We discuss various properties of the nonlinear $A$-proper operators as well as a generalized Leray-Schauder principle. Also, a method of approximating arbitrary continuous operators by $A$-proper mappings is described. We construct, via appropriate Browder-Petryshyn approximation schemes, approximative solutions for linear evolution equations in Banach spaces.

This paper is concerned with the internal and boundary stabilization of the steady-state solutions to quasilinear heat equations via internal linear feedback controllers provided by an LQ control problem associated with the linearized equation.

We establish a new form of the $3G$ theorem for polyharmonic Green function on the unit ball of ${ℝ}^n\left(n\ge 2\right)$ corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions $K_{m,n}$ containing properly the classical Kato class $K_n$. We exploit properties of functions belonging to $K_{m,n}$ to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order $2m$.