Paul W. Eloe

Initial value problems in discrete fractional calculus

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a v-th (0 < v ≤ 1) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.

Two-point boundary value problems for finite fractional difference equations

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Greens function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.

Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions

We discuss the existence of positive solutions of a nonlinear nth order boundary value problem u(n)+a(t)f(u)=0,t∈(0,1)u(0)=0,u′(0)=0,...,u(n- 2)(0)=0,αu(η)=u(1), where 0<η<1, 0<αηn-1<1. In particular, we establish the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones due to Krasnoselkiǐ and Guo.

Fractional q-Calculus on a time scale

Abstract The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale , where to ∈ ℝ and 0 < q < 1. By use of time scale calculus notation, we find the proof of many results more straight forward. We shall develop some properties of fractional q-calculus, we shall develop some properties of a q-Laplace transform, and then we shall employ the q-Laplace transform to solve fractional q-difference equations.

Gronwall's inequality on discrete fractional calculus

In this paper, we introduce discrete fractional sum equations and inequalities. We obtain the equivalence of an initial value problem for a discrete fractional equation and a discrete fractional sum equation. Then we give an explicit solution to the linear discrete fractional sum equation. This allows us to state and prove an analogue of Gronwalls inequality on discrete fractional calculus. We employ a nabla, or backward difference; we employ the Riemann-Liouville definition of the fractional difference. As a result, we obtain Gronwalls inequality for discrete calculus with the nabla operator. We illustrate our results with an application that gives continuous dependence of solutions of initial value problems on initial conditions.

Positive solutions of nonlinear functional difference equations

Abstract In this paper, we apply a cone theoretic fixed-point theorem and obtain sufficient conditions for the existence of positive solutions to some boundary value problems for a class of functional difference equations. We consider analogues of sublinear or superlinear growth in the nonlinear terms.

A generalization of concavity for finite differences

Abstract The concept of concavity is generalized to discrete functions, u , satisfying the n th -order difference inequality, (−1) n − k Δ n u ( m ) ≥ 0, m = 0, 1,..., N and the homogeneous boundary conditions, u (0) = ... = u (k−1) = 0, u ( N + k + 1) = ... = u ( N + n ) = 0 for some k ∈ “1, ..., n − 1”. A piecewise polynomial is constructed which bounds u below. The piecewise polynomial is employed to obtain a positive lower bound on u ( m ) for m = k , ..., N + k , where the lower bound is proportional to the supremum of u . An analogous bound is obtained for a related Greens function.

Inequalities based on a generalization of concavity

The concept of concavity is generalized to functions, y, satisfying nth order differential inequalities, (-I)n-ky(n)(t) > 0,0 IIyIi/4m, 1/4 0,0 IIYII/4, where Il = supo 2 is an integer, k E {1, ... , n 1}, and if (2) (l)n-k)y(n) > 0, 0 < t < 1, t3f YW (0 0, j,*** .. , 1Y(j O,j 1 . ,kthen for 1/4 < t < 3/4,

Existence and uniqueness of solutions for impulsive fractional differential equations

Motivated by some recent developments in the existence theory of impulsive fractional differential equations, in this paper we present a general method for converting an impulsive fractional differential equation to an equivalent integral equation. The applicability of the method is demonstrated by considering some boundary value problems for impulsive fractional differential equations.

Uniform asymptotic stability in nonlinear volterra discrete systems

Abstract We employ the notion of total stability to obtain new criteria for uniform asymptotic stability of the zero solution of a nonlinear Volterra discrete system. Resolvent equation methods are employed, and a summability criterion on the resolvent kernel is obtained. Also, we obtain a new difference equation that the resolvent R(n, s) satisfies.