Takaŝi Kusano

Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations

SummaryConditions are given for the nonlinear differential equation (1)Lny+f(t, y, ..., ...,y(n−1)=0to have solutions which exist on a given interval [t0, ∞)and behave in some sense like specified solutions of the linear equation (2)Lnz=0as t→∞.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee the existence of solutions of (1)only for sufficiently large t. The main theorem requires no assumptions regarding oscillation or nonoscillation of solutions of (2).A second theorem is specifically applicable to the situation where (2)is disconjugate on [t0, ∞),and a corollary of the latter applies to the case where Lz=zn.

Radial entire solutions of a class of quasilinear elliptic equations

will be examined under suitable conditions on the functions f: 8, x R -+ R and g: R, -+R+, where R, =(O, 00); R, = [0, co), and 2 is a real parameter. The capillarity equation and the equation of prescribed mean curvature are important special cases of (1.1) for which g(p) = (1 + P’)-“~ and fir, U) is suitably specialized [ 1, 2, 3, 4, 10, 11, 12, 131. An entire solution of (1.1) is defined to be a function u E C*(R”) satisfying (1.1) at every point XE RN. Our primary objective is to obtain sufficient conditions on f and g for (1.1) to have positive radial entire solutions of the following three types:

Oscillation Properties of Nonlinear Hyperbolic Equations

A variety of oscillation properties are established for solutions of characteristic initial value problems for the nonlinear telegraph equation with a forcing term. Some analogous questions are considered for initial boundary value problems for the forced nonlinear wave equation. The principal tool is an averaging technique which enables one to establish such oscillation properties in terms of related ordinary differential inequalities.

Radial entire solutions to even order semilinear elliptic equations in the plane

Semilinear elliptic equations of the form are considered, where Δ m is the m -th iterate of the two-dimensional Laplacian Δ, p ( t ) is continuous in [0, ∞), and f ( u is continuous and positive either in (0, ∞) or in ℝ. Our main objective is to present conditions on p and f which imply the existence of radial entire solutions to (*), that is, those functions of class C 2 m (ℝ 2 ) which depend only on |x| and satisfy (*) pointwise in ℝ 2 . First, necessary and sufficient conditions are established for equation (*), with p ( t ) > 0 in [0, ∞), to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of | x | 2 m −2 log | x as | x | → ∞. Secondly, it is shown that, in the case p ( t f ( u ) > 0 is nondecreasing in ℝ, equation (*) always has eventually negative radial entire solutions, all of which decrease at least as fast as negative constant multiples of | x | 2 m −2 log | x | as | x | → ∞. Our results seem to be new even when specialised to the prototypes where γ is a constant.

Decaying Entire Positive Solutions of Quasilinear Elliptic Equations

The quasilinear elliptic equation (*)Δu+f(x,u,∇u)=0 is considered in the whole Euclidean space ℝN,N≥3. 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|→∞.

Oscillation and a class of nonlinear differential systems with general deviating arguments

THE OSCILLATION property ofnonlinear functional differential equations with deviating arguments has drawn a great deal of attention in the last few years. An excellent survey of known results on the subject has been done by Mitropol’skii and Sevelo [l]. Most of the literature, however, has been devoted to the study of scalar differential equations and very little is known about vector (or systems of) differential equations. An effort in the direction of establishing oscillation results for systems of differential equations with deviating arguments was undertaken by Vareh, Gritsai and Sevelo [2] and by the present authors [3]. In this paper we proceed further in this direction to extend the theory developed in [3] to nonlinear differential systems with general deviating arguments of the form

Nonlinear oscillation of a fourth order elliptic equation

where 42 is the biharmonic operator in Euclidean n-space R”, E is an exterior domain in Iin, and c(x, u) is a continuous function defined on E x R1. Equation (1) is said to be oscillatory in E if every solution zc E P(E) of (1) that is nontrivial in any neighborhood of infinity has arbitrarily large zeros, that is, the set {x E E: U(X) = 0) is unbounded. Our objective here is to find sufficient conditions for equation (1) to be oscillatory in E. We observe that there is no oscillation result in the literature for higher-order nonlinear elliptic equations. In Section 1 with the use of the technique of Noussair and Swanson [3] a fourth order ordinary differential inequality of the form

On entire solutions of second order semilinear elliptic equations

Abstract Second order semilinear elliptic equations of the form Δu + ƒ(x, u, ▽u) = 0 , ▽u = ( ∂u ∂x 1 ,…, ∂u ∂x n ) , (∗) are considered in Rn, n ⩾ 3, where ƒ may depend quadratically on ▽u. With the aid of the method of super- and subsolutions fairly general sufficient conditions are given for equation (∗) to have infinitely many positive entire solutions which are bounded and bounded away from zero throughout Rn. Concrete examples illustrating the main results are also presented.

SLOWLY VARYING SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH RETARDED AND ADVANCED ARGUMENTS

Abstract Regularity, in the sense of Karamata, (with nonoscillation as a consequence) and the precise asymptotic behaviour of solutions of two functional differential equations are studied.

Oscillation criteria for a class of nonlinear partial differential equations

There is much current interest in studying the oscillatory behavior of solutions of hyperbolic equations. The problem of establishing oscillation criteria for characteristic initial value problems for hyperbolic equations has been investigated by several authors. The reader is referred to Kreith [3] and Pagan [8,9] for linear hyperbolic equations and to Travis and Yoshida [lo] and Yoshida [ 1 l] for nonlinear hyperbolic equations. Our objective here is to provide oscillation theorems for the characteristic initial value problem for the differential operator L defined by