Zuzana Došlá

Positive decreasing solutions of quasi-linear difference equations

The second order nonlinear difference equation is considered. A full characterization of limit behavior of all positive decreasing solutions is established. The obtained results answer some open problems formulated for Sturm-Liouville discrete operator. A comparison with the continuous case jointly with similarities and discrepancies is given as well.

An equivalence theorem on properties A, B for third order differential equations

Differential equations are often classified according to oscillatory/nonoscillatory properties of their solutions as equations having property A or property B. The aim of the paper is to state an equivalence theorem between property A and property B for third order differential equations. Some applications, to linear as well as to nonlinear equations, are given too. Particularly, we give integral criteria ensuring property A or B for nonlinear equations. Our only assumption on nonlinearity is its superlinearity in neighbourhood of infinity, hence our results apply also to Emden-Fowler type equations.

SOME PROPERTIES OF THIRD ORDER DIFFERENTIAL OPERATORS

AbstractConsider the third order differential operator L given by

Fourth-Order Differential Equation with Deviating Argument

L\left(\cdot\right) \equiv \frac{1}{{a_3 (t)}}\frac{d}{{dt}}\frac{1}{{a_2 (t)}}\frac{d}{{dt}}\frac{1}{{a_1 (t)}}\frac{d}{{d(t)}}\left(\cdot\right)

On third-order linear difference equations involving quasi-differences

and the related linear differential equation L(x)(t) + x(t) = 0. We study the relations between L, its adjoint operator, the canonical representation of L, the operator obtained by a cyclic permutation of coefficients ai, i = 1,2,3, in L and the relations between the corresponding equations.We give the commutative diagrams for such equations and show some applications (oscillation, property A).

Unbounded Solutions of Quasi-Linear Difference Equations

Recessive and dominant solutions for the half-linear difference equation where with {a n } and {b n } are positive real sequences for are studied. By the unique solvability of certain boundary value problems, recessive solutions are defined as “smallest solutions in a neighbourhood of infinity”. The equivalency with other properties, namely with the Riccati property and the convergence or divergence of a suitable series, is also proved.

Asymptotic problems for fourth-order nonlinear differential equations

We consider the fourth-order differential equation with middle-term and deviating argument , in case when the corresponding second-order equation is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.

Oscillation of a class of the fourth-order nonlinear difference equations

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argument . Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operator is oscillatory.