Solvability for Discrete Fractional Boundary Value Problems with a 𝑝-Laplacian Operator

Abstract

This paper is concerned with the solvability for a discrete fractional 𝑝-Laplacian boundary value problem. Some existence and uniqueness results are obtained by means of the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main results.

1. Introduction

For any number 𝑎∈ℝ and each interval 𝕀 of ℝ, we denote ℕ_𝑎={𝑎,𝑎+1,𝑎+2,…} and 𝕀_ℕ𝑎=𝕀∩ℕ_𝑎 throughout this paper. It is also worth noting that, in what follows, we appeal to the convention that the empty sum is taken to be 0.

In this paper, we will consider the existence and uniqueness of solutions for the following discrete fractional boundary value problem involving a 𝑝-Laplacian operator Δ𝜙_𝑝Δ^𝛼_𝐶𝑢(𝑡)=𝑓(𝑡+𝛼−1,𝑢(𝑡+𝛼−1)),𝑡∈[0,𝑏]_ℕ0,𝑢(𝛼−2)=𝛽₁𝑢(),()()𝛼+𝑏+1Δ𝑢𝛼−2=Δ𝑢𝛼−1=𝛽₂Δ𝑢(𝛼+𝑏), (1) where 1<𝛼≤2, 𝑏∈ℕ₁, 𝛽₁≠1, 𝛽₂≠1, Δ is the forward difference operator with stepsize 1,   Δ^𝛼_𝐶denotes the discrete Caputo fractional difference of order 𝛼,𝑓∶[𝛼−1,𝛼+𝑏−1]_ℕ𝛼−1×ℝ→ℝ is a continuous function, and 𝜙_𝑝 is the 𝑝-Laplacian operator; that is, 𝜙_𝑝(𝑢)=|𝑢|^𝑝−2𝑢, 𝑝>1. Obviously, 𝜙_𝑝 is invertible and its inverse operator is 𝜙_𝑞, where 𝑞>1 is a constant such that 1/𝑝+1/𝑞=1.

The theory of fractional differential equations has become a new important branch of mathematics (see, e.g., [1–8]). At the same time, boundary value problems for fractional differential equations have received considerable attention [9–18]. It is well known that discrete analogues of differential equations can be very useful in applications [19, 20], especially for using computer to simulate the behavior of solutions for certain dynamic equations. Compared to continuous case, significantly less is known about the discrete fractional calculus. However, within the recent years, a lot of papers have appeared on discrete fractional calculus and discrete fractional boundary value problems; see [21–37]. For example, in [25], Atıcı and Eloe explored a discrete fractional conjugate boundary value problem with the Riemann-Liouville fractional difference. To the best of our knowledge, this is pioneering work on discussing boundary value problems in discrete fractional calculus. After that, Goodrich studied discrete fractional boundary value problems involving the Riemann-Liouville fractional difference intensively and obtained a series of excellent results; see [26–31]. In [33, 34], Bastos et al. considered the discrete fractional calculus of variations and established several necessary optimality conditions for fractional difference variational problems. Abdeljawad introduced the conception of Caputo fractional difference and developed some useful properties of it in [35]. Ferreira in [37] initially investigated the existence and uniqueness of solutions for some discrete fractional boundary value problems of order less than one by the Banach fixed point theorem.

Very recently, some authors have focused their attention on the existence of solutions for fractional boundary value problems with the 𝑝-Laplacian operator in continuous case [38–44]. However, as far as we know, few papers can be found in the literature for the discrete fractional boundary value problems with the 𝑝-Laplacian operator [45].

Inspired by the aforementioned results, we will investigate the discrete fractional 𝑝-Laplacian boundary value problem (1) and establish some sufficient conditions for the existence and uniqueness of solutions to it by using the Banach contraction mapping principle.

The remainder of this paper is organized as follows. Section 2 preliminarily provides some necessary basic knowledge for the theory of discrete fractional calculus. In Section 3, the existence and uniqueness results for the solution to problem (1) will be established with the help of the contraction mapping principle. Finally, in Section 4, two concrete examples are provided to illustrate the possible applications of the established analytical results.

2. Preliminaries

For the convenience of the reader, we begin by presenting here some necessary basic definitions and lemmas on discrete fractional calculus theory.

Definition 1 (see [21]). For any 𝑡 and 𝜈, the falling factorial function is defined as 𝑡^𝜈=Γ(𝑡+1)Γ(𝑡+1−𝜈) (2) provided that the right-hand side is well defined. We appeal to the convention that if𝑡+1−𝜈is a pole of the Gamma function and 𝑡+1 is not a pole, then 𝑡^𝜈=0.

Definition 2 (see [46]). The 𝜈th fractional sum of a function 𝑓∶ℕ_𝑎→ℝ, for 𝜈>0, is defined by Δ^−𝜈𝑓(𝑡)=1Γ(𝜈)^𝑡−𝜈_𝑠=𝑎(𝑡−𝑠−1)^𝜈−1𝑓(𝑠),for𝑡∈ℕ_𝑎+𝜈. (3)

Definition 3 (see [35]). The 𝜈th Caputo fractional difference of a function 𝑓∶ℕ_𝑎→ℝ, for 𝜈>0, 𝜈∉ℕ, is defined by Δ^𝜈_𝐶𝑓(𝑡)=Δ^{−(𝑛−𝜈)}Δ^𝑛𝑓(𝑡)=1Γ(𝑛−𝜈)^{𝑡−𝑛+𝜈}_𝑠=𝑎(𝑡−𝑠−1)^{𝑛−𝜈−1}Δ^𝑛𝑓(𝑠),for𝑡∈ℕ_{𝑎+𝑛−𝜈}, (4) where 𝑛 is the smallest integer greater than or equal to 𝜈 and Δ^𝑛 is the 𝑛th forward difference operator. If 𝜈=𝑛∈ℕ, then Δ^𝜈_𝐶𝑓(𝑡)=Δ^𝑛𝑓(𝑡).

Lemma 4 (see [45]). Assume that 𝜈>0 and 𝑓 is defined on ℕ_𝑎. Then Δ^−𝜈Δ^𝜈_𝐶𝑓(𝑡)=𝑓(𝑡)+𝑐₀+𝑐₁𝑡+⋯+𝑐_𝑛−1𝑡^𝑛−1, (5) where 𝑐_𝑖∈ℝ, 𝑖=1,2,…,𝑛−1, and 𝑛 is the smallest integer greater than or equal to𝜈.

Now, we state and prove the following lemma, which provides a representation for the solution to (1) if the solution exists.

Lemma 5. Let ℎ∶[𝛼−1,𝛼+𝑏−1]_ℕ𝛼−1→ℝ, and let 𝛽₁,𝛽₂≠1. Then the following problem Δ𝜙_𝑝Δ^𝛼_𝐶𝑢(𝑡)=ℎ(𝑡+𝛼−1),𝑡∈[0,𝑏]_ℕ0,𝑢(𝛼−2)=𝛽₁𝑢(),()()𝛼+𝑏+1Δ𝑢𝛼−2=Δ𝑢𝛼−1=𝛽₂Δ𝑢(𝛼+𝑏), (6)(7) has a unique solution 𝑢(𝑡)=𝑎(𝑡)Γ(𝛼−1)^𝑏+1_𝑠=0(𝛼+𝑏−𝑠−1)^𝛼−2𝜙_𝑞^𝑠−1_𝜏=0ℎ()+𝛽𝜏+𝛼−1₁1−𝛽₁Γ(𝛼)×^𝑏+1_𝑠=0(𝛼+𝑏−𝑠)^𝛼−1𝜙_𝑞^𝑠−1_𝜏=0ℎ()+1𝜏+𝛼−1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^𝛼−1𝜙_𝑞^𝑠−1_𝜏=0ℎ(),[]𝜏+𝛼−1𝑡∈𝛼−2,𝛼+𝑏+1_ℕ𝛼−2, (8) where 𝑎(𝑡)=(𝛽₂{[𝛽₁(𝛼+𝑏+1)+2−𝛼]+(1−𝛽₁)𝑡})/((1−𝛽₁)(1−𝛽₂)).

Proof. The definition of the discrete Caputo fractional difference, together with condition Δ𝑢(𝛼−2)=Δ𝑢(𝛼−1), implies that Δ^𝛼_𝐶𝑢(0)=0. So from (6), we have 𝜙_𝑝Δ^𝛼_𝐶𝑢(𝑡)=𝜙_𝑝Δ^𝛼_𝐶𝑢(0)+^𝑡−1_𝑠=0ℎ()=𝑠+𝛼−1_{𝑡−1𝑠=0}ℎ(𝑠+𝛼−1), (9) and then Δ^𝛼_𝐶𝑢(𝑡)=𝜙_𝑞^𝑡−1_𝑠=0ℎ(𝑠+𝛼−1),𝑡∈[0,𝑏+1]_ℕ0. (10) Hence, in view of Lemma 4, we can get 𝑢(𝑡)=1Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^𝛼−1𝜙_𝑞^𝑠−1_𝜏=0ℎ(𝜏+𝛼−1)+𝑐₀+𝑐₁𝑡, (11) where 𝑡∈[𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2, 𝑐₀,𝑐₁∈ℝ.
Furthermore, we have (𝑡)=1Δ𝑢Γ(𝛼−1)^{𝑡−(𝛼−1)}_𝑠=0(𝑡−𝑠−1)^𝛼−2𝜙_𝑞×^𝑠−1_𝜏=0ℎ(𝜏+𝛼−1)+𝑐₁,𝑡∈[𝛼−2,𝛼+𝑏]_ℕ𝛼−2. (12) Then by conditions 𝑢(𝛼−2)=𝛽₁𝑢(𝛼+𝑏+1), Δ𝑢(𝛼−2)=𝛽₂Δ𝑢(𝛼+𝑏), we can get 𝑐₀=𝛽₁1−𝛽₁Γ(𝛼)×^𝑏+1_𝑠=0(𝛼+𝑏−𝑠)^𝛼−1𝜙_𝑞^𝑠−1_𝜏=0ℎ()+𝛽𝜏+𝛼−1₂𝛽₁(𝛼+𝑏+1)+2−𝛼1−𝛽₁1−𝛽₂Γ()×𝛼−1_𝑏+1𝑠=0(𝛼+𝑏−𝑠−1)^𝛼−2𝜙_𝑞^𝑠−1_𝜏=0ℎ(,𝑐𝜏+𝛼−1)₁=𝛽₂1−𝛽₂Γ()×𝛼−1_𝑏+1𝑠=0(𝛼+𝑏−𝑠−1)^𝛼−2𝜙_𝑞^𝑠−1_𝜏=0ℎ(𝜏+𝛼−1). (13) Substituting the values of𝑐₀and𝑐₁in (11), we get (8). This completes the proof.

Finally, we list below the following basic properties of the𝑝-Laplacian operator which will be used in the sequel.(1) If1<𝑝<2,𝑢𝑣>0 and|𝑢|,|𝑣|≥𝑚>0, then 𝜙_𝑝(𝑣)−𝜙_𝑝(𝑢)≤𝑝−1𝑚^𝑝−2|𝑣−𝑢|. (14)(2) If𝑝>2,|𝑢|,|𝑣|≤𝑀, then 𝜙_𝑝(𝑣)−𝜙_𝑝(𝑢)≤𝑝−1𝑀^𝑝−2|𝑣−𝑢|. (15)

3. Main Results

In this section, we will use the Banach contraction mapping principle to prove the existence and uniqueness for the solution to problem (1).

Let 𝔼 denote the Banach space of all functions from [𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2 into ℝ endowed with the norm defined by ‖𝑢‖=max{|𝑢(𝑡)|,𝑡∈[𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2}.

For the sake of convenience to the following discussion, we set 𝑎=max|𝑎(𝑡)|,𝑡∈[𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2, (16) where 𝑎(𝑡) is as given in Lemma 5. Also, for any 𝑢,𝑣∈𝔼, we denote 𝐴(𝑣,𝑢)(𝑡)=𝜙_𝑞^𝑡−1_𝑠=0𝑓(𝑠+𝛼−1,𝑣(𝑠+𝛼−1))−𝜙_𝑞^𝑡−1_𝑠=0𝑓(𝑠+𝛼−1,𝑢(𝑠+𝛼−1)), (17) for 𝑡∈[0,𝑏+1]_ℕ0. Obviously, 𝐴(𝑣,𝑢)(0)=0.

In view of Lemma 5, we transform problem (1) as 𝑢=ℱ𝑢, (18) where ℱ∶𝔼→𝔼 is defined by (𝑡)=𝑎(𝑡)ℱ𝑢)(Γ(𝛼−1)^𝑏+1_𝑠=0(𝛼+𝑏−𝑠−1)^𝛼−2×𝜙_𝑞^𝑠−1_𝜏=0𝑓((+𝛽𝜏+𝛼−1,𝑢𝜏+𝛼−1))₁1−𝛽₁Γ(𝛼)^𝑏+1_𝑠=0(𝛼+𝑏−𝑠)^𝛼−1×𝜙_𝑞^𝑠−1_𝜏=0𝑓((+1𝜏+𝛼−1,𝑢𝜏+𝛼−1))Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^𝛼−1×𝜙_𝑞^𝑠−1_𝜏=0𝑓(𝜏+𝛼−1,𝑢(𝜏+𝛼−1)), (19) for 𝑡∈[𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2. It is clear to see that𝑢is a solution of the problem (1) if and only if𝑢is a fixed point of ℱ.

Now, we state the main results as follows.

Theorem 6. Suppose 𝑝>2,𝛽₁≠1,𝛽₂≠1, and the following condition holds:(H₁) there exist positive numbers 𝜆 and 𝑘 with 𝑘<1−𝛽₁×()×𝑏!𝑞−1𝑏+11−𝛽₁(𝑎𝛼+𝛼+𝑏)+𝛽₁×[((𝛼+𝑏)𝜆Γ𝛼+1)]^{𝑞−2𝑏−1}_𝑖=1(𝛼+𝑖)⁻¹, (20)such that 𝜆𝛼𝑡^𝛼−1≤𝑓(𝑡,𝑢),𝑓𝑜𝑟(𝑡,𝑢)∈[𝛼−1,𝛼+𝑏−1]_ℕ𝛼−1×ℝ,𝑓()(||[]𝑡,𝑣−𝑓𝑡,𝑢)≤𝑘𝑣−𝑢,𝑓𝑜𝑟𝑡∈𝛼−1,𝛼+𝑏−1_ℕ𝛼−1,𝑢,𝑣∈ℝ. (21)(22) Then the problem (1) has a unique solution.

Proof. For any𝑢∈𝔼, by (21), we can get that ^𝑡−1_𝑠=0𝑓(𝑠+𝛼−1,𝑢(𝑠+𝛼−1))≥𝜆^𝑡−1_𝑠=0𝛼(𝑠+𝛼−1)^𝛼−1=𝜆(𝑡+𝛼−1)^𝛼≥𝜆Γ(𝛼+1),𝑡∈[1,𝑏+1]_ℕ0. (23) Due to 𝑝>2 and 1/𝑝+1/𝑞=1, we know that1<𝑞<2. By (14) and (22), for any𝑢,𝑣∈𝔼,𝑡∈[1,𝑏+1]_ℕ1, we have |𝐴(𝑣,𝑢)(𝑡)|≤𝑞−1[𝜆Γ(𝛼+1)]^𝑞−2×^𝑡−1_𝑠=0𝑓((((≤[(𝑠+𝛼−1,𝑣𝑠+𝛼−1))−𝑓𝑠+𝛼−1,𝑢𝑠+𝛼−1))𝑞−1𝜆Γ𝛼+1)]^{𝑞−2𝑡−1}_𝑠=0𝑘‖‖((𝑣−𝑢≤𝑘𝑞−1𝑏+1)[𝜆Γ𝛼+1)]^𝑞−2‖𝑣−𝑢‖. (24)
Next, for any 𝑢,𝑣∈𝔼 and for each 𝑡∈[𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2, together with the fact that 𝐴(𝑣,𝑢)(0)=0, we obtain 𝑡)−(𝑡=𝑎(𝑡)|(ℱ𝑣)(ℱ𝑢)()|Γ(𝛼−1)^𝑏+1_𝑠=0(𝛼+𝑏−𝑠−1)^𝛼−2𝐴(𝑠)+𝛽𝑣,𝑢)(₁1−𝛽₁Γ(𝛼)^𝑏+1_𝑠=0(𝛼+𝑏−𝑠)^𝛼−1𝐴(𝑠)+1𝑣,𝑢)(Γ(𝛼)^𝑡−𝛼_𝑠=0(𝑡−𝑠−1)^𝛼−1𝐴(𝑠=𝑎(𝑡)𝑣,𝑢)()Γ(𝛼−1)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠−1)^𝛼−2𝐴(𝑠)+𝛽𝑣,𝑢)(₁1−𝛽₁Γ(𝛼)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠)^𝛼−1𝐴(𝑠)+1𝑣,𝑢)(Γ(𝛼)^𝑡−𝛼_𝑠=1(𝑡−𝑠−1)^𝛼−1𝐴(𝑠≤|𝑎(𝑡𝑣,𝑢)())|Γ(𝛼−1)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠−1)^𝛼−2+𝛽₁1−𝛽₁Γ(𝛼)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠)^𝛼−1+1Γ(𝛼)^𝑡−𝛼_𝑠=1(𝑡−𝑠−1)^𝛼−1×𝑘𝑞−1(𝑏+1)[𝜆Γ(𝛼+1)]^𝑞−2‖‖≤𝑣−𝑢𝑎(𝛼+𝑏−1)^𝛼−1Γ(𝛼)+𝛽₁(𝛼+𝑏)^𝛼1−𝛽₁Γ()+()𝛼+1𝛼+𝑏^𝛼Γ()((𝛼+1×𝑘𝑞−1𝑏+1)[𝜆Γ𝛼+1)]^𝑞−2‖‖=𝑣−𝑢1−𝛽₁[𝑎𝛼+𝛼+𝑏]+𝛽₁(𝛼+𝑏)Γ(𝛼+𝑏)1−𝛽₁Γ()Γ()((𝛼+1𝑏+1×𝑘𝑞−1𝑏+1)[𝜆Γ𝛼+1)]^𝑞−2‖‖=𝑘(𝑣−𝑢𝑞−1𝑏+1)1−𝛽₁()+𝛽𝑎+𝛼+𝑏₁(𝛼+𝑏)[𝜆Γ(𝛼+1)]^𝑞−2×^𝑏−1_𝑖=1(𝛼+𝑖)×1−𝛽₁𝑏!⁻¹‖‖‖‖,𝑣−𝑢=𝐿𝑣−𝑢 (25) where𝐿=(𝑘(𝑞−1)(𝑏+1)[|1−𝛽₁|(𝑎𝛼+𝛼+𝑏)+|𝛽₁|(𝛼+𝑏)][𝜆Γ(𝛼+1)]^𝑞−2∏^𝑏−1_𝑖=1(𝛼+𝑖))/(|1−𝛽₁|𝑏!). From (20), we get that0<𝐿<1, which implies that ℱ is a contraction mapping. By means of the Banach contraction mapping principle, we get that ℱ has a unique fixed point in 𝔼; that is. the problem (1) has a unique solution. This completes the proof.

With a similar proof to that of Theorem 6, we can get the following theorem.

Theorem 7. Suppose 𝑝>2, 𝛽₁≠1, 𝛽₂≠1, and the following condition holds:(H₂) there exist constants𝜆>0and𝑘with 0<𝑘<1−𝛽₁×()×𝑏!𝑞−1𝑏+11−𝛽₁(𝑎𝛼+𝛼+𝑏)+𝛽₁×[((𝛼+𝑏)𝜆Γ𝛼+1)]^{𝑞−2𝑏−1}_𝑖=1(𝛼+𝑖)⁻¹, (26)such that 𝑓(𝑡,𝑢)≤−𝜆𝛼𝑡^𝛼−1,𝑓𝑜𝑟(𝑡,𝑢)∈[𝛼−1,𝛼+𝑏−1]_ℕ𝛼−1𝑓()(||[]×ℝ,𝑡,𝑣−𝑓𝑡,𝑢)≤𝑘𝑣−𝑢,𝑓𝑜𝑟𝑡∈𝛼−1,𝑏+𝛼−1_ℕ𝛼−1,𝑢,𝑣∈ℝ. (27) Then the problem (1) has a unique solution.

Theorem 8. Suppose1<𝑝<2,𝛽₁≠1,𝛽₂≠1, and the following condition holds:(H₃) there exists a nonnegative function𝑔∶[𝛼−1,𝛼+𝑏−1]_ℕ𝛼−1→ℝ and∑^𝑏_𝑠=0𝑔(𝑠+𝛼−1)=𝑀>0 such that 𝑓(𝑡,𝑢)≤𝑔(𝑡),(𝑡,𝑢)∈[𝛼−1,𝛼+𝑏−1]_ℕ𝛼−1×ℝ, (28)and there exists a positive constant𝑘such that 𝑓()(||[]𝑡,𝑣−𝑓𝑡,𝑢)≤𝑘𝑣−𝑢,𝑓𝑜𝑟𝑡∈𝛼−1,𝛼+𝑏−1_ℕ𝛼−1,𝑢,𝑣∈ℝ. (29) Then the problem (1) has a unique solution provided that 𝑘<1−𝛽₁×𝑏!1−𝛽₁(𝑎𝛼+𝛼+𝑏)+𝛽₁×()𝑀(𝛼+𝑏)𝑞−1𝑏+1^{𝑞−2𝑏−1}_𝑖=1(𝛼+𝑖)⁻¹. (30)

Proof. By (28), we can get that, for𝑡∈[1,𝑏+1]_ℕ1, ^𝑡−1_𝑠=0𝑓((≤𝑠+𝛼−1,𝑢𝑠+𝛼−1))_{𝑡−1𝑠=0}𝑓((≤𝑠+𝛼−1,𝑢𝑠+𝛼−1))_𝑏𝑠=0𝑔(𝑠+𝛼−1)=𝑀. (31) In view of 1<𝑝<2 and 1/𝑝+1/𝑞=1, we can get 𝑞>2. From (15) and (29), for any 𝑣,𝑢∈𝔼, we have |𝐴(𝑣,𝑢)(𝑡)|≤𝑞−1𝑀^𝑞−2×^𝑡−1_𝑠=0𝑓((−𝑠+𝛼−1,𝑣𝑠+𝛼−1))_{𝑡−1𝑠=0}𝑓((≤𝑀𝑠+𝛼−1,𝑢𝑠+𝛼−1))𝑞−1^𝑞−2×^𝑡−1_𝑠=0𝑓((((≤𝑀𝑠+𝛼−1,𝑣𝑠+𝛼−1))−𝑓𝑠+𝛼−1,𝑢𝑠+𝛼−1))𝑞−1^{𝑞−2𝑡−1}_𝑠=0𝑘‖‖𝑀𝑣−𝑢≤𝑘𝑞−1^𝑞−2𝑡‖𝑣−𝑢‖,𝑡∈[1,𝑏+1]_ℕ1. (32) Hence, for any 𝑡∈[𝛼−2,𝛼+𝑏+1]_ℕ𝛼−2, by 𝐴(𝑢,𝑣)(0)=0, we have 𝑡)−(𝑡=𝑎(𝑡)|(ℱ𝑣)(ℱ𝑢)()|Γ(𝛼−1)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠−1)^𝛼−2𝐴(𝑠)+𝛽𝑣,𝑢)(₁1−𝛽₁Γ(𝛼)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠)^𝛼−1𝐴(𝑠)+1𝑣,𝑢)(Γ(𝛼)^𝑡−𝛼_𝑠=1(𝑡−𝑠−1)^𝛼−1𝐴(𝑠≤|𝑎(𝑡𝑣,𝑢)())|Γ(𝛼−1)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠−1)^𝛼−2+𝛽₁1−𝛽₁Γ(𝛼)^𝑏+1_𝑠=1(𝛼+𝑏−𝑠)^𝛼−1+1Γ(𝛼)^𝑡−𝛼_𝑠=1(𝑡−𝑠−1)^𝛼−1×𝑘𝑞−1(𝑏+1)𝑀^𝑞−2‖‖≤𝑣−𝑢𝑎(𝛼+𝑏−1)^𝛼−1Γ(𝛼)+𝛽₁(𝛼+𝑏)^𝛼1−𝛽₁Γ()+()𝛼+1𝛼+𝑏^𝛼Γ())𝑀𝛼+1×𝑘𝑞−1(𝑏+1^𝑞−2‖‖=𝑘𝑣−𝑢1−𝛽₁(𝑎𝛼+𝛼+𝑏)+𝛽₁×()𝑀(𝛼+𝑏)𝑞−1𝑏+1^{𝑞−2𝑏−1}_𝑖=1()×𝛼+𝑖1−𝛽₁𝑏!⁻¹‖𝑣−𝑢‖=𝐿‖𝑣−𝑢‖, (33) where 𝐿=(𝑘[|1−𝛽₁|(𝑎𝛼+𝛼+𝑏)+|𝛽₁|(𝛼+𝑏)](𝑞−1)(𝑏+1)𝑀^𝑞−2∏^𝑏−1_𝑖=1(𝛼+𝑖))/(|1−𝛽₁|𝑏!). In view of (30), ℱ is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.

4. Examples

In this section, we will illustrate the possible application of the above established analytical results with the following two concrete examples.

Example 1. Consider the discrete fractional boundary value problem Δ𝜙₃Δ_𝐶^3/2𝑢𝑡)1(=3𝑡+2^1/2×12+sin²𝑢(𝑡+1/2)10√3+1+𝜃𝑢1390𝑡+2,𝑡∈[0,2]_ℕ0,−110𝑢29=𝑢2,−110Δ𝑢21=10Δ𝑢27=Δ𝑢2, (34) here𝜃is a real number.

Conclusion. Problem (34) has a unique nonnegative solution.

Proof. Corresponding to problem (1), 𝑝=3>2, 𝑞=3/2, 𝛼=3/2, 𝛽₁=1/10, 𝛽₂=1/10, 𝑏=2, and 𝑓(𝑡,𝑢)=3𝑡^1/2[1/2+sin²((𝑢/10√3)+𝜃)+(1/390)|𝑢|],(𝑡,𝑢)∈[1/2,5/2]_ℕ1/2×ℝ.
Choosing 𝜆=1 and 𝑘=3/100, by direct calculation, we can verify that 3𝑘=<1001−𝛽₁×()×𝑏!𝑞−1𝑏+11−𝛽₁(𝑎𝛼+𝛼+𝑏)+𝛽₁×[((𝛼+𝑏)𝜆Γ𝛼+1)]^{𝑞−2𝑏−1}_𝑖=1(𝛼+𝑖)⁻¹=18√3𝜋^1/4325≈0.1277. (35) It is easy to verify that 𝜆𝛼𝑡^𝛼−1=32𝑡^1/2≤3𝑡^1/212+sin²𝑢10√3+1+𝜃|𝑢(),()∈1390|=𝑓𝑡,𝑢𝑡,𝑢2,52_ℕ1/2𝑓()(×ℝ,𝑡,𝑣−𝑓𝑡,𝑢)=3𝑡^1/2sin²𝑣10√3+𝜃−sin²𝑢10√3+1+𝜃|𝑣|−1390|𝑢5390|≤32^1/21+1300||=390𝑣−𝑢69√4160𝜋||||||,𝑣−𝑢≈0.0294𝑣−𝑢<𝑘𝑣−𝑢 (36) for 𝑡∈[1/2,5/2]_ℕ1/2, 𝑢,𝑣∈ℝ. Therefore, by Theorem 6, the boundary value problem (34) has a unique solution. Furthermore, from the nonnegativeness of 𝑓 and the expression of ℱ, we also get that the unique solution of (34) is nonnegative.

Example 2. Consider the nonlinear discrete fractional boundary value problem Δ𝜙_3/2Δ_𝐶^3/2𝑢𝑡)=3(21𝑡+2^1/2×sin²𝑢(𝑡+1/2)[]40+𝜔,𝑡∈0,2_ℕ0,−12𝑢29=−𝑢2,−12Δ𝑢21=2Δ𝑢27=Δ𝑢2, (37) where 𝜔 is a real number.

Conclusion. Problem (37) has a unique solution.

Proof. The problem (37) can be regarded as problem (1), where 𝑝=3/2<2, 𝑞=3>2, 𝛼=3/2, 𝛽₁=−1/2, 𝛽₂=1/2,𝑏=2, and 𝑓(𝑡,𝑢)=(3/2)𝑡^1/2sin²((𝑢/40)+𝜔), (𝑡,𝑢)∈[1/2,5/2]_ℕ1/2×ℝ. Taking 𝑔(𝑡)=(3/2)𝑡^1/2, 𝑡∈[1/2,5/2]_ℕ1/2, then 𝑀=(105Γ(1/2))/32. Let 𝑘=1/600≈0.0017; we have 𝑘<1−𝛽₁×𝑏!1−𝛽₁(𝑎𝛼+𝛼+𝑏)+𝛽₁×()𝑀(𝛼+𝑏)𝑞−1𝑏+1^{𝑞−2𝑏−1}_𝑖=1(𝛼+𝑖)⁻¹=6415225√𝜋≈0.0024. (38) Moreover, we can verify that 𝑓((𝑡),()∈1𝑡,𝑢)≤𝑔𝑡,𝑢2,52_ℕ1/2𝑓()(≤3×ℝ,𝑡,𝑣−𝑓𝑡,𝑢)2𝑡^1/2140²||≤3𝑣−𝑢252^1/2140²||=9√𝑣−𝑢𝜋||||||,10240𝑣−𝑢≈0.0016𝑣−𝑢<𝑘𝑣−𝑢 (39) for 𝑡∈[1/2,5/2]_ℕ1/2, 𝑣,𝑢∈ℝ.
Therefore, by Theorem 8, problem (37) has a unique solution.

Acknowledgment

This work was supported by the Longdong University Grant XYZK-1010 and XYZK-1007.