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Recently, it was shown that the Euler type half-linear differential equations ****with periodic coefficients r and s are conditionally oscillatory and the critical oscillation constant was found. Nevertheless, the critical case remains unsolved. The objective of this article is to study the critical case. Thus, we consider the critical value of the coefficients and we prove that any considered equation is non-oscillatory. Moreover, we analyze the situation when the periods of coefficients r and s do not need to coincide. [ABSTRACT FROM AUTHOR]

We investigate oscillatory properties of second order Euler type half-linear differential equations whose coefficients are given by periodic functions and functions having mean values. We prove the conditional oscillation of these equations. In addition, we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case. The presented results are new for linear equations as well. [ABSTRACT FROM AUTHOR]

We investigate perturbed second order Euler type half-linear differential equations with periodic coefficients and with the perturbations given by the finite sums of periodic functions which do not need to have any common period. Our main interest is to study the oscillatory properties of the equations in the case when the coefficients give exactly the critical oscillation constant. We prove that any of the considered equations is non-oscillatory in this case. [ABSTRACT FROM AUTHOR]

For a differential equation of arbitrary even order with operator coefficients, a generalization of Arnold's theorems of alternation and non-oscillation is obtained. The oscillation theorem for a problem with general boundary conditions is proved. [ABSTRACT FROM AUTHOR]

We derive Wiman's asymptotic formula for the number of generalized zeros of (nontrivial) solutions of a second order dynamic equation on a time scale. The proof is based on the asymptotic representation of solutions via exponential functions on a time scale. By using the Jeffreys et al. approximation we prove Wiman's formula for a dynamic equation on a time scale. Further we show that using the Hartman-Wintner approximation one can derive another version of Wiman's formula. We also prove some new oscillation theorems and discuss the results by means of several examples [ABSTRACT FROM AUTHOR]

The aim of the paper is to give an oscillation theorem for inhomogeneous Stieltjes integro-differential equation of the form p(t)x′+∫atx(s)dσ=f(t). The paper generalizes the author's work [2].

In this paper, following a previous paper ([32] Permanence and extinction of a non-autonomous HIV-1 model with two time delays, preprint) on the permanence and extinction of a delayed non-autonomous HIV-1 within-host model, we introduce and investigate a delayed HIV-1 model including maximum homeostatic proliferation rate of CD4+ T-cells and varying coefficients. By applying the asymptotic analysis theory and oscillation theory, we show: (i) the system will be permanent when the threshold value R* > 1, and for this case we also obtain the explicit estimate of the eventual lower bound of the HIV-1 virus load; (ii) the threshold value R* < 1 implies the extinction of the virus. Furthermore, we obtain that the threshold dynamics is in agreement with that of the corresponding autonomous system, which extends the classic results for the system with constant coefficients. Numerical simulations are also given to illustrate our main results, and in particular, some sensitivity test of R* is established. [ABSTRACT FROM AUTHOR]

The basic boundary value oscillation problems for a three-dimensional elastic medium bounded by a closed surface are considered. Asymptotic formulas are derived for the eigenvalue and eigenfunction distributions in the problems. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies oscillation problems of Left-Definite regular self-adjoint SLPs with real coupled BCs. For a given eigenvalue of the Left-Definite SLPs with real coupled BCs, we conclude that is an eigenvalue of the SLPs with certain separated BCs. We also establish some inequalities among the eigenvalues for the SLPs with separated BCs and coupled BCs. With the help of the obtained results, we give an estimate of the index of eigenvalues and the number of zeros of eigenfunction for the SLPs with real coupled BCs. [Copyright &y& Elsevier]

We propose a new method for calculating eigenvalues of discrete symplectic boundaryvalue problems. This method is based on the discrete oscillation theory and on a modification of the Abramov double sweep method for discrete self-adjoint boundary-value problems. [ABSTRACT FROM AUTHOR]

In 1988, S. Bank showed that if { z} is a sparse sequence in the complex plane, with convergence exponent zero, then there exists a transcendental entire A( z) of order zero such that f″+ A( z) f=0 possesses a solution having { z} as its zeros. Further, Bank constructed an example of a zero sequence { z} violating the sparseness condition, in which case the corresponding coefficient A( z) is of infinite order. In 1997, A. Sauer introduced a condition for the density of the points in the zero sequence { z} of finite convergence exponent such that the corresponding coefficient A( z) is of finite order. In 2010, the second author proposed a unit disc analog of Bank's first result. In the analog, { z} is a sparse Blaschke sequence and A( z) belongs to the Korenblum space. The aim of the present paper is to introduce unit disc analogs of the two remaining results due to Bank and Sauer. [ABSTRACT FROM AUTHOR]

Abstract: We consider the fourth order quasilinear differential equation where α and β are positive constants, , , and , for , and establish necessary and sufficient integral conditions for the existence of eventually positive solutions of the fourth order quasilinear differential equations. [Copyright &y& Elsevier]

This research is partially a continuation of a 2007 paper by the author. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. Such Blaschke products are known as interpolating Blaschke products. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E. Fricain and J. Mashreghi in 2008. Finally, several aspects are pointed out to illustrate that interpolating Blaschke products appear naturally in studying the oscillation of solutions of a differential equation f″+ A( z) f=0, where A( z) is analytic in the unit disc. In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B. Schwarz on the case $A(z)=\frac{1+4\gamma^{2}}{(1-z^{2})^{2}}$ reveals that an infinite zero sequence is always a union of two exponential sequences. [ABSTRACT FROM AUTHOR]

Abstract: We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein''s spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators. [Copyright &y& Elsevier]

Abstract: Using relative oscillation theory and the reducibility result of Eliasson, we study perturbations of quasiperiodic Schrödinger operators. In particular, we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the oscillatory properties for the eigenfunctions of some fourth-order eigenvalue problems, where the boundary conditions are irregular in the sense of the classification of [S. Janczewski, Oscillation theorems for the differential boundary value problems of the fourth order, Ann. of Math. 29 (1928) 521–542]. In this case, we show that these oscillatory properties are different from those of the Sturm–Liouville problem. [Copyright &y& Elsevier]

Abstract: In this paper we develop an extension of the classical Sturm theory [C. Sturm, Sur une classe d''equations à derivée partielle, J. Math. Pures Appl. 1 (1836) 373–444], to study the oscillation properties for the eigenfunctions of some fourth-order two point boundary value problems on the interval . We are mainly interested in the case when these problems have negative eigenvalues induced by the sign of the parameters in the boundary conditions. In particular, we give an asymptotic estimate of the number of zeros in of the first eigenfunction in terms of the variation of parameters in the boundary conditions. [Copyright &y& Elsevier]

Abstract: We present a streamlined approach to relative oscillation criteria based on effective Prüfer angles adapted to the use at the edges of the essential spectrum. Based on this we provided a new scale of oscillation criteria for general Sturm–Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover and generalize the Gesztesy–Ünal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt. [Copyright &y& Elsevier]

The aim of this paper is to derive sufficient conditions for the linear delay differential equation ( r( t) y′( t))′ + p( t) y( τ( t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem. [ABSTRACT FROM AUTHOR]

We show that the half-linear differential equation (*)[r(t)θ(x')]' + s(t)/tpθ(x) = 0 with α-periodic positive functions r, s is conditionally oscillatory, i.e., there exists a constant K > 0 such that (*) with γs(t)/tp instead of s(t)/tp) is oscillatory for γ > K and nonoscillatory for γ < K. [ABSTRACT FROM AUTHOR]

Abstract: We extend relative oscillation theory to the case of Sturm–Liouville operators with different p''s. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein''s spectral shift function inside essential spectral gaps. [Copyright &y& Elsevier]

Abstract: Since 1939 when Thiele and Zeldovich developed a theory devoted to the reaction mechanism in a catalyst particle, nothing has disturbed the scientific paradigm in catalysis. In multiphase reactions, as is widely accepted, liquid occupies catalyst pores, remaining an immovable medium through which the reacting compounds “slowly” diffuse to active centres where the reaction takes place. Some years ago, it was predicted that in gas–liquid/liquid reactions with gas or heat evolution, liquid could chaotically move in catalyst pores with velocities up to 300m/s causing a great impact on the reaction performance in a catalyst particle and a reactor. The present paper deals with some experimental confirmation of the oscillatory mechanism and illustrates some phenomena that can purposefully be used for process and catalyst development. [Copyright &y& Elsevier]

Abstract: In this paper we present nonintegral criteria for oscillation of linear Hamiltonian matrix system , under the hypothesis (H): , , and are matrices of real valued continuous functions on the interval . These criteria are conditions of algebraic type only. Our results are also useful for the detection of the oscillation of particular matrix differential systems. [Copyright &y& Elsevier]

The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity. [ABSTRACT FROM AUTHOR]

Abstract: We extend old results dealing with non-oscillations of Wintner (1951) and Opial (1957) from scalar to vector differential equations with symmetric coefficients using the Schauder fixed point theorem. [Copyright &y& Elsevier]

We prove oscillation results for the real eigenvalues of Sturm–Liouville problems with an indefinite weight function. An essential role is played by the signature of an eigenvalue, which is shown to be related to the signs of the corresponding leading coefficients of the Titchmarsh–Weyl m-function and of the Prüfer angle at this eigenvalue. [Copyright &y& Elsevier]