Evolution equations with monotone operator and functional non-linearity at the time derivative

Conditions for the solubility of the so-called doubly non-linear equations
$$ Au+\frac\partial{\partial t}Bu=f, \qquad u(0)=u_0, $$
are investigated. Here $A$ is a monotone operator induced by a differential expression containing higher-order partial derivatives and $B$ is an operator induced by a monotone function. A theorem on the existence of a solution is proved. The method of monotone operators is used in combination with the method of compact operators. Examples of applications to parabolic differential equations are presented.