... nontrivialsolution of the BVP 1.1 and 1.2.Next, we show the uniqueness of solutions of the BVP 1.1 and 1.2. Assume, to the contrary, that there exist two nontrivial solutions y1and y2 of the BVP ... the proof of the theorem.Remark 3.2. From the proof of Theorem 3.1, we easily know that assume y Ay,x, x Ax, y,thus,lety0 y, x0 x, we haveyn y, xn x, n 1, 2, 3.42Thereforey ... uλsatisfies the properties (i)–(iii) stated inTheorem 1.1. The aim of this work is to relax the assumptions A1 and B1 on the nonlinear term,without demanding the existence of upper and...