Tariel Kiguradze

On conditions for linear singular boundary value problems to be well posed

For linear singular differential equations of higher order, we obtain necessary and sufficient conditions for nonlocal boundary value problems to be well posed or conditionally well posed.

Estimates for the Cauchy function of linear singular differential equations and some applications

For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.

ON THE CORRECTNESS OF THE DIRICHLET PROBLEM IN A CHARACTERISTIC RECTANGLE FOR FOURTH ORDER LINEAR HYPERBOLIC EQUATIONS

AbstractIt is proved that the Dirichlet problem is correct in the characteristic rectangle Dab = [0, a] × [0, b] for the linear hyperbolic equation

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

\begin{gathered} \frac{{\partial ^4 u}}{{\partial x^2 \partial y^2 }} = p_0 (x,y)u + p_1 (x,y)\frac{{\partial u}}{{\partial x}} + p_2 (x,y)\frac{{\partial u}}{{\partial y}} + \hfill \\ {\text{ }} + p_3 (x,y)\frac{{\partial ^2 u}}{{\partial x\partial y}} + q(x,y) \hfill \\ \end{gathered}

On the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Singular Hyperbolic Equations

with the summable in Dab coefficients p0, p1, p2, p3 and q if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal in some sense restrictions on p0, p1, p2 and p3 guaranteeing the correctness of the Dirichlet problem are established.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

The conditions of the existence and uniqueness of solutions, bounded in a strip together with their partial derivatives of the first order, have been established for the hyperbolic equations .

On Doubly Periodic Solutions of Nonlinear Hyperbolic Equations of Higher Order

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.

Analog of the first Fredholm theorem for higher-order nonlinear differential equations

Higher order partial differential equations with functional arguments including hyperbolic equations and beam equations are studied. Sufficient conditions are derived for every solution of certain boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional problem for higher order functional differential inequalities.