Order Scalar Differential Equations Research Articles

Impulsive system of ODEs with general linear boundary conditions

The paper provides an operator representation for a problem which consists of a system of ordinary differential equations of the first order with impulses at fixed times and with general linear boundary conditions z (t) = A(t)z(t) + f(t, z(t)) for a.e. t ∈ [a, b] ⊂ R, z(ti+)− z(ti) = Ji(z(ti)), i = 1, . . . , p, l(z) = c0, c0 ∈ R n . Here p, n ∈ N, a < t1 < . . . < tp < b, A ∈ L ([a, b];R), f ∈ Car([a, b] × R;R), Ji ∈ C(R ;R), i = 1, . . . , p, and l is a linear bounded operator on the space of left-continuous regulated functions on interval [a, b]. The operator l is expressed by means of the KurzweilStieltjes integral and covers all linear boundary conditions for solutions of the above system subject to impulse conditions. The representation, which is based on the Green matrix to a corresponding linear homogeneous problem, leads to an existence principle for the original problem. A special case of the n-th order scalar differential equation is discussed. This approach can be also used for analogical problems with state-dependent impulses. Mathematics Subject Classification 2010: 34B37, 34B15, 34B27

A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of N coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.

An existence result for first order periodic problems with discontinuous and singular nonlinearities

The lower and upper solution method is applied to prove existence of periodic solutions for first order scalar differential equations with discontinuous and singular right-hand sides.

Periodic and boundary value problems for second order differential equations

In this paper we study second order scalar differential equations with Sturm-Liouville and periodic boundary conditions. The vector fieldf(t,x,y) is Caratheodory and in some instances the continuity condition onx ory is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.

Qualitative behavior of the solutions of periodic first order scalar differential equations with weakly concave nonlinearity

Qualitative behavior of the solutions of periodic first order scalar differential equations with weakly concave nonlinearity

Periodic solutions of some vector retarded functional differential equations

has no nontrivial T-periodic solution (see the books of Krasnosel’skii [6], Reissig, Sansone, and Conti [14], and Roseau [15]). This result has been extended to the case of the vector retarded functional differential equation by Fennel1 [2]. When the requirement upon the linear part is not satisfied, supplementary conditions upon f are needed to ensure the existence of T-periodic solutions, as follows immediately from the Fredholm alternative [4] for the special case off independent of x. Such auxiliary conditions appear in papers of Lazer [7] (for a second order scalar differential equation), Ezeilo [l], Sedsiwy [16], Villari [19], and the author [8] (f or a third order scalar differential equation), Sedsiwy [17] and Reissig [ll, 121 (for the n-th order scalar case) and the same authors [18, 131 f or the corresponding vector case. Also, Fennel1 [2] has extended Lazer’s result and method to the retarded case. Those results are proved by various methods (Brouwer, Schauder, or Leray-Schauder fixed point theorems) which make the arguments rather tedious. On the other hand, all the equations considered in those papers are

On almost periodic solutions for undamped systems with almost periodic forcing

Using the method of averaging, we give sufficient conditions for the existence of an almost periodic solution of an undamped oscillatory system with almost periodic forcing, and show that these can be applied to a Duffing equation with almost periodic forcing provided the nonlinear term is sufficiently small, and the natural frequency of the linear part of the system is included in the set of Fourier exponents of the forcing function. It is well known that the real linear second order scalar differential equation + x = f (t), with f almost periodic (a.p. for short), cannot have almost periodic solutions, or even a solution bounded on the real line, whenever 1 is a Fourier exponent off It is the purpose of this note to obtain a result which will show that for such so-called resonance cases, there exist sufficiently small perturbations of this equation, involving only functions of x which have a.p. solutions. More precisely, iff has Fourier exponent 1 and one of the corresponding coefficients of the sine or cosine term in the Fourier series is zero, then there exist positive numbers v and E0 = E0(v) such that for 0 < E < E0 the equation (1) X +X-EVX+ E3X3f=(t) will have an a.p. solution. The magnitude of v is proportional to the absolute value of the Fourier coefficient corresponding to the Fourier exponent 1 in the series forf This a.p. solution is unstable in the sense of exhibiting the so-called saddle-point property, and the suprema of its absolute value and that of its derivative over all real t become unbounded as E --0. This property of (1) seems interesting for cases wheref has a sequence {2k}, k = 1, 2, * , of Fourier exponents, 2' < 1, and 2'-? 1 as k-xo. Received by the editors February 22, 1971. AMS 1970 subject classifications. Primary 34C25; Secondary 34C30.

Existence and uniqueness of solutions of boundary value problems for two dimensional systems of nonlinear differential equations

The paper considers the nonlinear system x ′ = f ( t , x , y ) , y ′ = g ( t , x , y ) x’ = f(t,x,y),y’ = g(t,x,y) with linear and nonlinear two point boundary conditions. With a Lipschitz condition, an interval of uniqueness for linear boundary conditions is determined using a comparison theorem. A corresponding existence theorem is established. Under the assumption of uniqueness, a general existence theorem is established for quite general nonlinearities in the functions and in the boundary conditions. Examples are provided. The results extend previous work on second order scalar differential equations.

Structure of the solution set of some first order differential equations of comparison type

The comparison principle is a powerful tool that has a wide variety of applications in ordinary differential equations. The results of this article describe the geometric structure of the solution space of some first order scalar differential equations that may arise in the comparison method. A quite general class of differential equations is found to have a similar solution set configuration as the differential equation of separable variable type. One of the main results establishes, under certain conditions, that there is a unique unbounded solution of the first order differential equation which exists on an interval of the form [ t 0 , ∞ {t_0},\infty ). Furthermore, this unbounded solution separates the solutions that are bounded on [ t 0 , ∞ {t_0},\infty ) from those that are not continuable to all t &gt; t 0 t &gt; {t_0} .

Solutions of systems of periodic differential equations

By a system of periodic differential equations referred to in the title we mean a system of the form f(t, U, u’) = 0, where u = u(t), u’ = du/dt, and f(t, u, v) = f(t + r, u, w). Our particular approach will be to break up the vector u into two vector components so that our system becomes f(t, X, y, x’, y’) = 0, where x is an m-dimensional vector, y is n-dimensional, and f is m + n dimensional. Either m or n may be zero, in which case the system is independent of x or y respectively. We shall impose conditions only on x which, however, will yield properties valid for y as well. Throughout the paper, we shall assume that there exists a unique solution through any prescribed point in a region in (t, X, y) space, and that the variables always remain in this region. These requirements will not be explicitly stated. In the first part of the paper we shall discuss systems of arbitrary order. In the second part, these results will be specialized to second order scalar differential equations.