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Nonlinear oscillation of higher-order functional differential equations with deviating arguments
... It is always assumed that functions p, ri, ro, r and g: [lo,oo) -* (0,oo) are continuous and g(t) -+ oo as t -> oo. We suppose that for / J> to (4) g(t) ^ t, (5) Ri(t) = / , x -->• oo as t -* oo for i = 1,2; Jto r *( 8 ) f l ds (6) R(t) = -^ oo as t -oo. Jt 0 r (s) ...
... The proof is found in Kitamura and Kusano [6]. ...
... This lemma is an analog of Theorem 1 of Kitamura and Kusano [6]. This lemma is a generalization of Taylor's formula with remainder encountered in calculus. ...
... n(t): [t 0 , oo) -> R, 1 ^ i ^ n -2, are continuous and satisfy(8) ~i(t) -> oo as t -» oo, Ti(t) ^ t, for i = 1,2,... ,n -2.Theorem 2. Suppose that(8) holds and the integrals in(6) converge. Assume that the second order delay equations ...
... Several $¥mathrm{p}¥mathrm{a}¥mathrm{p}¥mathrm{e}¥mathrm{r}¥mathrm{s}^{ [1][2][3][4][5]}$ have discussed the existence of the nonoscillatory solutions of $¥mathrm{n}¥mathrm{t}¥mathrm{h}$ order $¥mathrm{D}¥mathrm{D}¥mathrm{E}$ (delay differential equation) and obtained some interesting results. Recently, Grove, Kulenovic and $¥mathrm{L}¥mathrm{a}¥mathrm{d}¥mathrm{a}¥mathrm{s}^{ [6]}$ has given some sufficient conditions for the first order NDDE to have nonoscillatory solutions. ...
... (4) $0<K/2¥leq x(t)¥leq K$, $0<K/2¥leq y(t)¥leq K$ .By the condition that $|f(t, x)|¥leq|f(t, y)|$ for $|x|¥leq|y|$ , $xy$ $>0$ , we have(5) $f(t, x(t-¥sigma(t)))¥leq f(t, K)$ ,for $t¥geq T$ ;(6) $f(t, y(t-¥sigma(t)))¥leq f(t, K)$ , for $t¥geq T$ .For $t¥geq T$ , $0¥leq c<1$ , by (3), (4), (6), we have$(U¥mathrm{x})(t)+(Sy)(t)=¥frac{3(1+c)K}{4}-cx(t-¥tau)$ $+¥int_{T}^{t}¥int_{s_{n-1}}^{+¥infty}¥ldots¥cdots¥int_{S1}^{+¥infty}f(s, y(s-¥sigma(s)))dsds_{1}¥cdots¥cdots ds_{n-1}$ $¥geq¥frac{3(1+c)K}{4}-cx(t-¥tau)¥geq¥frac{3(1+c)K}{4}-c¥cdot K=(¥frac{3}{4}-¥frac{c}{4})K¥geq K/2$ , $(Ux)(t)+(Sy)(t)=¥frac{3(1+c)K}{4}-c¥mathrm{x}(t-¥tau)$ $+¥mathrm{H}_{Ts_{n}-1}^{t+¥infty}¥ldots¥cdots¥int_{S1}^{+¥infty}f(s, y(s-¥sigma(s)))dsds_{1}¥cdots¥cdots ds_{n-1}$ $¥leq¥frac{3(1+c)K}{4}-c¥cdot¥frac{K}{2}+¥int_{T}^{t}¥int_{S_{n-1}}^{+¥infty}¥ldots¥cdots¥int_{s_{1}}^{+¥infty}f(s, K)dsds_{1}¥cdots¥cdots ds_{n-1}$ $¥leq¥frac{3(1+c)K}{4}-c¥cdot¥frac{K}{2}+¥frac{(1-c)K}{4}=K$ .Similarly, for $t¥geq T$ , ? $1<c<0$, we have $(Ux)(t)+(Sy)(t)¥geq¥frac{3(1+c)K}{4}-c¥cdot¥frac{K}{2}=(¥frac{3}{4}+¥frac{c}{4})K¥geq¥frac{K}{2}$ , $(Ux)(t)+(Sy)(t)¥leq¥frac{3(1+c)K}{4}-c¥cdot K+¥frac{(1+c)K}{4}=K$ .For $T^{*}¥leq t<T$ , $|c|<1$ , it is easy to see that ...
... as a special case; see for example, the papers [4], [5] . As the time goes by considerable attention of the researchers seems to have been directed toward the oscillation of differential equations whose principal part involve the so-called " higher order nonlinear Sturm–Liouville differential operators " , a typical example of which is ...
... Since the oscillation of all solutions of (A) is equivalent to the absence of nonoscillatory solutions of (A), to prove the above theorems it suffices to show that the above conditions eliminate all possible positive solutions of (A) and conversely. In the process of proving the theorems use is made of the oscillation criteria of Kitamura and Kusano [4], stated below, for differential equations of form (1.2) which is rewritten here as ...
We are concerned with the oscillatory and nonoscillatory behavior of solutions of differential equations involving an even order nonlinear Sturm–Liouville operator of the form
where α and β are distinct positive constants. We first give the criteria for the existence of nonoscillatory solutions with specific asymptotic behavior on infinite intervals, and then derive necessary and sufficient conditions for all solutions of (∗) to be oscillatory by eliminating all nonoscillatory solutions of (∗).
... This theorem is a special case of Theorem 1 of Kitamura and Kusano (see [5]). (4) and (5) ...
... where L n is the general disconjugate operator with (3) Pi > 0 and p { e C[0, oo), 0 < ί < n, and F is some functional of u. As examples, we cite [1], [7], [8], [9], [10], [11], [13], and [17]. ...
... Many of the results obtained are given in [1] and [2]. Of the more recent results, we mention in particular the works [3]- [8]-These papers investigate equations in which the higher-order derivative participates only for a single value of the independent variable. There have also been some contributions dealing with similar problems concerning functional-differential inequalities, for example, [2], [9]- [ll]. ...
The present paper studies some asymptotic (including oscillatory) properties of the solutions of operator-differential inequalities of the form [(Lx)(t)](n)+ (Fx)(t)≦0 ([(Lx)(r)](n)+ (Fx)(t)≧ 0), where [formula omitted] (the latter symbol denotes the space of locally summable functions). As an application of the results obtained, theorems are proved for the asymptotic behaviour of the solutions of certain classes of functional-differential and integro-differential neutral-type equations.
... The study of oscillation of higher order nonlinear functional differential equations with deviating arguments was attempted for the first time by Onose [28,29]. A typical generalization of Onose's oscillation theorem can be found in [11]. Recently, wide attention of the researchers has been attracted to the investigation of oscillation (or nonoscillation) of differential equations whose principal differential operators involve nonlinear Sturm-Liouville type differential operators [12, 20, 22, 30 -34]. ...
... The sufficiency proof is similar to that of Theorem 2.2. We can choose T>a large enough so that $rh(t, cR(t)) dt<c, and replace (2.5) and (2.6), respectively, by ? and furthermore that L has the factorized form [4,11] (3 3) The following theorem is proved by virtually the same procedure, applying Theorem 2.3 instead of Theorem 2.2. As an example of (1.1), consider the semilinear equation Case II: k>0. ...
Necessary and sufficient conditions are found for the existence of two positive solutions of the semilinear elliptic equation Δ u + q (|x|) u = f ( x , u ) in an exterior domain Ω⊂ℝ ⁿ , n ≥ 1, where q , f are real-valued and locally Hölder continuous, and f ( x , u ) is nonincreasing in u for each fixed x ∈Ω. An example is the singular stationary Klein-Gordon equation Δ u — k ² u = p ( x ) u -λ where k and λ are positive constants. In this case NASC are given for the existence of two positive solutions u i ( x ) in some exterior subdomain of Ω such that both |x| m exp[(-l) i-1 k | x |] u i ( x ) are bounded and bounded away from zero in this subdomain, m = ( n —1)/2, i = 1, 2.
... For typical results regarding (5) we refer to [3], [4], [8], [12], [16], [18], [23], [28] and [36]. There is, however, much current interest in the study of the oscillation properties of higher-order differential equations of the forms (LE) and (NE) involving general disconjugate operators L n defined by (1); see, for example, the papers [1] [2] [5] [6] [7] [9] [10] [11] [13] [14] [15] [17] [19] [20] [21] [22] [23] [24] [25] [26] [27] [29] [30] [31] [32] [33] [34] [35] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]. In the present paper we proceed further in this direction to establish new oscillation results for equations (LE) and (NE) with general deviating arguments g h (f) which extend and unify many of the previous results obtained in the above-mentioned papers. ...
... . For the details the reader is referred to Kitamura and Kusano [2]. ...
... A recently published book by Shevelov [lo] gives a rich set of references on the general area of oscillation theory. Related work on disconjugate L, can also be found in Kitumura and Kusano [2], Granata [l], Kusano and Naito [3], and Staikos and Philos [9]. In order to make our analysis less cumbersome, we introduce notations for repeated integrals as used in Singh and Kusano [8]. ...
For the equation (A), where Ln is a disconjugate nonlinear operator, sufficient conditions have been found to ensure that all proper solutions of (A) are slowly oscillating. The operator Ln has the form .
... With the operator L, as in (16) Eq. (4) can then be rewritten as &I Y = -Ye-, Y)* (19) The theorem below is a special case of a recent result of Kitamura and Kusano [5]; see also Granata [4]. THEOREM 3.1. ...
... One problem consists of showing that known results regarding oscillation and asymptotic behavior remain valid when delays are introduced. For example, Kitamura and Kusano [7] have considered L,u+ /It, u(g(t)))=O ( 1.1) (where L, is a general disconjugate nth order differential operator) and shown that a variety of qualitative properties for L,u +f(r, u(t)) = 0 remain valid for ( 1.1) with bounded delays. In a different direction Ladas and Stavroulakis [9] have shown that the introduction of a sufficiently large delay will transform a nonoscillatory differential equation such as U' + p( t ) u = 0 into a functional equation u'+p(t)u(t-r)=O (1.2) all of whose solutions are oscillatory. ...
... then the bounded solution can be extended to x/> 0, y/> 0. This result, as they note, is a generalization of the existence of an asymptotically constant solution of d"u +_f(x,y)=O, 0~<x<~, dx . which goes back to Atkinson [11] and Nehari [12] for n = 2, with more general ODE results being due to several other authors131415. Generalizations to elliptic equations have been given by Swanson [16, 17] and Kreith and Swanson [7]. ...
Oscillation criteria and asymptotic behavior of solutions of characteristic initial-value problems have been extended to the Goursat problem.
Soit p i :[a,∞)→(0,∞), 0≤i≤n, des fonctions continues, on definit les operateurs differentiels L i , 0≤i≤n, par L 0 y(t)=y(t)/p 0 (t), L i y(t)=1/p i (t)•dL i−1 y(t)/dt, 1≤i≤n et on considere les equations L n y(t)+f(t,y(g(t)))=0, L n y(t)−f(t,y(g(t)))=0 ou f:[a,∞)×(0,∞)→(0,∞) et g:[a,∞)→(0,∞) sont des fonctions continues et lim t→∞ g(t)=∞. On etudie la structure des solutions de ces equations definies et positives au voisinage de l'infini
Synopsis
The equation to be considered is
where p i ( t ), 0≦ i ≦ n , and q ( t ) are continuous and positive on some half-line [ a , ∞). It is known that (*) always has “strictly monotone” nonoscillatory solutions defined on [ a , ∞), so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of (*). In this paper sufficient conditions are given for this situation to hold for (*). The structure of the solution space of (*) is also studied.
Some new criteria for the bounded oscillation of a fourth or-der functional differential equation are established. Comparison results with first/second order equations as well as necessary and sufficient conditions are developed.
The present paper studies some asymptotic (including oscillatory) properties of the solutions of operator-differential inequalities of the form
[(Lx](t)](n) + (Fx)(t) \leqslant 0,[(Lx](t)]^{(n)} + (Fx)(t) \leqslant 0,
where
L:[(C)\tilde]n - 1 [t0 ,¥] ® [(C)\tilde]n - 1 [t0 ,¥],F:[(C)\tilde]n - 1 [t0 ,¥] ® [(C)\tilde] - 1 [t1 ,¥),t1 \geqslant t0 L:\tilde C^{n - 1} [t_0 ,\infty ] \to \tilde C^{n - 1} [t_0 ,\infty ],F:\tilde C^{n - 1} [t_0 ,\infty ] \to \tilde C^{ - 1} [t_1 ,\infty ),t_1 \geqslant t_0
(the latter is the spaceof locally summable functions). As an application of the obtained results, theorems are proved]or the asymptotic behaviour of the solutions of certain classes of functional-differential andintegro-differential neutral type equations.
Lovelady has recently proved the following oscillation theorem. © 1981, University of California, Berkeley. All Rights Reserved.
Some new oscillation theorems forhigher-order nonlinear functional differential equations of the formare established.
It is shown that for a class of n-th order differential equations the existence of one solution converging at infinity is equivalent to the existence of other solutions with appropriate asymptotic behaviours
Oscillation conditions are given for the equation Lu + f(t, u ) = 0, where Id,1,dldu
It is shown that the disconjugate equation (1) Lx = (l/βn)(d/dt) • (1/βn-1) • • • (d/dt)(x/β, )(d/dt)(x/βo)=0, a<t<b, where βi > 0, and (2)βi € C(a, b), can be written in essentially unique canonical forms so that (Formula Present). From this it follows easily that (1) has solutions jci,;…, which arc positive in (u, b) near b(a) and satisfy lim, (Formula Present). Necessary and sufficient conditions are given for (1) to have solutions such that y (Formula Present). Usingdifferent methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition (2') (Formula Present) (a, b).
Oscillation conditions are given for the equation L u + f ( t , u ) = 0 {L_u} + f(t,u) = 0 , where with β 0 , … , β n {\beta _0}, \ldots ,{\beta _n} positive and continuous on ( 0 , ∞ ) , ∫ ∞ β i d t = ∞ ( 1 ⩽ i ⩽ n − 1 ) (0,\infty ),\int {^\infty {\beta _i}dt = \infty (1 \leqslant i \leqslant n - 1)} , and f f subject to conditions which include u f ( t , u ) ⩾ 0 uf(t,u) \geqslant 0 . The results obtained include previously known oscillation conditions for the equation u ( n ) + f ( t , u ) = 0 {u^{(n)}} + f(t,u) = 0 for both linear and nonlinear cases.
Summary This paper is devoted to the study of oscillation of all solutions of the fourth order functional differential equation with general deviating arguments
where
2necessary and sufficient conditions are established for all solutions of equation (1)to be oscillatory when L4is in canonical form, that is,
, 1?i?3.In Section3,on the basis of Trench's theory, a characterization of oscillation of equation (1)is obtained when L4is not necessarily in canonical form.
Utilizing the quadratic functional criteria and positive functionals, the author develops a comparison theorem for oscillation of partial differential equations. This result parallels a comparison of the oscillation of scalar ordinary differential equations to oscillation of vector-matrix differential equations established by LEWIS and WRIGHT.
We want to look at the matrix partial differential equation.
Synopsis
This paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of n th order ( n > 1) non-linear differential equations with deviating arguments involving the so called n th order r -derivative of the unknown function x defined by
where r 1 , ( i = 0,1,…, n – 1) are positive continuous functions on [ t 0 , ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r ⁰ = r 1 = … = r n–1 = 1.
Oscillation equations in exterior domains
- Y Kitamura
Y. KITAMURA AND T. KUSANO, Oscillation equations in exterior domains, Arch. Rational Mech. Anal., in press.
1 I. W. F. TRENCH, Oscillation properties of perturbed disconjugate equations and asymptotic behavior of bounded solutions of higher order of positive solutions of differential and asymptotic behavior of all solutions of differential Sect and principal systems for general disconjugate
- W F Trench
- Canonical
- Equations
- Trans
- Amer
- Math
- Sot
W. F. TRENCH, Canonical forms equations, Trans. Amer. Math. Sot. 189 (1974), 319-327. 1 I. W. F. TRENCH, Oscillation properties of perturbed disconjugate equations, Proc. Amer. Mafh. Sot. 52 (1975), 147-155. behavior Equations 19 (1975), 165-175. and asymptotic behavior of bounded solutions of higher order of positive solutions of differential and asymptotic behavior of all solutions of differential Sect. A 81 (1978), and principal systems for general disconjugate