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Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach

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Using an elementary approach, we prove the existence of three positive and concave solutions of the second-order two-point boundary-value problem

$$ \begin{array}{lll}&& x^{\prime\prime}(t)=\alpha(t)f(t,x(t),x^{\prime}(t)),\qquad 0 < t < 1,\\&& \qquad \qquad \qquad x(0)=x(1)=0.\end{array}$$

We rely on the analysis of the corresponding vector field on the phase space, Kneser-type properties of the solution funnel, and the Schauder fixed-point theorem. The obtained results demonstrate the simplicity and efficiency (one could study a problem with more general boundary conditions) of our new approach as compared with the commonly used ones, such as the Leggett–Williams fixed-point theorem and its generalizations.

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Authors and Affiliations

  1. Naval Academy of Greece, Piraeus, Greece

    P. K. Palamides

  2. Technological Educational Institute of Piraeus, Athens, Greece

    A. P. Palamides

Authors
  1. P. K. Palamides
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  2. A. P. Palamides
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Correspondence to P. K. Palamides.

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Published in Neliniini Kolyvannya, Vol. 15, No. 2, pp. 233–243, April–June, 2012.

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Palamides, P.K., Palamides, A.P. Triple positive solutions for a class of two-point boundary-value problems. A fundamental approach. J Math Sci 189, 823–833 (2013). https://doi.org/10.1007/s10958-013-1222-z

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