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The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear...

We study the question of the unique solvability of the periodic type problem for the second order linear integro-differential equation with distributed argument deviation $$u^{\text{'}\text{'}}\left(t\right)=p_0\left(t\right)u\left(t\right)+{\int }_0^{\omega }p(t,s)u\left(\tau (t,s)\right)\mathrm{d}s+q\left(t\right),$$ and on the basis of the obtained results by the a priori boundedness principle we prove the new results on the solvability of periodic type problem for the second order nonlinear functional differential equations, which are close to the linear integro-differential equations. The proved results are optimal in some sense.