Ivan Kiguradze

On periodic solutions of higher-order functional differential equations.

For higher-order functional differential equations and, particularly, for nonautonomous differential equations with deviated arguments, new sufficient conditions for the existence and uniqueness of a periodic solution are established.

Positive solutions of periodic type boundary value problems for first order singular functional differential equations

Abstract For first order singular functional differential equations, optimal sufficient conditions for the existence of positive solutions of periodic type boundary value problems are established.

Conditions for the well-posedness of nonlocal problems for higher order linear differential equations with singularities

Abstract Criteria of the conditional well-posedness of nonlocal problems for linear partial differential equations of hyperbolic type and for higher order linear ordinary differential equations with singular coefficients are established.

A priori estimates of solutions of nonlinear boundary value problems for singular in a phase variable second order differential inequalities

Abstract. For singular in a phase variable second order differential inequalities, a priori estimates of positive solutions, satisfying nonlinear nonlocal boundary conditions, are established.

The Cauchy problem for singular in phase variables nonlinear ordinary differential equations

Abstract. The optimal sufficient conditions for local solvability of the Cauchy problem for singular in phase variables nonlinear ordinary differential equations of higher orders are established.

On a boundary value problem on an infinite interval for nonlinear functional differential equations

Abstract Sufficient conditions are found for the solvability of the following boundary value problem: u ( n ) ( t ) = f ( u ) ( t ) , u ( i - 1 ) ( 0 ) = φ i ( u ( n - 1 ) ( 0 ) )   ( i = 1 , … , n - 1 ) , lim inf t → + ∞ | u ( n - 2 ) ( t ) | < + ∞ , u^{(n)}(t)=f(u)(t),\qquad u^{(i-1)}(0)=\varphi_{i}(u^{(n-1)}(0))\quad(i=1,% \dots,n-1),\qquad\liminf_{t\to+\infty}\lvert u^{(n-2)}(t)|<+\infty, where f : C n - 1 ⁢ ( ℝ + ) → L loc ⁢ ( ℝ + ) {f\colon C^{n-1}(\mathbb{R}_{+})\to L_{\mathrm{loc}}(\mathbb{R}_{+})} is a continuous Volterra operator, and φ i : ℝ → ℝ {\varphi_{i}\colon\mathbb{R}\to\mathbb{R}} ( i = 1 , … , n {i=1,\dots,n} ) are continuous functions.

On nonlinear boundary value problems for higher order functional differential equations

Abstract For higher order nonlinear functional differential equations, sufficient conditions for the solvability and unique solvability of some nonlinear nonlocal boundary value problems are established.