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Solvability of generalized third-order coupled systems with two-point boundary conditions

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Abstract

In this paper we consider the nonlinear third-order coupled system composed by the differential equations \(\begin{cases}-u'''(t) = f(t,u(t),u'(t),u''(t),v(t),v'(t),v''(t)),\\-v'''(t)= h(t,u(t),u'(t),u''(t),v(t),v'(t),v''(t)),\end{cases}\) with f,h: [0, 1] × ℝ6 → ℝ continuous functions, and the boundary conditions \(\begin{cases}u(0) =u'(0)=u'(1)=0,\\v(0) =v'(0)=v'(1)=0.\end{cases}\)

We remark that the nonlinearities can depend on all derivatives of both unknown functions, which is new in the literature, as far as we know. This is due to an adequate auxiliary integral problem with a truncature, applying lower and upper solutions method with bounded perturbations.

The main theorem is an existence and localization result, which provides some qualitative data on the system solution, such as, sign, variation, bounds, etc., as it can be seen in the example.

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Authors and Affiliations

  1. Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações (CIMA), and Instituto de Investigação e Formação Avançada, Universidade de Évora., Rua Romão Ramalho, 59, 7000-671, Évora, Portugal

    Feliz Minhós

  2. Escola Superior Politécnica de Malanje, Malanje, Angola

    Infeliz Coxe

Authors
  1. Feliz Minhós
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  2. Infeliz Coxe
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Correspondence to Feliz Minhós.

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Communicated by T. Krisztin

Supported by National Funds through FCT-Fundação para a Ciência e a Tecnologia, project SFRH/BSAB/114246/2016.

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Minhós, F., Coxe, I. Solvability of generalized third-order coupled systems with two-point boundary conditions. ActaSci.Math. 84, 659–672 (2018). https://doi.org/10.14232/actasm-017-785-0

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  • DOI: https://doi.org/10.14232/actasm-017-785-0

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