Positive Solutions of Some Three-point Boundary Value Problems via Fixed Point Index for Weakly Inward A-proper Maps

in which the second derivative may occur nonlinearly. Positive solutions for the case f (t,u,u′,u′′) = g(t)h(u) have been studied by Ma [15] and Webb [20, 21], when f (t,u,u′,u′′) = h(t,u) by He and Ge [5] and also by Lan [11]. The case f (t,u,u′,u′′) = g(t)h(u,u′) has been studied by Feng [4]. The results in [4, 15] are obtained by means of Krasnosel’skiı̆’s theorem [8], the ones in [5] use Leggett and Williams’ theorem [14] and the results in [11, 20, 21] are achieved via the classical fixed point index for compact maps, see for example [1]. Lafferriere and Petryshyn [9] and Cremins [2] studied existence of positive solutions of the so-called Picard boundary value problem