Kurt Kreith

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.

A Picone identity for first order differential systems

Abstract The Picone identity for solutions of Sturm-Liouville equations is generalized to solutions of certain first order nonlinear differential inequalities, to first order vector and matrix systems, and to certain first order systems of partial differential equations. These identities lead to generalizations of the Sturm-Picone Theorem, a disconjugacy criterion for a nonselfadjoint fourth order differential equation, and to a generalized maximum principle for elliptic equations.

A class of comparison theorems for non-linear hyperbolic initial value problems

An exponential form of the Riccati transformation is used to establish zeros of solutions of a class of characteristic initial value problems.

Higher order Wirtinger inequalities

Wirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesacks type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

Stability criteria for conjugate points of indefinite second order differential systems

Abstract Conjugate points of vector differential equations x″ + A(t) x = 0 are called hyper-conjugate points if they are realized by a positive solution. Conditions are established for the existence of stable hyperconjugate points, and these are related to a systems form of the Sturm comparison theorem.

Kiguradze classes for characteristics initial value problems

Abstract A well known technique for classifying the asymptotic behavior of solutions of ordinary differential equations is extended to hyperbolic characteristics initial value problems

A SELF-ADJOINT "SIMULTANEOUS CROSSING OF THE AXIS"

AbstractBy constructing the corresponding Greens function in a trapezoidal domain, we establish the existence of self-adjoint realizations of

VARIATIONAL SYSTEMS DISCONJUGACY CRITERIA FOR WNSELFADJOINT DIFFERENTIAL EQUATIONS

A = \frac{{\partial ^2 }}{{\partial t^2 }} - \frac{{\partial ^2 }}{{\partial s^2 }}