Takasi Kusano

Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type

This paper is concerned with asymptotic analysis of positive solutions of the second-order nonlinear differential equation (A)x^^(t)+q(t)@f(x(t))=0, where q:[a,~)->(0,~) is a continuous function which is regularly varying and @f:(0,~)->(0,~) is a continuous increasing function which is regularly varying of index @c@?(0,1). An application of the theory of regular variation gives the possibility of determining precise information about the asymptotic behavior at infinity of intermediate solutions of Eq. (A).

Existence of regularly varying solutions with nonzero indices of half-linear differential equations with retarded arguments

Sharp conditions are established for the existence of a pair of regularly varying solutions with nonzero indices of half-linear functional differential equations of the type (|x^(t)|^@asgnx^(t))^=q(t)|x(g(t))|^@asgnx(g(t)),@a>0,q(t)>0,g(t)

Positive decreasing solutions of systems of second order singular differential equations

are considered in an interval [a, ), where c, 3, A, # are positive constants andp, q, qo, b are positive continuous functions on [a, o). A positive decreasing solution of (,) is called proper or singular according to whether it exists on [a,)or it ceases to exist at a finite point of(a, ). First, conditions are given under which there does exist a singular solution of(,). Then, conditions are established for the existence of proper solutions of (,) which are classified into moderately decreasing solutions and strongly decreasing solutions according to the rate of their decay as .

Oscillation properties of solutions of a class of nonlinear parabolic equations

Oscillations of solutions of a class of nonlinear parabolic equations are investigated, and the unboundedness of solutions is also studied as corollaries. Our approach is to employ the modifications of Picone-type identities for half-linear elliptic operators.

OSCILLATION CRITERIA FOR A CLASS OF PARTIAL FUNCTIONAL-DIFFERENTIAL EQUATIONS OF HIGHER ORDER

Higher order partial differential equations with functional arguments including hyperbolic equations and beam equations are studied. Sufficient conditions are derived for every solution of certain boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce the multi-dimensional oscillation problem to a one-dimensional problem for higher order functional differential inequalities.

Elbert-type comparison theorems for a class of nonlinear Hamiltonian systems

Picone-type identities are established for a pair of solutions (x, y) and (X, Y) of the respective systems of the form x′ = r(t)x + p(t)φ1/α(y), y′ = −q(t)φα(x)− r(t)y, (1.1) and X′ = R(t)X + P(t)φ1/α(Y), Y′ = −Q(t)φα(X)− R(t)Y, (1.2) where α is a positive constant, p, q, r, P, Q and R are continuous functions on an interval J and φγ(u) denotes the odd function in u ∈ R defined by φγ(u) = |u| sgn u = |u|γ−1u, γ > 0. The identities are used to prove Sturmian comparison and separation results for components of solutions of systems (1.1) and (1.2).