Christopher S. Goodrich

Existence of a positive solution to a class of fractional differential equations

Abstract In this paper, we consider a (continuous) fractional boundary value problem of the form − D 0 + ν y ( t ) = f ( t , y ( t ) ) , y ( i ) ( 0 ) = 0 , [ D 0 + α y ( t ) ] t = 1 = 0 , where 0 ≤ i ≤ n − 2 , 1 ≤ α ≤ n − 2 , ν > 3 satisfying n − 1 ν ≤ n , n ∈ N , is given, and D 0 + ν is the standard Riemann–Liouville fractional derivative of order ν . We derive the Green’s function for this problem and show that it satisfies certain properties. We then use cone theoretic techniques to deduce a general existence theorem for this problem. Certain of our results improve on recent work in the literature, and we remark on the consequences of this improvement.

Continuity of solutions to discrete fractional initial value problems

In this paper, we consider a fractional initial value problem (IVP) in the case where the order @n of the fractional difference satisfies 0<@n@?1. We show that solutions of this IVP satisfy a continuity condition both with respect to the order of the difference, @n, and with respect to the initial conditions, and we deduce several important corollaries from this theorem. Thus, we address a complication that arises in the fractional case but not in the classical (integer-order) case.

Discrete fractional calculus

Preface.- 1. Basic Difference Calculus.- 2. Discrete Delta Fractional Calculus and Laplace Transforms.- 3. Nabla Fractional Calculus.- 4. Quantum Calculus.- 5. Calculus on Mixed Time Scales.- 6. Fractional Boundary Value Problems.- 7. Nonlocal BVPs and the Discrete Fractional Calculus.-Solutions to Selected Problems.- Bibliography.- Index.

On a discrete fractional three-point boundary value problem

In this paper, we analyse a ν-th order, , discrete fractional three-point boundary value problem (BVP). We show that Greens function associated to this problem satisfies certain conditions. We demonstrate that the range of admissible boundary conditions depends upon the order ν of the difference equation, and we give explicit formulae for this dependence. By using both the Brouwer fixed point theorem and the Krasnoselskiĭ fixed point theorem, we then show that a solution to this problem exists. Our results extend recent results on discrete fractional BVPs (FBVPs), and they also provide an initial set of results on the theory of multipoint FBVPs on the time scale of integers.

On a fractional boundary value problem with fractional boundary conditions

Abstract In this paper, we consider a discrete fractional boundary value problem, for t ∈ [ 0 , b + 1 ] N 0 , of the form − Δ ν y ( t ) = f ( t + ν − 1 , y ( t + ν − 1 ) ) , y ( ν − 2 ) = 0 , [ Δ α y ( t ) ] t = ν + b − α + 1 = 0 , where f : [ ν − 1 , … , ν + b ] N ν − 2 × R → R is continuous, 1 ν ≤ 2 , and 0 ≤ α 1 . We prove that this problem can be interpreted as a discrete multipoint problem. We also show that the problem is a generalization of some recent results. Our results provide some basic analysis of discrete fractional boundary conditions.

On semipositone discrete fractional boundary value problems with non-local boundary conditions

We consider the existence of at least one positive solution to the discrete fractional equation , where and , equipped with a two-point boundary condition that can possibly be both non-local and nonlinear. Due to the fact that f is allowed to be negative for some values of t and y, we consider here the semipositone problem. In addition to discussing conditions under which this problem is guaranteed to have at least one positive solution for small values of , we provide an example to illustrate the use of our results. Due to the generality of our results, we include many boundary conditions as special cases such as the conjugate- and multipoint-type conditions.

Some new existence results for fractional difference equations

In this paper, we introduce several existence theorems for a discrete fractional boundary value problem with Dirichlet boundary conditions in the case where the order ν of the fractional difference satisfies 1 < ν ≤ 2. We use cone theoretic techniques to deduce the existence of one or more positive solutions. We then deduce uniqueness theorems for the same problem by assuming a Lipschitz condition. We show that many of the classical existence and uniqueness theorems for second-order discrete boundary value problems extend to the fractional-order case.

Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions

We consider the coupled system of discrete fractional boundary value problemswhere and . In particular, we consider the case of nonlocal and possibly nonlinear boundary conditions, and demonstrate that this problem has at least one positive solution by imposing growth conditions only on the nonlocal terms and . Of special note is that no growth conditions of any kind are imposed on the nonlinearities and .

Coupled systems of boundary value problems with nonlocal boundary conditions

Abstract We consider the coupled system − x ″ = λ 1 f ( t , y ( t ) ) , − y ″ = λ 2 g ( t , x ( t ) ) , t ∈ ( 0 , 1 ) , subject to the coupled boundary conditions x ( 0 ) = H 1 ( φ 1 ( y ) ) , x ( 1 ) = 0 and y ( 0 ) = H 2 ( φ 2 ( x ) ) , y ( 1 ) = 0 . Since H 1 and H 2 are nonlinear functions and φ 1 and φ 2 are linear functionals realized as Stieltjes integrals, the boundary conditions may be nonlocal and nonlinear in character. By assuming that φ 1 and φ 2 satisfy a particular decomposition hypothesis together with some growth assumptions on H 1 and H 2 at 0 and + ∞ , we show that this system can possess at least one positive solution even if no growth conditions are imposed on f and g .

Monotonicity results for delta fractional differences revisited

Abstract In this paper, by means of a recently obtained inequality, we study the delta fractional difference, and we obtain the following interrelated theorems, which improve recent results in the literature. Theorem A Assume that f : ℕa → ℝ and that Δaνf(t)