Nontrivial solution of third-order nonlinear eigenvalue problems

In this paper, we study the existence and uniqueness of nontrivial solution for the following third-order eigenvalue problems (TEP):u‴=λf(t,u,u′),0 0 is a parameter, 12⩽η<1 is a constant, f:[0,1]×R×R→R is continuous, R=(-∞,+∞). Without any monotone-type and nonnegative assumption, we obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of TEP when λ in some interval. Our approach is based on Leray-Schauder nonlinear alternative.