous self-map on X and send it into a relatively compact subset of C[t, ∞). That such choices of X are feasible is guaranteed by the existence of three types of regularly varying functions that determine exactly the asymptotic behavior of all possible solutions of ( . )
Summary
It is of particular interest in the theory of qualitative analysis of differential equations to determine the exact asymptotic behavior at infinity of the solutions under the appropriate assumptions for the coefficients of an equation This problem is extremely complex when the coefficients are general continuous functions. In this paper we establish the necessary and sufficient conditions for the existence of intermediate solutions for (E) and precisely determine their behavior at infinity, using the theory of regularly varying functions. By an only regularly or a slowly varying function, we mean regularity at infinity To help the reader we present here some elementary properties of regularly varying functions and a fundamental result, called Karamata’s integration theorem, which will be used throughout the paper.
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