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Nonoscillatory solutions of forced differential equations of the second order
... The literature on this topic is very rich. The reader can consult, for instance, the survey paper by Kartsatos [15] where several oscillation criteria for forced and perturbed equations along with extensive bibliography on the subject can be found, as well as the papers by Grace [6], Grace and Lalli [7,8], Graef and Spikes [9], Graef et al. [10], El-Sayed [11], Kartsatos [13,14,16], Kawano et al. [17], Rankin [27], Rogovchenko [29], Wong [34], Yeh [36], and the references cited there. ...
... have been derived by Kawano et al. [17]. Recent results for Eq. ...
... Remark 4.2 As it has been mentioned in the Introduction, most results on oscillation and nonoscillation of perturbed differential equations require that the associated unforced equation has the same property which then remains preserved (Grace and Lalli [8], Graef and Spikes [9], Kartsatos [13,16], Kawano et al. [17], and Rankin [27]). Furthermore, results of Kartsatos [13] and Wong [32] depend heavily on the choice of the forced term which satisfies condition (H) mentioned above. ...
We give constructive proof of the existence of vanishing at infinity oscillatory solutions for a second-order perturbed nonlinear differential equation. In contrast to most results reported in the literature, we do not require oscillatory character of the associated unperturbed equation, monotonicity of nonlinearity, and we establish global existence of oscillatory solutions without assuming it a priori. Furthermore, as our example demonstrates, existence of bounded oscillatory solutions does not exclude existence of unbounded nonoscillatory solutions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
... In this spirit our main result unifies some earlier Kartsatos' results on the maintenance of oscillations under the effect of a "small" or "periodic-like" forcings and at the same time extends them to more general forcing functions. For other related results concerning Eq. (1) and corresponding functional differential equations and inequalities we refer the reader to the papers of Chen and Yeh [2,3], Foster [4], Grace and Lalli [5,6], Jaros [7,8], Kawano, Kusano and Naito [13], Kusano et al. [14][15][16], McCann [17], Onose [18,19] and True [21]. ...
... Since lim^^ L 0 p 2 (0 = 0, there is a ίa^ί* such that (13) c = T + L°P2(0 > °f or t ^ t 3 . From (12), (13) and (c) we obtain ...
... They presented the so-called "supersolution-subsolution" method (see Section 4 in detail). Using the supersolution-subsolution approach, many authors showed the existence of positive solution for equations of elliptic type (see, for example, [2][3][4]6,9,10]). In particular, Constantin [2,3] applied the supersolution-subsolution existence theory in [8] to the equation ∆u + φ(x, u) + ψ |x| x · ∇u = 0 in an exterior domain of R p with p 3. ...
The existence of decaying, positive solutions of quasilinear elliptic equations of the form Δu+φ(x,u)+x·∇u/|x|2=0 is established in an exterior domain , under suitable smoothness and growth conditions. The main result is proved by means of a supersolution–subsolution method given by Noussair and Swanson. By using phase plane analysis of a system of Liénard type, a supersolution and a subsolution of the above equation are found out. An extension of the main result to more general case is also attempted. Finally, some examples are attached.
... In this regard, see [84] for an extensive complete study of a specific nonlinear equation and [85] for a bibliographical study of unforced equations of the form y ′′ (x) + F (x, y(x), y ′ (x)) = 0. For results which compare the non-oscillatory behavior of forced equations of the form (1) with those of the associated unforced equation, (6) below, and possible equations with delays, we refer the reader to [1], [33] and [73]. One should not forget that even though the literature is filled with sufficient criteria for oscillation/non-oscillation of unforced equations like ...
Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented.
This chapter contains some special results. In Section 9.1, we shall extend the Sturm-Picone theorem, obtain nonoscillation theorems for perturbed second order nonlinear differential equations, and present a nonlinear Picone type identity to enable us to prove some Sturm-Picone type comparison theorems for nonlinear equations. Section 9.2 is devoted to the study of nonoscillatory solutions of forced differential equations of second order. In Section 9.3 we shall present some limit cycle criteria and discuss its related properties for nonlinear second order differential equations. Finally, in Section 9.4 we shall present some properties of solutions of very general second order differential equations.
A necessary and sufficient condition is given for the oscillation of all solutions of the differential equation x(+ Pit, x, x',--, xâ-1') = Q(t) where xâP(t, x„ x2, â- â- â-, x„) 0 for every x O, and Q is a continuous periodic function. This result answers a question recently raised by J. S. W. Wong. It is also shown that a well-known sufficient condition for the existence of at least one nonoscillatory solution of the unperturbed equation guarantees, for a large class of equations, the nonexistence of bounded oscillatory solutions.
We are concerned with the oscillatory behavior of the second order elliptic equation
1
where Δ is the Laplace operator in n -dimensional Euclidean space R ⁿ , E is an exterior domain in R ⁿ , and c : E × R → R and f:E → R are continuous functions.
A function v : E − R is called oscillatory in E if v ( x ) has arbitrarily large zeros, that is, the set { x ∈ E : v ( x ) = 0} is unbounded. For brevity, we say that equation (1) is oscillatory in E if every solution u ∈ C ² ( E ) of (1) is oscillatory in E .
We establish new oscillation theorems for the nonlinear differential equation
where are continuous, and , for . These criteria involve the use of averaging functions.
This paper is devoted to a study of differential equations and inequalities of the form
and
The results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.
Synopsis
Sufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain of E ⁿ . Such results apply in particular to the n -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.
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In this paper, we study the oscillation of n th-order differential equations. Recently, Atkinson and the present authors studied (separately) the comparison properties of differential inequalities. Kartsatos treated the nth-order ordinary case and proposed several open problems.
The purpose of this paper is to answer one of them in the affirmative concerning more general functional differential equations. The result is that if under several conditions, the equation
is oscillatory for n even or a solution x(t) of (1) is oscillatory or for n odd, then this is also the case for the equation
A necessary and sufficient condition is given for the oscillation of all solutions of the differential equation where for every , and Q is a continuous periodic function. This result answers a question recently raised by J. S. W. Wong. It is also shown that a well-known sufficient condition for the existence of at least one nonoscillatory solution of the unperturbed equation guarantees, for a large class of equations, the nonexistence of bounded oscillatory solutions.
Consider the n -th order delay differential equation
In the last few years, the oscillatory behavior of delay differential equations has been the subject of intensive investigations. But much less is known about the equation (A) with small forcing term q ( t ). The only papers devoted to this problem are by Kartsatos, Kusano, and the present author. The purpose of this paper is to prove some new oscillation theorems which contain the previous results.
In this paper functions satisfying the inequality in a
domain are studied. Here is a linear second order elliptic
operator with positive definite characteristic form, , and
is defined in an interval , in
which , and .
Bibliography: 13 titles.
Oscillation and nonoscillation theorems are presented for the nonlinear retarded differential equation
[r(t)x'(t)]' + x\left( (g(t)) \right)F\left( ([x\left( (g(t)) \right)]^2 ,t) \right) = 0 \left( (r(t) > 0) \right)
((A))
where
\mathop \smallint ¥ r-1 (t)dt < ¥\mathop \smallint \limits^\infty r^{-1} (t)dt< \infty
. The results obtained show that there is a remarkable difference between the oscillatory property of(A) in the case
\mathop \smallint ¥ r-1 (t)dt < ¥\mathop \smallint \limits^\infty r^{-1} (t)dt< \infty
and that of(A) in the case
\mathop \smallint ¥ r-1 (t)dt = ¥\mathop \smallint \limits^\infty r^{-1} (t)dt = \infty
.
The quasilinear elliptic equation (*)u+f(x,u,u)=0 is considered in the whole Euclidean space
N
,N3. Under suitable structure hypotheses it is shown that (*) has an entire positive solution which decays to zero at infinity. In particular, conditions are established for the existence of an entire positive solution of (*) which behaves like a constant multiple of |x|
2–N
as |x|.
We are concerned with uniqueness and existence theorems for two point boundary value problems for the nonlinear differential equation Ly = f(x, y), where L is the classical nth order linear differential operator. In proving our results interesting comparison theorems are proven for linear differential equations.
On the equation Au + k(x)e" = 0, Uspekhi Mat
- A Oleinik
A. OLEINIK, On the equation Au + k(x)e" = 0, Uspekhi Mat. Nuuk 33 (1978), 203-204
(English transl.: Russian Math. Surveys 33 (1978), 243-244).
Positive solutions of semilinear Schriidinger equations in exterior domains, fndiuna Univ
- E S A Noussair And C
- Swanson
E. S. NOUSSAIR AND C. A. SWANSON, Positive solutions of semilinear Schriidinger equations in exterior domains, fndiuna Univ. Math. J. 28 (1979), 993-1003.
Positive solutions of semilinear Schrödinger equations in exterior domains
- Noussair
On the equation Δu + k(x)eu = 0
- Oleinik