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We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.

One of the obstacles that arises in the generalization of Sturm's oscillation theorem to the case of general linear Hamiltonian systems is the need to associate a sign with each crossing point, necessitating a signed count rather than a direct count of such points. It's known, however, that this difficulty does not arise for all system/boundary-condition combinations, and so it can be overcome in some cases by exchanging one boundary condition for another while also keeping track of any ancillary counts that accompany the exchange. The primary tool for making such an exchange is Hormander's index, and in this analysis we develop a straightforward method for computing Hormander's index, and employ our method to formulate oscillation-type theorems for linear Hamiltonian systems on both bounded and unbounded intervals.

We find very general classes of linear and half-linear difference equations which are conditionally oscillatory. We identify the critical oscillation constant whose value implies the oscillation or non-oscillation of studied equations. Our results are divided into oscillatory and non-oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values. In addition, our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values. We point out that the obtained results are new even for linear equations with periodic coefficients. Such consequences are formulated at the end of this paper.

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.

By theorems of Sturm and Lyapunov, the L ∞ and L 1 norms of the coefficient q in a differential equation u ″ + q ( t ) u = 0 limit the oscillation of nontrivial solutions. This paper establishes similar results for the L p norms when 1 p ∞ .

We compare oscillatory behavior of two half-linear differential equations with different powers by the homogeneous functions appearing in these equations. Our results generalize and extend several previously proved power comparison theorems.

The main purpose of this paper is to investigate the oscillation theory of meromorphic solutions of the second order linear differential equation f ″ + A ( z ) f = 0 for the case where A is meromorphic in the unit disc D = { z : | z | 1 } .

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., Comment: 13 pages

This paper is concerned with the oscillation of a class of partial functional population model, and some sufficient conditions for all positive solutions of the model to oscillate about the positive equilibrium are obtained. The approach is based on the upper- and lower-solution method of the partial functional differential equations and the oscillation theory of the functional differential equation. Moreover, some numerical simulations are also given to illustrate our results.

In this paper we develop an extension of the classical Sturm theory [C. Sturm, Sur une classe d'equations a derivee 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 [ 0 , 1 ] . 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 ( 0 , 1 ) of the first eigenfunction in terms of the variation of parameters in the boundary conditions.

In the paper we study the damped half-linear partial differential equation div ( A ( x ) ‖ ∇ u ‖ p − 2 ∇ u ) + 〈 b → ( x ) , ‖ ∇ u ‖ p − 2 ∇ u 〉 + c ( x ) | u | p − 2 u = 0 . Using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations. Using careful radialization we improve several known oscillation criteria.

In this paper we present nonintegral criteria for oscillation of linear Hamiltonian matrix system U ′ = A ( x ) U + B ( x ) V , V ′ = C ( x ) U − A * ( x ) V under the hypothesis (H): A ( x ) , B ( x ) = B * ( x ) > 0 , and C ( x ) = C * ( x ) are 2 × 2 matrices of real valued continuous functions on the interval I = [ a , ∞ ) , ( − ∞ a ) . These criteria are conditions of algebraic type only. Our results are also useful for the detection of the oscillation of particular matrix differential systems.

We establish some uniqueness and existence results for first-order ordinary differential equations with constant-signed discontinuous nonlinear parts. Several examples are given to illustrate the applicability of our work.

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.

We consider the structure of the solution set of a nonlinear Sturm–Liouville boundary value problem defined on a general time scale. Using global bifurcation theory we show that unbounded continua of nontrivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and we show that certain nodal properties of the solutions are preserved along these continua. These results extend the well-known results of Rabinowitz for the case of Sturm–Liouville ordinary differential equations.

In this paper, a non-variational version of a max-min principle is extended, and some unique existence results are obtained for the periodic boundary value problem of the higher order ordinary differential equations under a resonant condition.

Letf1, f2be two linearly independent solutions of the linear differential equationf″ + A(z)f = 0, whereA(z) is transcendental entire, and assume that the exponents of convergence for the zero-sequences off1, f2satisfy max(λ(f1), λ(f2)) = ∞. Our main result proves that the zeros ofE ≔ f1f2are uniformly distributed in the sense that quite arbitrary large areas of the complex plane can be removed in such a way that if only zeros outside of these areas will be counted for the exponents of convergences, their maximum still remains infinite.

Divergent series of Dirac delta functions have been used in the solution of ordinary and functional differential equations. Such divergent series also form the basic block in the asymptotic expansion of generalized functions. In this article we combine these two ideas and show that the series of deltas arising as solutions of ordinary differential equations can be realized as the asymptotics of distributional solutions of certain associated equations.

In this paper an initial value problem for a nonlinear ordinary differential equation of second order is considered. Some nonlinear operator methods are used to prove the existence and uniqueness of nontrivial solutions. Moreover, an approximation of the nontrivial solution is given.

In this paper, we investigate the complex oscillation theory of ƒ (k) + B k−1 ƒ (k−1) + ··· + B1ƒ′ + Aƒ = F, where, B 1 , ..., B k −1 , F ≢ 0 are entire functions, and A is a transcendental entire function. We obtain some precise results.

In this paper, we give precise estimates of the exponent of convergence of the zero sequence of solutions for the linear differential equation f (k) + a k − 1 f (k − 1) + … + a 0 f = P 1 e P 0 , where a 0 , …, a k − 1 , P 0 , P 1 ≢ 0 are polynomials such that k ⩾ 1 , deg P 0 >1+ max 1⩾j⩾k deg a k −j j . (1) In addition, we also obtain general estimates of the exponent of convergence of the zero sequence of solutions for the above equation under the restriction contrary to (1).

This paper is a companion paper to “An Oscillation Theory for Second-Order Integral Differential Equations.” The underlying theme is that both topic (oscillation theory and numerical oscillation theory) follow as a corollary to an approximation theory of quadratic forms given previously by the author. In Section I we give the mathematical preliminaries; some of which are not included in the earlier paper. This includes the relationship between the fundamental quadratic form and its integral differential equation (the Euler-Lagrange equations). In Section II the approximating quadratic forms are defined on the approximating Hilbert space. In Section III we show that our approximating hypothesis are satisfied and give the fundamental inequality relationships [Eqs. (12) and (13)]. We also show that the mth oscillation point is a continuous function of our approximating parameter. Finally in Section IV we show how that the approximating indices may be easily obtained by computer algorithms.

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting. In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results. To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”