TRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT BOUNDARY-VALUE PROBLEMS. A FUNDAMENTAL APPROACH

摘要

Using an elementary approach, we prove the existence of three positive and concave solutions of the second-order two-point boundary-value problem x"(t) = α(t)f(t,x(t),x'(t)), 0 < t < 1, x(0) = x(1) = 0. 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.