Eric R. Kaufmann

A multiplicity result for a boundary value problem with infinitely many singularities

Abstract We consider the second order boundary value problem −u″(t)=a(t)f(u(t)), 0 where a(t)∈Lp[0,1] for some p⩾1 and has countably many singularities in [0,1/2). We show that there exist countably many positive solutions using Holders inequality and Krasnoselskiis fixed point theorem for operators on a cone.

Discretization scheme in Volterra integro-differential equations that preserves stability and boundedness

A nonstandard discretization scheme is applied to continuous Volterra integro-differential equations. We will show that under our discretization scheme the stability of the zero solution of the continuous dynamical system is preserved. Also, under the same discretization, using a combination of Lyapunov functionals, Laplace transforms and z-transforms, we show that the boundedness of solutions of the continuous dynamical system is preserved.

Singular Conjugate Boundary Value Problems on a Time Scale

Let 𝕊 be a time scale symmetric about 1/2. Let be right dense and define The conjugate nonlinear boundary value problem where a(t) is singular at and f satisfies certain growth conditions, is shown to have infinitely many solutions using Krasnoselskis fixed point theorem.

Impulsive Periodic Boundary Value Problems for Dynamic Equations on Time Scale

Let T be a periodic time scale with period such that , and . Assume each is dense. Using Schaeffers theorem, we show that the impulsive dynamic equation where , , and is the -derivative on T, has a solution.