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\(L^p\)-solutions of a nonlinear third order differential equation and the Poincaré–Perron problem

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Abstract

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory \(L^p\)-solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (1) the characteristic polynomial associated with the linear part has simple and real roots, (2) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (3) the polynomial coefficients are in \(L^p([t_0,\infty [)\). These results are applied to study a fourth order linear differential equation of Poincaré type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems cannot be applied.

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Acknowledgements

A. Coronel, F. Huancas and L. Friz would like to thank the support of research projects at Universidad del Bío-Bío (Chile): DIUBB 172409 GI/C, DIUBB 103309 4/R, the program “Fondo de Apoyo a la Participacin a Eventos Internacionales” (FAPEI), and “Fortalecimiento del postgrado” of the project “Instalación del Plan Plurianual UBB 2016-2020”. M. Pinto thanks the support of Fondecyt project 1120709.

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Authors and Affiliations

  1. GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile

    Aníbal Coronel, Luis Friz & Fernando Huancas

  2. Departamento de Matemática, Facultad de Ciencias, Universidad de Chile, Santiago, Chile

    Manuel Pinto

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  1. Aníbal Coronel
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  2. Luis Friz
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  3. Fernando Huancas
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  4. Manuel Pinto
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Correspondence to Aníbal Coronel.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Coronel, A., Friz, L., Huancas, F. et al. \(L^p\)-solutions of a nonlinear third order differential equation and the Poincaré–Perron problem. J. Fixed Point Theory Appl. 21, 3 (2019). https://doi.org/10.1007/s11784-018-0641-3

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  • DOI: https://doi.org/10.1007/s11784-018-0641-3

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