Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

New oscillation and nonoscillation criteria are established for the equation

Existence of conjugate points for second-order linear differential equations

The sufficient conditions are established under which the second-order linear differential equation is conjugate.     

Nonpositive solutions of one functional differential inequality

We establish efficient conditions guaranteeing that every solution of the problem

Oscillation and Nonoscillation Criteria for Second-Order Linear Differential Equations

Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established.     

Periodic solutions of nonautonomous ordinary differential equations

For higher order ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions are established. Results obtained cover the case when the right-hand side of the equation is not of a constant sign with respect to an independent variable.     

On a Two-Point Boundary Value Problem for Second Order Singular Equations

The problem on the existence of a positive in the interval ]a, b[ solution of the boundary value problem is considered, where the functions f and satisfy the local Carathéodory conditions. The possibility for the functions f and g to have singularities in the first argument (for t = a and t = b) and in the phase variable (for u = 0) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.     

On a Boundary-Value Problem of Periodic Type for First-Order Linear Functional Differential Equations

We establish unimprovable sufficient conditions for the unique solvability of the boundary-value problem

On the Cauchy problem for linear hyperbolic functional-differential equations

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.