The Dirichlet problem for nonsymmetric augmented k-Hessian type equations

To solve the Dirichlet problem for nonsymmetric augmented $k$-Hessian type equations, $2\le k\le n$, we first of all solve this problem for corresponding symmetric augmented $k$-Hessian type ones. Then, by using the Banach fixed point theorem, we prove the existence of $\delta$-admissible solution in $C^{2,\alpha }$ of the problem, provided that the skew-symmetric matrices entering the equations are sufficiently small in some sense. Some necessary conditions for existence and sufficient conditions for uniqueness of this kind of solution are given.