Multiplicity results to nonlinear Hammerstein integral equations and applications

Abstract

Under certain assumptions on the functions $f$, $G$ and $h$, we establish one new criterion on the operator $L$ defined on $C\left(I\right)$ by

$$Lu\left(t\right)={\int }_0^{w}G\left(t,s\right)h\left(s\right)f\left(u\left(s\right)\right)ds,t\in I,\omega \in \left\{1,\infty \right\},$$

to guarantee that the operator equation has at least three solutions, where $I=\left[0,1\right]$ if $\omega =1$ and $I=\left[0,\infty \right)$ if $\omega =\infty$, via the Leguette–Williams fixed point theorem. Two examples are given to demonstrate our main result.