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Je uvažovana obecna třida linearnich diferencnich rovnic. Pomoci modifikace diskretni Riccatiho metody, je ziskano přimocare neoscilacni kriterium pro uvažovane rovnice. V rozporu s poctem znamých kriterii nemuseji být uvažovany žadne pomocne posloupnosti. Použity jsou přimo koeficienty uvažovaných rovnic. K ilustraci novosti a využiti prezentovaneho kriteria jsou poskytnuty důsledky a přiklady.

We prove nonoscillation theorems for the second order Emden-Fowler equation (E): y + a ( x ) | y | γ − 1 y = 0 y+a(x)|y|^{\gamma -1}y=0 , γ > 0 \gamma >0 , where a ( x ) ∈ C ( 0 , ∞ ) a(x)\in C(0,\infty ) and γ ≠ 1 \gamma \not =1 . It is shown that when x ( γ + 3 ) / 2 + δ a ( x ) x^{(\gamma +3)/2+\delta }a(x) is nondecreasing for any δ > 0 \delta >0 and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when 0 > γ > 1 0>\gamma >1 and 0 > δ > ( 1 − γ ) / 2 0>\delta >(1-\gamma )/2 .

The differential equation \[ d x ( t ) = a ( x ( t ) , t ) d Z ( t ) + b ( x ( t ) , t ) d t dx(t) \, = \, a(x(t),t) \,dZ(t) \:+\: b(x(t),t) \,dt \] for fractal-type functions Z ( t ) Z(t) is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation x ( t ) = h ( y ( t ) + Z ( t ) , t ) x(t)\, =\, h(y(t)+Z(t),t) for certain C 1 C^1 -functions h h and y y . The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.

We consider the difference system a n + 1 − a n + ∑ i = 1 m Q i a n − k i = 0 {a_{n + 1}} - {a_n} + \sum \nolimits _{i = 1}^m {{Q_i}{a_{n - {k_i}}} = 0} where k i ∈ Z , Q i ∈ R r × r , i = 1 , … , m {k_i} \in \mathbb {Z},{Q_i} \in {\mathbb {R}^{r \times r}},i = 1, \ldots ,m , and systems of more general forms. We establish explicit conditions for the systems to be oscillatory. Those conditions are necessary and sufficient for special cases.

We prove an existence theorem for positive solutions of nonlinear second order ordinary differential equations by using the perturbation method and the Schauder fixed point theorem. We consider the following problem on the nonlinear second order ordinary differential equation in R+ = (0, oc): {(la) U" +f(x,u)=0, (1) j (lb) u>0 inR+, (Ic) limb~o+ u(x) = 0, where f(x, y) is a Borel measurable function in R+ x R+ and dominated by a positive function F: (2) If(x, y)I 0 such that for each c E (0, b], (1) has a solution u satisfying u(x) (5) lim ux = c. Too0 x Received by the editors September 14, 1992. 1991 Mathematics Subject Classification. Primary 34A34, 47H 10; Secondary 60J45. ( 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

The paper is devoted to an oscillation theorem for the second-order forced linear differential equation of the form ( p ( t ) x ′ ) ′ + q ( t ) x = g ( t ) (p(t)x’)’ + q(t)x = g(t) . The sign of the coefficient q q is not definite, and the function g g is not necessarily the second derivative of an oscillatory function. The question raised by J. Wong in Second order nonlinear forced oscillations (SIAM J. Math. Anal. 19 (1988), 667-675) is answered. A region of oscillation of Mathieu’s equation is specified.

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system. Bounds for oscillations. In [1] it was shown that a linear ordinary differential equation of order n, with real analytic coefficients bounded in a neighborhood of the interval [−1, 1], admits a uniform upper bound for the number of isolated zeros of a solution defined on this interval. The analyticity condition can be relaxed; only the boundedness of the coefficients matters. Probably, the simplest result in this spirit is the following theorem for the linear ordinary differential equation y(t) + a1(t)y(t) + · · ·+ an(t)y(t) = 0 (1) with continuous coefficients on [α, β] ⊂ R. Theorem 1 ([3, 4]). If the coefficients of the differential equation (1) are uniformly bounded by the constant C ≥ 1 (that is, max{|ai(t)| : i = 1, . . . , n} ≤ C), then a solution defined on [α, β] cannot have more than n−1 + n ln 2C|β−α| isolated zeros. An analog of this result for a system of ordinary differential equations, viewed as a vector field in space, would concern the number of isolated intersections between integral trajectories of the vector field and hyperplanes (or, more generally, hypersurfaces). For polynomial systems of degree d on R of the form ẋi = vi(t, x), i = 1, . . . , n, vi(t, x) = ∑ k+|α|≤d vikαt x, (2) and algebraic hypersurfaces given by {P = 0} where P = P (t, x) is a polynomial of degree d, the following theorem, proved in [3] (see also [2]), gives a bound for the number of isolated intersections in case the magnitude of the domain of the solution and the amplitude of the solution are controlled by the height of the polynomial system, that is, the number max{|vikα| : k + |α| ≤ d, i = 1, . . . , n}. Received by the editors July 31, 2000 and, in revised form, September 11, 2000. 1991 Mathematics Subject Classification. Primary 34C10, 34M10; Secondary 34C07.

An oscillation criterion is given for the second order sublinear differential equation x + a ( t ) | x | γ sgn ⁡ x = 0 , 0 > γ > 1 x + a(t){\left | x \right |^\gamma }\operatorname {sgn} x = 0,0 > \gamma > 1 , where the coefficient a ( t ) a(t) is not assumed to be nonnegative for all large values of t t . The result extends a condition recently discovered by Butler, Erbe, and Mingarelli for the linear equation.

We consider the second order matrix differential systems (1) (P(t)Y')' + Q(t)Y = 0 and (2) Y″ + Q(t)Y = 0 where Y, P, and Q are n × n real continuous matrix functions with P(t), Q(t) symmetric and P(t) positive definite for t ∈ [t 0 , ∞) (P(t) > 0, t ≥ t 0 ). We establish sufficient conditions in order that all prepared solutions Y(t) of (1) and (2) are oscillatory. The results obtained can be regarded as generalizing well-known results of Kamenev in the scalar case

A condition is given which guarantees that each solution of y ( n ) = f ( x , y , y ′ , … , y ( n − 1 ) ) {y^{(n)}} = f(x,y,y’, \ldots ,{y^{(n - 1)}}) either extends or becomes unbounded on its maximal interval of existence.

Some new integral criteria for the oscillation of the nonlinear second order differential equation with damped term y ( t ) + p ( t ) y ′ ( t ) + q ( t ) f ( y ( t ) ) = 0 y(t) + p(t)y’(t) + q(t)f(y(t)) = 0 are given.

Positive solutions of y ( n ) ± f ( t , y ) = 0 {y^{(n)}} \pm f(t,y) = 0 are characterized as maximal or minimal depending on whether their asymptotic behavior is O ( t n − 1 ) O({t^{n - 1}}) or O ( 1 ) O(1) , respectively. Necessary and sufficient conditions for the existence of such extremal solutions are established.

We present here some results pertaining to the oscillatory behavior at infinity of the vector differential equation \[ y + Q ( t ) y = 0 , t ∈ [ 0 , ∞ ) y + Q(t)y = 0,\quad t \in [0,\infty ) \] , where Q ( t ) Q(t) is a real continuous n × n n \times n symmetric matrix function. It has been conjectured (cf., e.g. [6]) that the criterion \[ lim t → ∞ λ 1 { ∫ 0 t Q ( s ) d s } = ∞ \lim \limits _{t \to \infty } {\lambda _1}\left \{ {\int _0^t {Q(s)\;ds} } \right \} = \infty \] where λ 1 ( ⋅ ) {\lambda _1}( \cdot ) denotes the maximum eigenvalue of the matrix concerned, implies oscillation. We show that this is so under the tacit assumption \[ lim inf t → ∞ t − 1 tr { ∫ 0 t Q ( s ) d s } > − ∞ \lim \inf \limits _{t \to \infty } {t^{ - 1}}{\text {tr}}\left \{ {\int _0^t {Q(s)\;ds} } \right \} > - \infty \] where tr ( ⋅ ) {\text {tr}}( \cdot ) represents the trace of the matrix under consideration.

This article is concerned with the oscillatory behavior at infinity of the solution y : [ a , ∞ ) → R n y:[a,\infty ) \to {{\mathbf {R}}^n} of a system of second-order differential equations, y ( t ) + Q ( t ) y ( t ) = 0 y\left ( t \right ) + Q\left ( t \right )y\left ( t \right ) = 0 , t ∈ [ a , ∞ ) t \in [a,\infty ) ; Q Q is a continuous matrix-valued function on [ a , ∞ ) [a,\infty ) whose values are real symmetric matrices of order n n ; it is assumed that the largest eigenvalue of the matrix ∫ a t Q ( s ) d s \int _a^t {Q\left ( s \right )ds} tends to infinity as t → ∞ t \to \infty . Various sufficient conditions are given which guarantee oscillatory behavior at infinity; these conditions generalize those of Mingarelli [C.R. Math. Rep. Acad. Sci. Canada 2 (1980), 287-290, and Proc. Amer. Math. Soc. 82 (1981), 593-598].

New oscillation criteria are given for the differential equation \[ u + a ( t ) | u | α sgn ⁡ u = 0 , 0 > α > 1 , u + a(t)|u{|^\alpha }\operatorname {sgn} u = 0,\quad 0 > \alpha > 1, \] where a ( t ) a(t) is allowed to take on negative values for arbitrarily large t t .

Conditions will be placed on the m × m m \times m matrices G ( t ) G(t) and G i ( t ) {G_i}(t) to assure that for any integer k = 1 , … , n k = 1, \ldots ,n , the linear differential system \[ x i ′ = G i ( t ) x i + 1 , i = 1 , … , n − 1 , x n ′ = G ( t ) x 1 , {x’_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x’_n} = G(t){x_1}, \] where the x i {x_i} are m × m m \times m matrices, has a solution ( x 1 , … , x n ) ({x_1}, \ldots ,{x_n}) with the property that x k ( t ) → I {x_k}(t) \to I (the identity matrix) and if k > n k > n , x i ( t ) → 0 {x_i}(t) \to 0 , i = k + 1 , … , n i = k + 1, \ldots ,n , as t → ∞ t \to \infty . Furthermore, important bounds on the x i ( t ) {x_i}(t) will be given. Some of these conditions will require that ∫ a ∞ | G | > ∞ \int _a^\infty {\left | G \right | > \infty } while others will not. Corollaries will be given for special cases such as ( R ( t ) x ) = G ( t ) x (R(t)x) = G(t)x . No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.

In this paper we shall prove the following theorem: Let K K be a differential field of characteristic zero. Let φ \varphi and ψ \psi be elements of a differential field extension of K K such that (i) φ ≠ 0 \varphi \ne 0 and ψ ≠ 0 \psi \ne 0 ; (ii) φ \varphi and ψ \psi satisfy nontrivial linear differential equations with coefficients in K K , say, P ( φ ) = 0 P(\varphi ) = 0 and Q ( ψ ) = 0 Q(\psi ) = 0 ; (iii) φ = ψ n \varphi = {\psi ^n} for some positive integer n n such that n ⩾ ord P n \geqslant {\text { ord }}P . Then the logarithmic derivatives of φ \varphi and ψ \psi are algebraic over K K . (Note that φ ′ / φ = n ( ψ ′ / ψ ) \varphi ’/\varphi = n(\psi ’/\psi ) .)

In separate papers, D. L. Lovelady has related oscillation of solutions of certain linear differential equations of odd order ⩾ 3 \geqslant 3 and even order ⩾ 4 \geqslant 4 to oscillation of an associated second order equation. This paper presents a unified proof of Lovelady’s results for equations of arbitrary order ⩾ 3 \geqslant 3 . The results are somewhat more detailed and the equations need not be linear.

The ordinary differential equation(p(t)y′)′+q(t)f(y)=r(t)(p(t)y’)’ + q(t)f(y) = r(t)and its companion functional differential equation are considered. Sufficient conditions are given which ensure that all oscillatory solutions tend to zero ast→∞t \to \infty.

New criteria for the oscillatory or nonoscillatory behavior of ordinary, linear differential equations of second order and the location of their roots are obtained with the aid of new types of Sturmian majorant comparison equations. Several known results are obtained as special cases under less stringent conditions.

We shall assume that the coefficients of all equations of the form (1) satisfy (i) aij(x) differentiable, c(x) continuous; (ii) aij=d2ji, c =c; (iii) Eaijtijj > 0 for all complex n-tuples ((, , n) # (0, , 0) in a smooth bounded domain G CE'. If the coefficients of (1) can be extended into a larger domain G'DG so that (i)-(iii) hold in G as well as G, then we say that (1) is nonsingular in G. Points of aG at which such an extension is not possible comprise the singular boundary S. For n = 1, these considerations will reduce to the well-known oscillation theory for the Sturm-Liouville equation

(ii) the given series in (2) and (3) converge wherever \x\ Xo, (iii) » and p are positive integers and the m/s are non-negative integers, and (iv) x is a complex variable while X is a complex parameter. In the present treatment (1) and (2) are considered in a certain neighborhood of x = 0 and, for the parameter X, in a neighborhood of X= ». The point at x = 0 is said to be a regular point of (1), and whenever one or more mi is positive, x = 0 is also said to be a turning point of (1). The transformations

THEOREM. Suppose G (t, xo, , Xn-1) is a function which is (n + 1) dimensionally continuous for t _ 0, oo O, i =0, n 2. There then exists a unique solution of the initial value problem

We obtain sufficient conditions for the existence of periodic solutions to differential equations of the form x(m) + am_ ix(m ) + * * * +alx' + g(t, x) = f(t) (m > 1). The conditions are necessary for some classes of functions, and require no growth condition on g(t, x) for x > 0 (x < 0).

This article is concerned with the oscillatory behavior at infinity of the solution y: (a, infinity) ..-->.. R/sup n/ of a system of n second-order differential equations, y''(t) + Q(t)y(t) = 0, t epsilon (a, infinity); Q is a continuous matrix-valued function on (a, infinity) whose values are real symmetric matrices of order n. It is shown that the solution is oscillatory at infinity if (at least) n - 1 eigenvalues of the matrix integral/sub a//sup t/Q(t) dt tend to infinity as t ..-->.. infinity. 7 references.

The object of this note is to discuss the existence under fairly general conditions of solutions of the two-parameter eigenvalue problem defined by the pair of differential equations d2y,Idx' + {qj(xi; i,u) + r,(x,)}yi=O, i = 1, 2, x C[O, 1], and associated two-point Sturm-Liouville end conditions.

The object of this paper is to present sufficient conditions for the oscillation of certain solutions of the second order, nonlinear matrix differential equation. The oscillation criteria obtained here improve the recent results of the author and E. C. Tomastik. The methods employed in the paper extend a technique introduced by H. C. Howard and for the special linear version of the nonlinear equation, the resulting oscillation criteria represent improvement of Howard's work. Using an extension of the unitary transformation introduced by F. V. Atkinson, estimates of the oscillation of solutions ai-e obtained. Introduction. This paper is concerned with the following special systemii of 2n2 nonlinear first order differential equations

This paper is concerned with the relationship between the concepts of oscillation, nonoscillation and disconjugacy of the general third order linear differential equation y-+p(x)y"+q(x)y'+r(x)y=O. Two cases are considered: (i) q(x) nonpositive and r(x) nonnegative and (ii) both q(x) and r(x) nonpositive. Sufficient conditions for the disconjugacy of the equation are presented, improving the recent work of W. J. Kim. The relationship between disconjugacy and oscillation is described in each case, and these results provide a partial answer to a question raised by J. H. Barrett.