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First multi-colour complete light curves and low-resolution spectra of short-period eclipsing binary TT Cet are presented. The stellar atmospheric parameters of the primary star were derived through spectra fitting as:$T_{eff}=7\,091\pm124\,{\text{K}}$,$\log g = 4.15\pm0.33\,{\text{cm}}/\text{s}^2$, and$[Fe/H]=-0.23\pm0.04\,\text{dex}$. The light curves were analysed using the Wilson–Devinney code. The photometric solution suggests that this target should be a near-contact binary with the primary component filling its critical Roche lobe (i.e. SD1-type NCB). The luminosity enhancement around the primary light maximum (phase 0.10–0.40) on the light curve was detected like other SD1-type NCBs, which could be caused by a hot spot near the facing surface of the secondary component due to mass transfer. Long-term decrease of the orbital period at a rate of$dP/dt=-5.01\,({\pm}0.06)\times 10^{-8}\,{{\text{d}} \cdot{yr}}^{-1}$was detected by the O–C analysis, which supports the mass transfer from the primary to the secondary and is consistent with its primary filling configuration. No third body was found through the light curve and O–C analysis. TT Cet may locate in the broken contact stage predicted by the thermal relaxation oscillation theory (TRO) and will evolve to the contact stage eventually. It is another good observational example supporting the TRO theory. We have collected all known SD1-type NCBs with absolute parameters from the literatures. The relations of these parameters are summarised for these rare systems.

An eigenvalue problem for k Sturm–Liouville equations coupled by k parameters λ1,…,λk is considered. In contrast to the standard case, for each r, the second-order derivative in the rth equation is multiplied by λr. This problem presents various interesting features. For example, the existence of eigenvalues with oscillation counts beyond a certain (computable) value is obtained without any of the restrictive definiteness conditions known from the standard case. Uniqueness is also analysed, and the results are given greater precision via eigencurve methods in the case of two equations coupled by two parameters.

New necessary and sufficient conditions are given for the quantization of a class of periodic second-order non-homogeneous ordinary differential equations in the complex plane. The problem is studied from the viewpoint of complex oscillation theory first developed in works by Bank and Laine and Gundersen and Steinbart. We show that, when a solution is complex non-oscillatory (finite exponent of convergence of zeros), then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has finitely many zeros in every branch, and this is a type of quantization for the non-homogeneous differential equation. The degenerate solutions are polynomial/rational-type functions, which are of independent interest. In particular, this shows that complex non-oscillatory solutions of this class of differential equations are equivalent to the subnormal solutions considered in a recent paper by Chiang and Yu. In addition to the asymptotics of special functions, the other main idea that we apply in our proof is a classical result by Wright that gives precise asymptotic locations of large zeros of a functional equation.

In this paper we present sufficient conditions for all trajectories of the system to cross the vertical isocline h(y) = F(x), which is very important in the global asymptotic stability of the origin, oscillation theory and existence of periodic solutions. Also we give sufficient conditions for all trajectories which start at a point on the curve h(y) = F(x), to cross the y-axis which is closely connected with the existence of homoclinic orbits, stability of the zero solution, oscillation theory and the centre problem. The obtained results extend and improve some of the authors' previous results and some other theorems in the literature.

In this note we study the zeros of solutions of differential equations of the form u′′+pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.

We prove the existence of a positive solution to an equation of the form (1/Φ(t)) (Φ(t)u′(t))′ = f(u(t)) with Dirichlet conditions where the friction term Φ′/Φ is increasing. Our method combines variational and topological arguments and provides an L∞ estimate of the solution.

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

We discuss the exact number of almost periodic solutions of certain ordinary differential equations of the second order. The class of equations under consideration is inspired by a well-known result in the area of elliptic boundary value problems.

Oscillation and related results are given for the problem under separated end conditions, assuming 1/p, q and r ∈ L1(I). Attention focuses on the two cases (i) p > 0 with r indefinite, and (ii) p indefinite with r > 0.

We consider an ordinary differential equation arising in the study of the Ricci flow on R2. The existence and uniqueness of solutions of this equation are derived. We then study the asymptotic behaviour of these solutions at ±∞.

W investigate the e growth and th frequence oy zerof os f th solutione osf th differentiae lequation / + /^(z)/^- 1 ' + • • • + P 0 (z)f = H(Z), (1.1)where Po(z),P\(z),...,P n -i(z) are polynomials with Po(z) ^ 0, and H(z) ^ 0 is anentire function of finite order. It is well known that every solution / of equation (1.1) isan entire function.For an entire function g, let p(g) denote the order of growth of g, and when g ^ 0, let\{g) denote the exponent of convergence of the sequence of zeros of g.For solutions / of equation (1.1), we investigate the possible values of />(/), and therelationships between the values of />(/), A(/), p(H) and X(H). We give examples toillustrate the sharpness of our results.2. Statement of resultsTo discuss the possible orders of solutions of equation (1.1), we need to know the possibleorders of the solutions of the corresponding homogeneous equation/ + P

We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations Y″ = 0 and Y‴ = 0. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.

We consider the half linear Sturm-Liouville problemon the interval [0,1] subject to separated boundary conditions (which may be eigenparameter dependent at x = 1) and use Prüfer techniques to produce an oscillation theory for this problem. Both right definite (r > 0) and left definite (r of both signs) cases are discussed.

In previous papers we have studied oscillation properties of Sturm–Liouville problems (−Py′)′ + qy = λry, with λ-dependent boundary conditions, under various ‘definiteness’ conditions. Here we present a new, unified, approach which also covers cases previously untreated, e.g. of semidefinite weight, and also the fully indefinite problem.

In this paper, we investigate the complex oscillation theory ofwhere A, F≢0 are entire functions, and obtain general estimates of the exponent of convergence of the zero-sequence and of the order of growth of solutions for the above equation.

Non-linear stochastic systems driven by white noise are analysed from the viewpoint of non-linear oscillation theory. Under various familiar hypotheses concerning dissipative and restorative dynamical forces, the existence and uniqueness, asymptotic growth, and oscillatory behavior of the solutions are demonstrated.

We are interested in two parameter eigenvalue problems of the formsubject to Dirichlet boundary conditionsThe weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurvesin the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

Precise decay estimates as t → ∞ are derived for a class of nonlinear second order ordinary differential equations of the formwhere h, g and f are functions likeWith α > −1, β > −1 and γ > −1.

SynopsisOne considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

SynopsisAbsolutely square integrable solutions are determined for the equation= λywhere the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions asxapproaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

SynopsisWe give some results on a boundary-value problem for an ordinary differential equation whose coefficients are in the B*-algebra C(K), where K is a compact metric space. We deduce the existence of a countable number of eigenvalues and corresponding eigenfunctions, the latter being complete in a certain sense. There follows an expansion result and some remarks on a self-adjoint realisation of the associated differential operator.

If a second order ordinary differential equation has a simple or a double pole at a point zfl, then the standard Liouville-Green approximation could sometimes be valid near that point. In this paper we present an asymptotic series solution that is always valid near a double pole. A solution for that of a simple pole is also indicated. Asymptotic validity is proved.

The purpose of this paper is to establish comparison criteria, by which the oscillatory and asymptotic behavior of linear retarded differential equations of arbitrary order is inherited from the oscillation of an associated second order linear ordinary differential equation. These criteria are new even in the case of ordinary differential equations.

SynopsisIntegral inequalities are used to obtain comparison theorems for a class of second-order differential equations which includes the Emden-Fowler equation, certain Liénard equations, and linear equations of the form d2y/dx2+f(x)y = 0. For these linear equations the results below imply Sturm's classical comparison theorem.

SynopsisThis paper studies the stability regions associated with the multi-parameter systemwhere the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

SynopsisSufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain ofEn. Such results apply in particular to then-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.

A new result is obtained for the oscillation of second order linear ordinary differential equations with alternating coefficients. This oscillation result extends a recent oscillation criterion due to Kamenev [.Mat. Zametki 23 (1978), 249–251].

It has long been realised that there is a precise analogy between the properties of vibrating mechanical systems and those of alternating current electric circuits. It was from this analogy that the concept of “ mechanical impedance ” (more appropriately termed “ dynamic stiffness ”) was derived, thus enabling the theory of certain types of mechanical vibration, notably torsional motion, to be put on a more rational basis.By means of analogous elements it is possible to represent a mechanical vibrating system, provided it includes no bending elements, by an equivalent electric circuit. The author has been unable to discover any full description of the process, and the object of the present paper is to set out the principles underlying the procedure and to illustrate them briefly by examples.

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

In his recent book on ordinary differential equations Hille (3) devotes a chapter to complex oscillation theory. Drawing upon his own work in this area and the work of Nehari, Schwarz, Taam, and others, he gives a variety of oscil-lation and nonoscillation theorems for solutions of the differential equationwhere z is a complex variable and p is regular in some appropriate domain. There are a number of results for (1.1) with an arbitrary coefficient o and some discussions for special cases of classical interest, such as the Bessel and Mathieu equations. There is a bibliography at the end of the chapter. For other recent work in this area attention is directed to papers by Herold (1, 2) Kim (4, 5) and Lavie (6) where other references are given.