  In this paper, we study the existence of solutions of some kinds of operator
equations via operator inequalities. First, we investigate characterizations of
operator order $A\geqslant B >0$ and chaotic operator order log $A \geqslant$
log $B$ for positive definite operators $A$, $B$ in terms of operator
equations, and generalize the results in \cite{CSLin}. Then, we introduce
applications of complete form of Furuta inequality in operator equations. Some
kinds of operator equations are researched and related characterizations of
solutions are proved.
