# ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY SOLUTIONS OF FOURTH ORDER LINEAR DIFFERENTIAL EQUATIONS

## Authors

• Anibal Coronel
• Fernando Huancas
• Manuel Pinto

## Keywords:

Poincaré-Perron problem, asymptotic behavior, Riccati type equations

## Abstract

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of linear constant coefficient equation. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypothesis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the
perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hypotheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied.

## References

A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl.116(2) (1986), 415–426.

R. Bellman, A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Non-linear Differential and Difference Equations, Office of Naval Research of United States, Department of the Navy, NAVEXOS,P-596,1949

R. Bellman, On the asymptotic behavior of solutions of u''(1 + f(t))u = 0, J. Math. Anal. Appl., 31(1) (1950), 83–91.

R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.

E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations. D. C. Heath and Co., Boston, Mass., 1965.

A. Coronel, F. Huancas and M. Pinto; Asymptotic integration of a linear fourth order differential equation of Poincar´e type, Electron. J. Qual. Theory Differ. Equ., 76 ( 2015) 1–24.

A. R. Davies, A. Karageorghis, T. N. Phillips, Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic flows, Int. J. Numer. Methods Engng., 26 (1988), 647–662.

M.S.P. Eastham, The Asymptotic Solution of Linear Differential Dystems, Applications of the Levinson theorem. London Mathematical Society Monographs, volume 4, Oxford University Press, New York, 1989.

M.V. Fedoryuk, Asymptotic Analysis:Linear ordinary differential equations (Translated from the Russian by Andrew Rodick). Springer-Verlag, Berlin, 1993.

2016-06-30

## How to Cite

Coronel, A., Huancas, F., & Pinto, M. (2016). ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY SOLUTIONS OF FOURTH ORDER LINEAR DIFFERENTIAL EQUATIONS. Selecciones Matemáticas, 3(01), 47-54. https://doi.org/10.17268/sel.mat.2016.01.07

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