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

Operator inequalities dealing with operator equations