The nonlinear Kneser problem for singular in phase variables second-order differential equations

For the singular in phase variables differential equation u'' = f (t, u, u') ), sufficient conditions are found for the existence of a solution satisfying the conditions Phi(u) = c, u(t) > 0, u'(t) 0, where Phi : C([0, a]; R+) to R+ is a continuous nondecreasing functional, c > 0, and a > 0

The focal boundary value problem for strongly singular higher-order nonlinear functional-differential equations

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

Lasota–Opial type conditions for periodic problem for systems of higher-order functional differential equations

In the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations. In the paper in some sense optimal conditions that guarantee the unique solvability of the linear problem are obtained, and on the basis of these results the optimal conditions of the solvability and unique solvability for the nonlinear problem are proved