The paper discusses continuous and generalized solutions of equations with partial derivatives having the operator coefficients which operate in Banach spaces. The operator with the elder derivative with respect to time is Fredholm. We apply Lyapunov--Schmidt's ideas and the generalized Jordan sets techniques to reduce partial differential-operator equations with the Fredholm operator in the main part to regular problems. In addition this technique has been exploited to prove the theorem of existence and uniqueness for a singular initial-value problem, as well as to construct the left and right regularizators of singular operators in Banach spaces and to construct fundamental operators in the theory of generalized solutions of singular equations.
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