Existence of local solutions for fractional difference equations with left focal boundary conditions

Johnny Henderson; Jeffrey T. Neugebauer

doi:10.1515/fca-2021-0014

Publicly Available Published by De Gruyter January 29, 2021

Existence of local solutions for fractional difference equations with left focal boundary conditions

Johnny Henderson and Jeffrey T. Neugebauer

From the journal Fractional Calculus and Applied Analysis

https://doi.org/10.1515/fca-2021-0014

Abstract

For 1 < ν ≤ 2 a real number and T ≥ 3 a natural number, conditions are given for the existence of solutions of the νth order Atıcı-Eloe fractional difference equation, Δ^νy(t) + f(t + ν − 1, y(t + ν − 1)) = 0, t ∈ {0, 1, …, T}, and satisfying the left focal boundary conditions Δy(ν − 2) = y(ν + T) = 0.

MSC 2010: Primary 26A33; Secondary 39A12

Key Words and Phrases: Atıcı-Eloe fractional sums and differences; left focal boundary value problem; fixed point; local solution

1 Introduction

In this paper, for 1 < ν ≤ 2 a real number and T ≥ 3 a natural number, we are concerned with local solutions of the nonlinear νth order fractional difference equation,

(1.1)Δνy(t)+f(t+ν−1,y(t+ν−1))=0, t∈{0,1,…,T},

satisfying the left focal boundary conditions,

(1.2)Δy(ν−2)=0, y(ν+T)=0,

where Δ is the forward difference operator with step-size 1, Δ^ν is the Atıcı-Eloe fractional difference, and f(x,r):{ν−1,ν,…,ν+T−1}×ℝ→ℝ is continuous. Our methods rely on application of the Leray-Schauder Nonlinear Alternative [17], which will involve imposing growth restrictions on f as well as restrictions on the natural number T. Since the paper involves restrictions on the size of T, classically such a result is called a local existence result as opposed to a global existence result. Primary motivation for this work is the recent paper by Henderson [18] devoted to local solutions of Dirichlet boundary value problems for fractional difference equations.

A good deal of the current interest in fractional difference equations, devoted to both their theoretical development and their applications, was spawned by the Atıcı and Eloe definitions [1, 2], in the context of discrete domains, of fractional sums and fractional differences. Soon afterwards, Goodrich further developed and extended those papers [9, 10, 11]. Atıcı and Eloe dealt in their first paper with initial value problems for fractional difference equations, and then followed in their second paper with applications of their definitions in obtaining positive solutions for Dirichlet boundary value problems for fractional difference equations. Their second work involved a Guo-Krasnosel’skii fixed point argument requiring the construction of a Green’s function for their fractional problem. Some of Goodrich’s subsequent research that required construction of appropriate Green’s functions has focused on questions extending the Atıcı and Eloe results and can be found in [12, 13, 14, 15, 16]. For some other researchers' closely related works involving boundary value problems for fractional difference equations, we cite [4, 5, 8, 19, 20, 27, 28, 29, 30, 31].

It is natural that discrete fractional calculus and fractional difference equations often appear in modeling natural processes such as found in the paper by Atıcı and Şengül [3] and in the paper by Metzler et al. [22]. Especially prominent is the current use of boundary value problems for discrete fractional difference equations in its applications to discrete control processes; see, for example, this list of monographs devoted to discrete fractional control [6, 7, 21, 23, 24, 25].

2 Some preliminaries and the Green function

We begin this section with the Atıcı-Eloe definitions of fractional sum and fractional difference in the context of a discrete domain.

Definition 2.1

Let n ∈ ℕ and n − 1 < κ ≤ n be a real number, and let a ∈ ℝ. For t ∈ {a + κ, a + κ + 1, …}, the κth order Atıcı-Eloe fractional sum, Δ^−κu, of the function u is defined by

Δ−κu(t):=1Γ(κ)∑s=at−κ(t−s−1)(κ−1)u(s),

where t(κ):=Γ(t+1)Γ(t+1−κ) is the falling function.

For t ∈ {a + n − κ, a + n + 1 − κ, …}, the κth order Atıcı-Eloe fractional difference, Δ^κu, of the function u is defined by

Δκu(t):=Δn−(n−κ)u(t):=Δn(Δ−(n−κ)u(t)),

where Δ is the forward difference defined by Δu(t) = u(t + 1) − u(t), and Δⁱu(t) = Δ (Δ^{i − 1}u(t)), i = 2, 3, ….

Remark 2.1

We note that, for u defined on {a, a + 1, …}, then Δ^−κu is defined on {a + κ, a + κ + 1, …}.

In [11], for 1 < ν ≤ 2, Goodrich constructed the Green function, G(t, s), for −Δ^νy(t) = 0, t ∈ {0, 1, …, T}, satisfying the left focal boundary conditions (1.2). In particular, by direct computation, Goodrich obtained

G(t,s)=c(ν,T)Γ(ν){Γ(ν)(ν+T−s−1)(ν−1), t=ν−2,ν−1,[(2−ν)t(ν−1)+(ν−1)(ν−2)]×(ν+T−s−1)(ν−1), 0≤t−ν≤s≤T,[(2−ν)t(ν−1)+(ν−1)(ν−2)]×(ν+T−s−1)(ν−1)−1c(ν,T)(t−s−1)(ν−2),0≤s<t−ν≤T,

with

c(ν,T)=1(ν+T)(ν−2)[(ν−1)+(2−ν)(T+2)].

Goodrich also obtained the following properties of G(t, s) which will be of importance to us:

For each s ∈ {0, …, T},
ΔtG(ν−2,s)=0 and G(ν+T,s)=0.
G(t, s) > 0, for (t, s) ∈ {ν − 2, …, ν + T − 1} × {0, …, T}.
max_{t∈{ν−2, …, ν+T}}G(t, s) = G(s + ν − 1, s), for s ∈ {0, …, T}.

We remark that y is a solution of the linear discrete fractional difference equation, Δ^νy(t) + h(t + ν − 1) = 0, t ∈ {0, …, T}, and satisfying (1.2), if and only if y:{ν−2,…,ν+T}→ℝ has the form

y(t)=∑s=0TG(t,s)h(s+ν−1), t∈{ν−2,…,ν+T}.

In the next section, G(t, s) will play the role of the kernel of a completely continuous summation operator.

3 Local existence of solutions

In this section, we make application of the the Leray-Schauder Nonlinear Alternative [17] in obtaining local solutions of (1.1)-(1.2). In doing so, we impose restrictions on f and restrictions on the natural number T.

Theorem 3.1

[Leray-Schauder Nonlinear Alternative]. Let (E, ‖ · ‖) be a Banach space, Kbe a closed and convex subset ofE, Ube a relatively open subset ofKsuch that 0 ∈ U, andN:U¯→Kbe completely continuous. Then, either

u = Nuhas a solution inU¯,

There existu ∈ ∂Uand λ ∈ (0, 1) such thatu = λNu.

For 1 < ν ≤ 2 a real number and T ≥ 3 a natural number, assume f(x,r):{ν−1,ν,…,ν+T−1}×ℝ→ℝ is continuous, and let

m:=maxs∈{0,…,T}G(s+ν−1,s).

We now present the result of this paper.

Theorem 3.2

Assume

There existσ : {ν − 1, …, T + ν − 1} → [0, ∞) and a nondecreasing functionψ : [0, ∞) → [0, ∞) such that
|f(x,r)|≤σ(x)ψ(|r|), (x,r)∈{ν−1,…,T+ν−1}×ℝ,

and

There existsℓ>0such that
ℓmψ(ℓ)(T+1)maxx∈{ν−1,…,T+ν−1}σ(x)>1.

Then, (1.1)-(1.2) has a solution defined on {ν − 2, …, T + ν}.

Proof

Let the Banach space E:={h:{ν−2,…,T+ν}→ℝ | Δh(ν−2)=h(T+ν)=0} be equipped with the norm

‖h‖:=maxx∈{ν−2,…,T+ν}|h(x)|.

We seek fixed points of the mapping N : E → E defined by

(Nh)(t):=∑s=0TG(t,s)f(s+ν−1,h(s+ν−1)), h∈E, t∈{ν−2,…,T+ν},

where G(t, s) is the Green function of Section 2. We note that u ∈ E is a solution of (1.1)-(1.2) if and only if u is a fixed point of N.

We first show that N maps bounded sets into bounded sets. In that direction, for r > 0, let

Br:={h∈E | ‖h‖≤r}

be a bounded subset of E. Then, by (A), for t ∈ {ν − 2, …, T + ν} and h ∈ B_r,

|(Nh)(t)|≤∑s=0TG(t,s)|f(s+ν−1,h(s+ν−1))|≤∑s=0TG(s+ν−1,s)|f(s+ν−1,h(s+ν−1))|≤∑s=0Tmσ(s+ν−1)ψ(|h(s+ν−1)|)≤m∑s=0Tmaxx∈{ν−1,…,T+ν−1}σ(x)ψ(r)=mψ(r)maxx∈{ν−1,…,T+ν−1}σ(x)(T+1).

Hence, ‖Nh‖ ≤ mψ(r) max_{x∈{ν−1,…,T+ν−1}}σ(x)(T + 1), and so N maps B_r into a bounded set.

Since {ν − 2, …, T + ν} is a discrete set, it follows immediately that N maps B_r into an equicontinuous set. Therefore, by the Arzelà-Ascoli Theorem, N is completely continuous.

Next, we suppose h ∈ E and that for some 0 < λ < 1, h = λNh. Then, for t ∈ {ν − 2, …, T + ν}, and again by (A),

|h(t)|=|λ(Nh)(t)|≤∑s=0TG(t,s)|f(s+ν−1,h(s+ν−1))|≤∑s=0TG(s+ν−1,s)|f(s+ν−1,h(s+ν−1))|≤m∑s=0Tσ(s+ν−1)ψ(‖h‖)≤mmaxx∈{ν−1,…,T+ν−1}σ(x)ψ(‖h‖)(T+1),

which yields

‖h‖mmaxx∈{ν−1,…,T+ν−1}σ(x)ψ(‖h‖)(T+1)≤1.

It follows from (B) that ‖h‖≠ℓ. If we set

U:={h∈E | ‖h‖<ℓ},

then the operator N:U¯→E is completely continuous. From the choice of U, then there is no h ∈ ∂U such that h = λNh, for some 0 < λ < 1. It follows from Theorem 3.1 that N has a fixed point y∈U¯, which is a desired solution of (1.1)-(1.2).

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Received: 2020-04-06

Revised: 2020-12-01

Published Online: 2021-01-29

Published in Print: 2021-02-23

Existence of local solutions for fractional difference equations with left focal boundary conditions

Abstract

1 Introduction

2 Some preliminaries and the Green function

Definition 2.1

Remark 2.1

3 Local existence of solutions

Theorem 3.1

Theorem 3.2

Proof

References

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