On a nonlinear Hammerstein integral equation with a parameter

This paper, following theories for asymptotically linear operators, the Schaefer fixed point theorem, decomposition of operators, and critical point theory, is mainly concerned with the existence and multiplicity of solutions to a nonlinear Hammerstein integral equation with a parameter. The results show that when the nonlinearity satisfies certain conditions, different parametric intervals lead to different existence results; however, in some cases only the sign of the parameter makes a contribution to the existence of solutions for the problem. Our results can be applied to some well known boundary value problems, and some examples are given.