# 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
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.

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

## Keywords

Multiple Positive Solution Symmetric Positive Solution Nonlinear Singular Boundary Triple Positive Solution Triple Solution
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