Journal of Mathematical Sciences

, Volume 189, Issue 5, pp 823–833

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

Article
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.

## Keywords

Multiple Positive Solution Symmetric Positive Solution Nonlinear Singular Boundary Triple Positive Solution Triple Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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