Positive solutions to some equations with homogeneous operator

In this paper, we discuss the positive solutions to the equation φ(u)u=λaAu+Bu+u0, where A is a positive linear completely continuous operator, B is an α-homogeneous operator defined on a cone in a real Banach space and φ(u)=a+b‖u‖β. By using the fixed point index theory, when u0 is sufficiently small, the spectral radius λr(A)<1 and α−γβ>1, where γ=sgnb, we obtain a positive solution to the above equation under some appropriate conditions. The new results generalize the previous research about the homogeneous operator equation. As an application, by using our main theorem we can obtain a symmetrical positive solution to the one dimensional Kirchhoff equation.