Lynn Erbe

Positive solutions for a nonlinear differential equation on a measure chain

We are concerned with proving the existence of positive solutions of general two point boundary value problems for the nonlinear equation Lx(t) := -[r(t)x^@D(t)]^@D=/tf(t, x(t)). We will use fixed point theorems concerning cones in a Banach space. Important results concerning Greens functions for general two point boundary value problems for Lx(t) := -[r(t)x^@D(t)]^@D=0 will also be given.

OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR DYNAMIC EQUATIONS ON TIME SCALES

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.

Oscillation Results for Second-order Linear Equations on a Time Scale

We are interested in extensions of certain averaging techniques for the second-order scalar differential equation ( r ( t ) x j ( t )) j + q ( t ) x σ ( t )=0, on a time scale (measure chain) T . These results include some earlier criteria for the difference equations case.

Comparison theorems for linear dynamic equations on time scales

We obtain several comparison theorems for second order linear dynamic equations on a time scale. These results extend comparison theorems for the continuous case and provide some new results in the discrete case, as well as other more general situations.

Boundedness and oscillation for nonlinear dynamic equations on a time scale

We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation (p(t)x Δ (t)) Δ + q(t)(f o x σ ) = 0, on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient q(t). We illustrate the results by several examples, including a nonlinear Emden-Fowler dynamic equation.

Oscillation criteria for second-order matrix dynamic equations on a time scale

We obtain oscillation criteria for a second-order self-adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function. We illustrate our results with some nontrivial examples.

Oscillation criteria for nonlinear damped dynamic equations on time scales

Abstract We present new oscillation criteria for the second order nonlinear damped delay dynamic equation ( r ( t ) ( x Δ ( t ) ) γ ) Δ + p ( t ) ( x Δ σ ( t ) ) γ + q ( t ) f x ( τ ( t ) ) = 0 . Our results generalize and improve some known results for oscillation of second order nonlinear delay dynamic equation. Our results are illustrated with examples.

Oscillation criteria for nonlinear functional neutral dynamic equations on time scales

This paper is concerned with the oscillation of the second-order nonlinear functional dynamic equations on a time scale where γ is the quotient of odd positive integers, r(t), p(t), and are positive rd-continuous functions on , , and . We establish some new sufficient conditions for oscillation for the above equation. Several examples illustrating our results will be given.

Eigenvalue conditions and positive solutions

We are concerned with proving the existence of positive solutions of general two point boundary value problems for the nonlinear equation . Here the independent variable t is in a “measure chain”. We will use fixed point theorems for operators on a Banach space.

Oscillation results for a dynamic equation on a time scale

This paper is dedicated to Calvin Ahlbrandt