On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

On inequalities of Lyapunov for linear Hamiltonian systems on time scales

In this paper, we establish several new Lyapunov type inequalities for linear Hamiltonian systems on an arbitrary time scale T when the end-points are not necessarily usual zeroes, but rather, generalized zeroes, which generalize and improve all related existing ones including the continuous and discrete cases.     

On weighted Iyengar type inequalities on time scales

In this study, we establish some new weighted Iyengar type integral inequalities using Steffensen’s inequality on time scales.     

On the exponential stability of dynamic equations on time scales

In this paper, we deal with some theorems on the exponential stability of trivial solution of time-varying non-regressive dynamic equation on time scales with bounded graininess. In particular, well-known Perron's theorem is generalized on time scales. Under rather restrictive condition, that is, integral boundedness of coefficient operators, we obtain a characterization of the uniformly exponential stability.     

On the topological classification of dynamic equations on time scales

This paper considers the topological classification of non-autonomous dynamic equations on time scales. In this paper we show, by a counterexample, that the trivial solutions of two topologically conjugated systems may not have the same uniform stability. This is contrary to the expectation that two topologically conjugated systems should have the same topological structure and asymptotic behaviors. To counter this mismatch in expectation, we propose a new definition of strong topological conjugacy that guarantees the same topological structure, and in particular the same uniform stability, for the corresponding solutions of two strongly topologically conjugated systems. Based on the new definition, a new version of the generalized Hartman–Grobman theorem is developed. We also include some examples to illustrate the feasibility and effectiveness of the new generalized Hartman–Grobman theorem.     

On the oscillation for third-order nonlinear neutral delay dynamic equations on time scales

In this paper, we consider the third-order nonlinear neutral delay dynamic equations

$$\begin{aligned} \left( b(t)\left( \left( ((x(t)-p(t)x(\tau (t)))^\Delta )^{\alpha _1}\right) ^\Delta \right) ^{\alpha _2}\right) ^\Delta +f(t,x(\delta (t)))=0 \end{aligned}$$

on a time scale \(\mathbb {T}\), where \(\alpha _i\) are quotients of positive odd integers, \(i=1\), 2, \(|f(t,u)|\ge q(t)|u|\), \(b,\ p\) and q are real-valued positive rd-continuous functions defined on \(\mathbb {T}\). By using the Riccati transformation technique and integral averaging technique, some new sufficient conditions which ensure that every solution oscillates or tends to zero are established. Our results are new for third-order nonlinear neutral delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given here to illustrate our main results.

     

A new periodicity concept for time scales

By means of the shift operators we introduce a new periodicity concept on time scales. This new approach will enable researchers to investigate periodicity notion on a large class of time scales whose members may not satisfy the condition there exists a P > 0 such that $t \pm P \in \mathbb{T}$ for all $t \in \mathbb{T}$ , which is being currently used. Therefore, the results of this paper open an avenue for the investigation of periodic solutions of q-difference equations and more.     

Estimates for the singular values of stieltjes type transforms

We obtain two-sided estimates for the Schatten-von Neumann norms of a selfadjoint Stieltjes type transform in the space of square integrable functions.     

Stability criteria for impulsive systems on time scales

In this paper we study stability of impulsive system on time scales. By using comparison method, Lyapunov function and analysis method, the asymptotic stability criteria for system with impulses at fixed times and impulses at variable times on time scales are obtained, respectively. An example is presented to illustrate the efficiency of proposed result.     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

Solutions for higher-order dynamic equations on time scales

Let T be a time scale. We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order p-Laplacian dynamic equations on time scales. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.     

Instability results for slowly time varying linear dynamic systems on time scales

We develop eigenvalue criteria under which the solutions of a “slowly” time varying linear dynamic system of the form xΔ(t)=A(t)x(t) are unstable.     

Bounds for the stieltjes transform of spectral functions of singular eigenvalues

The main problem consists in obtaining G-estimates for singular eigenvalues of real matrices A=(a_ij),i = ,i = if the only things known are X_i, i = , independent observations over the matrix A+, where is a stochastic matrix. Under certain conditions on , A, n, m, and s results are obtained for the Stieltjes transform of spectral functions of the singular eigenvalues of the matrix A.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 464–469, April, 1990.     

Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation

(r(t)(xΔγ⁾Δ⁾+p(t)xγ(t)=0,     

Comparison theorems for linear dynamic equations on time scales

We obtain several comparison theorems for second order linear dynamic equations on a time scale. These results extend comparison theorems for the continuous case and provide some new results in the discrete case, as well as other more general situations.     

Some mean value theorems for integrals on time scales

In this short paper, we present time scales version of mean value theorems for integrals in the single variable case.     

Oscillation criteria for second-order dynamic equations on time scales

Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results.