The Existence of Solutions to a System of Discrete Fractional Boundary Value Problems

Abstract

We study the existence of solutions for the boundary value problem −Δ^𝜈𝑦₁(𝑡)=𝑓(𝑦₁(𝑡+𝜈−1),𝑦₂(𝑡+𝜇−1)), −Δ^𝜇𝑦₂(𝑡)=𝑔(𝑦₁(𝑡+𝜈−1),𝑦₂(𝑡+𝜇−1)), 𝑦₁(𝜈−2)=Δ𝑦₁(𝜈+𝑏)=0, 𝑦₂(𝜇−2)=Δ𝑦₂(𝜇+𝑏)=0, where 1<𝜇,𝜈≤2, 𝑓,𝑔∶ℝ×ℝ→ℝ are continuous functions, 𝑏∈ℕ₀. The existence of solutions to this problem is established by the Guo-Krasnosel'kii theorem and the Schauder fixed-point theorem, and some examples are given to illustrate the main results.

1. Introduction

In recent years, fractional differential equations have been of great interest. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. The continuous fractional calculus has seen tremendous growth within the last ten years or so. Some of the recent progress in the continuous fractional calculus has included a paper [1] in which the authors explored a continuous fractional boundary value problem of conjugate type. Using cone theory, they then deduced the existence of one or more positive solutions. Other recent work in the direction of those articles may be found, see [2–14].

Recently, there appeared a number of papers on the discrete fractional calculus, such as [15–32], which has helped to build up some of the basic theories of this area. For example, Atici and Eloe discussed properties of the generalized falling function, a corresponding power rule for fractional delta-operators, and the commutivity of fractional sums in [15]. And Goodrich studied a two-point fractional boundary value problem in [24], which gave the existence results for a certain two-point boundary value problem of right-focal type for a fractional difference equation.

From the above works, we can see a fact, although the discrete fractional boundary value problem has been studied by some authors, to the best of our knowledge, systems of discrete fractional boundary value problem are seldom considered.

Goodrich in [24] considered a discrete fractional boundary value problem−Δ^𝜈𝑦(𝑡)(,𝑦()(),=𝑓t+𝜈−1,𝑦𝑡+𝜈−1)𝜈−2=0=Δ𝑦𝜈+𝑏 (1.1) where 𝑡∈[0,𝑏+1]_ℕ0, 𝑓∶[𝜈−1,𝜈+𝑏−1]_ℕ𝜈−1×ℝ→ℝ, and 1<𝜈≤2, which gave a representation for the solution to this problem and established the existence and uniqueness of solution to this problem by the Guo-Krasnosel’kii theorem.

Goodrich in [25] examined a system of discrete fractional difference equations subject to nonlocal boundary conditions−Δ^𝜈1𝑦₁(𝑡)=𝜆₁𝑎₁𝑡+𝜈₁−1𝑓₁𝑦₁𝑡+𝜈₁−1,𝑦₂𝑡+𝜈₂−1,−Δ^𝜈2𝑦₂(𝑡)=𝜆₂𝑎₂𝑡+𝜈₂−1𝑓₂𝑦₁𝑡+𝜈₁−1,𝑦₂𝑡+𝜈₂−1, (1.2) for 𝑡∈[0,𝑏]_ℕ0, subject to the conditions𝑦₁𝜈₁−2=𝜓₁𝑦₁,𝑦₂𝜈₂−2=𝜓₂𝑦₂,𝑦₁𝜈₁+𝑏=𝜙₁𝑦₁,𝑦₂𝜈₂+𝑏=𝜙₂𝑦₂, (1.3) where 𝜆_𝑖>0,𝑎_𝑖∶ℝ→[0,+∞),𝜈_𝑖∈(1,2], and for each 𝑖=1,2, 𝜓_𝑖,𝜙_𝑖∶ℝ^𝑏+3→ℝ are given functionals, and 𝑓_𝑖∶[0,+∞)×[0,+∞)→[0,+∞) are continuous for each admissible 𝑖. This paper gives the existence of a positive solution on discrete fractional boundary value problems.

Motivated by all the works above, in this paper, we discuss the existence of solutions to a system of discrete fractional boundary value problem−Δ^𝜈𝑦₁(𝑡)=𝑓𝑦₁(𝑡+𝜈−1),𝑦₂𝑡+𝜇−1,for𝑡∈[0,𝑏+1]_ℕ0,−Δ^𝜇𝑦₂(𝑡)=𝑔𝑦₁(𝑡+𝜈−1),𝑦₂𝑡+𝜇−1,for𝑡∈[0,𝑏+1]_ℕ0,𝑦₁(𝜈−2)=Δ𝑦₁()𝑦𝜈+𝑏=0,₂𝜇−2=Δ𝑦₂𝜇+𝑏=0, (1.4) where 1<𝜇,𝜈≤2, 𝑓,𝑔∶ℝ×ℝ→ℝ are continuous functions, 𝑏∈ℕ₀.

The plan of the paper is as follows. In Section 2, we will present some lemmas in order to prove our main results. Section 3, establishes and proves our main results. In Section 4 we give some examples to illustrate the main results.

2. Preliminaries

For the convenience of the reader, we give some background materials from fractional difference theory to facilitate the analysis of problem (1.4). These and other related results and their proof can be found in [15–18].

Definition 2.1. One defines 𝑡^𝜈∶=Γ(𝑡+1)/Γ(𝑡+1−𝜈), for any 𝑡 and 𝜈 for which the right-hand side is defined. One also appeals to the convention that if 𝑡+1−𝜈 is a pole of the Gamma function and 𝑡+1 is not a pole, then 𝑡^𝜈=0.

Definition 2.2. The 𝜈th fractional sum of a function 𝑓, for 𝜈>0, is defined by Δ^−𝜈𝑓()1𝑡;𝑎∶=Γ(𝜈)^𝑡−𝜈_𝑠=𝑎(𝑡−𝑠−1)^𝜈−1𝑓(𝑠), (2.1) for 𝑡∈{𝑎+𝜈,𝑎+𝜈+1,…}∶=ℕ_𝑎+𝜈. One also defines the 𝜈th fractional difference for 𝜈>0 by Δ^𝜈𝑓(𝑡)∶=Δ^𝑁Δ^𝜈−𝑁𝑓(𝑡), where 𝑡∈ℕ_𝑎+𝜈 and 𝜈∈ℕ is chosen so that 0≤𝑁−1<𝜈≤𝑁.

Lemma 2.3. Let 𝑡 and 𝜈 be any numbers for which 𝑡^𝜈 and 𝑡^𝜈−1 are defined, then Δ𝑡^𝜈=𝜈𝑡^𝜈−1. (2.2)

Lemma 2.4. Let 0≤𝑁−1<𝜈≤𝑁, where 𝑁∈ℕ and 𝑁−1≥0, then Δ^−𝜈Δ^𝜈𝑦(𝑡)=𝑦(𝑡)+𝐶₁𝑡^𝜈−1+𝐶₂𝑡^𝜈−2+⋅⋅⋅+𝐶_𝑁𝑡^𝜈−𝑁, (2.3) for some 𝐶_𝑖∈ℝ, with 1≤𝑖≤𝑁.

Before stating the next useful lemma, let us introduce the following notation, which will be important in the sequel:𝑇₁∶=(𝑡,𝑠)∈[𝜈−1,𝜈+𝑏+1]_ℕ𝜈−1×[0,𝑏+1]_ℕ0,𝑇∶0≤𝑠<𝑡−𝜈+1≤𝑏+2₂∶=(𝑡,𝑠)∈[𝜈−1,𝜈+𝑏+1]_ℕ𝜈−1×[0,𝑏+1]_ℕ0,𝐿∶0≤𝑡−𝜈+1≤𝑠≤𝑏+2₁∶=(𝑡,𝑠)∈𝜇−1,𝜇+𝑏+1_ℕ𝜇−1×[0,𝑏+1]_ℕ0,𝐿∶0≤𝑠<𝑡−𝜇+1≤𝑏+2₂∶=(𝑡,𝑠)∈𝜇−1,𝜇+𝑏+1_ℕ𝜇−1×[0,𝑏+1]_ℕ0∶0≤𝑡−𝜇+1≤𝑠≤𝑏+2. (2.4)

Lemma 2.5. ([24]) The unique solution of the discrete fractional boundary value problem −Δ^𝜈𝑦(𝑡)=ℎ(𝑡+𝜈−1),𝑦(𝜈−2)=0=Δ𝑦(𝜈+𝑏) (2.5) is given by 𝑦(𝑡)∶=^𝑏+1_𝑠=0𝐺₁(𝑡,𝑠)ℎ(𝑠+𝜈−1), (2.6) where 1<𝜈≤2,𝐺₁(𝑡,𝑠) is given by 𝐺₁()1𝑡,𝑠∶=Γ(𝜈)Γ(𝑏+3)𝑡^𝜈−1Γ()()𝜈+𝑏+1𝜈+𝑏−𝑠−1^𝜈−2−(𝑡−𝑠−1)^𝜈−1,(𝑡,𝑠)∈𝑇₁,Γ(𝑏+3)𝑡^𝜈−1Γ()()𝜈+𝑏+1𝜈+𝑏−𝑠−1^𝜈−2,(𝑡,𝑠)∈𝑇₂. (2.7)

Remark 2.6. Let us note that in case we put 𝜇 replacing 𝜈 in Lemma 2.5, it follows that 𝑦(𝑡)∶=^𝑏+1_𝑠=0𝐺₂(𝑡,𝑠)ℎ𝑠+𝜇−1, (2.8) where 1<𝜇≤2,𝐺₂(𝑡,𝑠) is given by 𝐺₂()1𝑡,𝑠∶=Γ𝜇Γ(𝑏+3)𝑡^𝜇−1Γ𝜇+𝑏+1𝜇+𝑏−𝑠−1^𝜇−2−(𝑡−𝑠−1)^𝜇−1,(𝑡,𝑠)∈𝐿₁,Γ(𝑏+3)𝑡^𝜇−1Γ𝜇+𝑏+1𝜇+𝑏−𝑠−1^𝜇−2,(𝑡,𝑠)∈𝐿₂. (2.9)

Lemma 2.7. ([24]) The Green function 𝐺₁(𝑡,𝑠) given in Lemma 2.5 satisfies
(i)𝐺₁(𝑡,𝑠)≥0 for (𝑡,𝑠)∈[𝜈−1,𝜈+𝑏+1]_ℕ𝜈−1×[0,𝑏+1]_ℕ0(ii)max_{𝑡∈[𝜈−1,𝜈+𝑏+1]ℕ𝜈−1}𝐺₁(𝑡,𝑠)=𝐺₁(𝑠+𝜈−1,𝑠), for 𝑠∈[0,𝑏]_ℕ0(iii) there exists a number 𝛾₁∈(0,1) such thatmin_{)()]𝑡∈[(𝑏+𝜈/4,3𝑏+𝜈/4}𝐺₁(𝑡,𝑠)≥𝛾₁max_{[]𝑡∈𝜈−1,𝜈+𝑏+1ℕ𝜈−1}𝐺₁(𝑡,𝑠)=𝛾₁𝐺₁(𝑠+𝜈−1,𝑠),for𝑠∈[0,𝑏+1]_ℕ0. (2.10)
Similarly, 𝐺₂(𝑡,𝑠) satisfies(I)𝐺₂(𝑡,𝑠)≥0 for (𝑡,𝑠)∈[𝜇−1,𝜇+𝑏+1]_ℕ𝜇−1×[0,𝑏+1]_ℕ0(II)max_{𝑡∈[𝜇−1,𝜇+𝑏+1]ℕ𝜇−1}𝐺₂(𝑡,𝑠)=𝐺₂(𝑠+𝜇−1,𝑠), for 𝑠∈[0,𝑏]_ℕ0(III) there exists a number 𝛾₂∈(0,1) such thatmin_{𝑡∈(𝑏+𝜇)/4,3𝑏+𝜇/4}𝐺₂(𝑡,𝑠)≥𝛾₂max_{𝑡∈𝜇−1,𝜇+𝑏+1ℕ𝜇−1}𝐺₂(𝑡,𝑠)=𝛾₂𝐺₂𝑠+𝜇−1,𝑠,for𝑠∈[0,𝑏+1]_ℕ0. (2.11)

Let ℬ₁ and ℬ₂ represent the Banach space of all maps from [𝜈−1,𝜈+𝑏]_ℕ𝜈−1 into ℝ and [𝜇−1,𝜇+𝑏]_ℕ𝜇−1 into ℝ, respectively. And ‖⋅‖ is the usual maximum norm. Define𝒳∶=ℬ₁×ℬ₂. (2.12) By equipping 𝒳 with the norm𝑦₁,𝑦₂∶=𝑦₁+𝑦₂, (2.13) it follows that (𝒳,‖⋅‖) is a Banach space, see [26, 27].

Next, we wish to develop a representation for a solution of (1.4) as the fixed point of an appropriate operator on 𝒳. To accomplish this, we present some straightforward adaptations of results from [24] that will be of use here. Since the proofs of these adaptations are evident, we do not include them.

Now, consider the operator 𝒮∶𝒳→𝒳 defined by𝒮𝑦₁,𝑦₂𝑡₁,𝑡₂∶=𝒮₁𝑦₁,𝑦₂𝑡₁,𝒮₂𝑦₁,𝑦₂𝑡₂, (2.14) where we define 𝒮₁∶𝒳→ℬ₁ by𝒮₁𝑦₁,𝑦₂𝑡₁∶=^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑓𝑦₁(𝑠+𝜈−1),𝑦₂𝑠+𝜇−1, (2.15) and 𝒮₂∶𝒳→ℬ₂ by𝒮₂𝑦₁,𝑦₂𝑡₂∶=^𝑏+1_𝑠=0𝐺₂𝑡₂,𝑠𝑓𝑦₁(𝑠+𝜈−1),𝑦₂𝑠+𝜇−1. (2.16) We claim that whenever (𝑦₁,𝑦₂)∈𝒳 is a fixed point of the operator 𝒮, it follows that the pair of functions 𝑦₁(𝑡) and 𝑦₂(𝑡) are a solution to problem (1.4).

Lemma 2.8. Let 𝑓,𝑔∶ℝ×ℝ→ℝ. If (𝑦₁,𝑦₂)∈𝒳 is a fixed point of 𝒮, then (𝑦₁,𝑦₂) is a solution to problem (1.4).

Proof. Suppose that the operator 𝒮 has a fixed point, say (𝑦₁,𝑦₂)∈𝒳. Let (𝑡₁,𝑡₂)∈ℕ_𝜈−1×ℕ_𝜇−1, then we have 𝑦₁=𝒮₁𝑦₁,𝑦₂𝑡₁, (2.17) where 𝒮₁ is defined as in (2.15). It is easy to check that 𝒮₁(𝑦₁,𝑦₂)(𝜈−2)=0 and Δ𝒮₁𝑦₁,𝑦₂(𝜈+𝑏)=𝑆₁𝑦₁,𝑦₂(𝜈+𝑏+1)−𝑆₁𝑦₁,𝑦₂()=𝜈+𝑏_𝑏+1𝑠=0Γ(𝑏+3)(𝜈+𝑏+1)^𝜈−1Γ()()𝜈+𝑏+1𝜈+𝑏−𝑠−1^𝜈−2−(𝜈+𝑏−𝑠)^𝜈−1×𝑓𝑦₁(𝑠+𝜈−1),𝑦₂−𝑠+𝜇−1_𝑏+1𝑠=0Γ(𝑏+3)(𝜈+𝑏)^𝜈−1Γ()()𝜈+𝑏+1𝜈+𝑏−𝑠−1^𝜈−2−(𝜈+𝑏−𝑠−1)^𝜈−1×𝑓𝑦₁(𝑠+𝜈−1),𝑦₂=𝑠+𝜇−1_𝑏+1𝑠=0(𝜈+𝑏+1)(𝜈+𝑏−𝑠−1)^𝜈−2−(𝑏+2)(𝜈+𝑏−𝑠−1)^𝜈−2×𝑓𝑦₁(𝑠+𝜈−1),𝑦₂−𝑠+𝜇−1_𝑏+1𝑠=0(𝜈+𝑏−𝑠)Γ(𝜈+𝑏−𝑠)()Γ()−Γ()𝑏−𝑠+1𝑏−𝑠+1𝜈+𝑏−𝑠Γ()𝑦𝑏−𝑠+1×𝑓₁(𝑠+𝜈−1),𝑦₂=𝑠+𝜇−1_𝑏+1𝑠=0(𝜈−1)(𝜈+𝑏−𝑠−1)^𝜈−2−(𝜈−1)Γ(𝜈+𝑏−𝑠)()Γ()𝑦𝑏−𝑠+1𝑏−𝑠+1×𝑓₁(𝑠+𝜈−1),𝑦₂=𝑠+𝜇−1_𝑏+1𝑠=0()Γ()𝜈−1𝜈+𝑏−𝑠Γ()−()Γ()𝑏−𝑠+2𝜈−1𝜈+𝑏−𝑠Γ()𝑦𝑏−𝑠+2×𝑓₁(𝑠+𝜈−1),𝑦₂𝑠+𝜇−1=0, (2.18) so the boundary conditions are satisfied. Furthermore, we can check that 𝒮₁(𝑦₁,𝑦₂)(𝑡₁) satisfies the difference equation in (1.4). Since similar verifications may be made for 𝒮₂(𝑦₁,𝑦₂)(𝑡₂), the claim follows, which completes the proof.

3. Main Results

In this section, we will show that under certain conditions, problem (1.4) has at least one solution.

Let us now present the conditions that we will assume henceforth. We note that conditions (F1)-(F2) are similar to the conditions given by Henderson et al. [26]()),[()F1∶𝑓,𝑔∈𝐶([0,∞0,∞))F2∶lim_{𝑦1,𝑦2→(0+,0+)}𝑓𝑦₁,𝑦₂𝑦₁+𝑦₂=𝑓^∗,lim_{(𝑦1,𝑦2)→(0+,0+)}𝑔𝑦₁,𝑦₂𝑦₁+𝑦₂=𝑔^∗,lim_{𝑦1,𝑦2→()+∞,+∞}𝑓𝑦₁,𝑦₂𝑦₁+𝑦₂=𝑓^∗∗,lim_{(𝑦1,𝑦2)→(+∞,+∞)}𝑔𝑦₁,𝑦₂𝑦₁+𝑦₂=𝑔^∗∗, (3.1) satisfying^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗≤1+𝜀2,^𝑏+1_𝑠=0𝐺₂𝑠+𝜇−1,𝑠𝑔^∗≤1+𝜀2_𝑏+1𝑠=0,𝛾𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗∗≥1−𝜀2,^𝑏+1_𝑠=0𝛾𝐺₂𝑠+𝜇−1,𝑠𝑔^∗∗≥1−𝜀2, (3.2) for some 0<𝜀<min{𝑓^∗∗,𝑔^∗∗}.

Define the cone by𝒦∶=𝑦₁,𝑦₂∈𝒳∶𝑦₁,𝑦₂≥0,min_{𝑡1,𝑡2∈)()]×[(𝜈−1/4,3𝜈+𝑏/4𝜇−1/4,3𝜇+𝑏/4}×𝑦₁𝑡₁+𝑦₂𝑡₂≥𝛾𝑦₁,𝑦₂, (3.3) where 𝛾=min{𝛾₁,𝛾₂}, for 𝑖=1,2,𝛾_𝑖 is defined as in Lemma 2.7. Clearly, 𝒦⊆𝒳. In order to show that 𝒮 has a fixed point in 𝒦, we must first demonstrate that 𝒦 is invariant under 𝒮, that is, 𝒮∶𝒦→𝒦. We show this now.

Lemma 3.1. Assume that 𝑓,𝑔 are nonnegative functions. Let 𝒮∶𝒳→𝒳 be the operator defined as (2.14), then 𝒮∶𝒦→𝒦.

Proof. Suppose that (𝑦₁,𝑦₂)∈𝒦. We claim that min_{𝑡1,𝑡2∈[()]×(𝜈−1)/4,3𝜈+𝑏/4(𝜇−1)/4,3𝜇+𝑏/4}𝒮₁𝑦₁,𝑦₂𝑡₁+𝒮₂𝑦₁,𝑦₂𝑡₂≥𝛾𝒮𝑦₁,𝑦₂. (3.4) In fact, min_{𝑡1∈[()](𝜈−1)/4,3𝜈+𝑏/4}𝒮₁𝑦₁,𝑦₂𝑡₁≥min_{𝑡1∈[()](𝜈−1)/4,3𝜈+𝑏/4}^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑓𝑦₁(𝑠+𝜈−1),𝑦₂𝑠+𝜇−1≥𝛾₁^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓𝑦₁(𝑠+𝜈−1),𝑦₂𝑠+𝜇−1=𝛾₁max_{𝑡1∈[]𝜈−1,𝜈+𝑏ℕ𝜈−1}^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑓𝑦₁(𝑠+𝜈−1),𝑦₂𝑠+𝜇−1=𝛾₁𝒮₁𝑦₁,𝑦₂. (3.5) Similarly, we get min_{𝑡2∈𝜇−1/4,3𝜇+𝑏/4}𝒮₂𝑦₁,𝑦₂𝑡₂≥𝛾₂𝒮₂𝑦₁,𝑦₂. (3.6) Put 𝛾=min{𝛾₁,𝛾₂}. Consequently, from (3.5) and (3.6), we obtain min_{𝑡1,𝑡2∈)()][(𝜈−1/4,3𝜈+𝑏/4×[𝜇−1/4,3𝜇+𝑏/4]}𝒮₁𝑦₁,𝑦₂𝑡₁+𝒮₂𝑦₁,𝑦₂𝑡₂≥min_{𝑡1∈)()][(𝜈−1/4,3𝜈+𝑏/4}𝒮₁𝑦₁,𝑦₂𝑡₁+min_{𝑡2∈𝜇−1/4,3𝜇+𝑏/4}𝒮₂𝑦₁,𝑦₂𝑡₂≥𝛾₁𝒮₁𝑦₁,𝑦₂+𝛾₂𝒮₂𝑦₁,𝑦₂≥𝛾𝒮₁𝑦₁,𝑦₂+𝒮₂𝑦₁,𝑦₂=𝛾𝒮𝑦₁,𝑦₂. (3.7) So, for (𝑦₁,𝑦₂)∈𝒦, we find min_{𝑡1,𝑡2()()∈[𝜈−1/4,3𝜈+𝑏/4]×[𝜇−1/4,3𝜇+𝑏/4]}𝒮₁𝑦₁,𝑦₂𝑡₁+𝒮₂𝑦₁,𝑦₂𝑡₂≥𝛾𝒮𝑦₁,𝑦₂. (3.8) Therefore, from (3.8), whenever we get (𝑦₁,𝑦₂)∈𝒦, it follows that 𝒮(𝑦₁,𝑦₂)∈𝒦. This completes the proof.

We next recall the Guo-Krasnosel’kii fixed-point theorem and Schauder fixed-point theorem, which we will use to prove the main results.

Lemma 3.2. ([21]) Let ℬ be a Banach space, and let 𝒦⊆ℬ be a cone. Assume that Ω₁ and Ω₂ are open subsets contained in ℬ such that 0∈Ω₁ and Ω₁⊆Ω₂. Assume, further, that 𝑇∶𝒦∩(Ω₂⧵Ω₁)→𝒦 is a completely continuous operator. If either(1)‖𝑇𝑦‖≤‖𝑦‖ for 𝑦∈𝒦∩𝜕Ω₁ and ‖𝑇𝑦‖≥‖𝑦‖ for 𝑦∈𝒦∩𝜕Ω₂, (2)or ‖𝑇𝑦‖≥‖𝑦‖ for 𝑦∈𝒦∩𝜕Ω₁ and ‖𝑇𝑦‖≤‖𝑦‖ for 𝑦∈𝒦∩𝜕Ω₂,then 𝑇 has at least one fixed point in 𝒦∩(Ω₂⧵Ω₁).

Lemma 3.3. ([22]) (Schauder fixed-point theorem) Suppose that 𝒳 is a Banach space. Let 𝒦 be a bounded closed-convex set of 𝒳, and let 𝑇∶𝒦→𝒦 be a completely continuous operator, then 𝑇 has at least one fixed point in 𝒦.

Theorem 3.4. Suppose that conditions (F1)-(F2) hold, then problem (1.4) has at least one solution.

Proof. From Lemma 3.1, we know that 𝒮∶𝒦→𝒦. Note that 𝒮 is a summation operator on a discrete finite set. Hence, 𝒮 is a completely continuous operator.
By conditions (F1) and (F2), for 𝜀 given in (F2), there exists some constant 𝑟₁>0, such that 𝑓𝑦₁,𝑦₂≤𝑓^∗+𝜀𝑦₁+𝑦₂,𝑔𝑦₁,𝑦₂≤𝑔^∗+𝜀𝑦₁+𝑦₂, (3.9)(3.10) whenever ‖(𝑦₁,𝑦₂)‖<𝑟₁.
Setting Ω₁∶=𝑦₁,𝑦₂∈𝒳∶𝑦₁,𝑦₂<𝑟₁, (3.11) Then, for (𝑦₁,𝑦₂)∈𝒦∩𝜕Ω₁, by Lemma 2.7, (F2) and (3.9), we find that 𝒮₁𝑦₁,𝑦₂=max_{𝑡1∈[]𝜈−1,𝜈+𝑏ℕ𝜈−1}^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑓𝑦₁(𝑠+𝜈−1),𝑦₂≤𝑠+𝜇−1_𝑏+1𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗+𝜀𝑦₁+𝑦₂≤𝑦₁,𝑦₂^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗≤1+𝜀2𝑟₁=12𝑦₁,𝑦₂. (3.12) Similar, by (3.10), it can be shown that 𝒮₂𝑦₁,𝑦₂≤12𝑦₁,𝑦₂. (3.13) Thus, putting (3.12) and (3.13) together, for (𝑦₁,𝑦₂)∈𝒦∩𝜕Ω₁, we have 𝒮𝑦₁,𝑦₂≤12𝑦₁,𝑦₂+12𝑦₁,𝑦₂=𝑦₁,𝑦₂. (3.14)
Now, for the above 𝜀, by (F2), we can find a constant 𝑟^∗∗>0 such that 𝑓𝑦₁,𝑦₂≥𝑓^∗∗−𝜀𝑦₁+𝑦₂,𝑔𝑦₁,𝑦₂≥𝑔^∗∗−𝜀𝑦₁+𝑦₂, (3.15)(3.16) whenever ‖𝑦₁+𝑦₂‖≥𝑟^∗∗. Let 12𝑟^∗∗≤𝑟₂≤min^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗∗−𝜀𝛾𝑟^∗∗,^𝑏+1_𝑠=0𝐺₂𝑠+𝜇−1,𝑠𝑔^∗∗−𝜀𝛾𝑟^∗∗. (3.17) Moreover, put Ω₂∶=𝑦₁,𝑦₂∈𝒳∶𝑦₁,𝑦₂<2𝑟₂. (3.18) If (𝑦₁,𝑦₂)∈𝒦∩𝜕Ω₂, then 𝑦₁𝑡₁+𝑦₂𝑡₂≥min_{𝑡1,𝑡2∈)()][(𝜈−1/4,3𝜈+𝑏/4×[(𝜇−1)/4,3(𝜇+𝑏)/4]}𝑦₁𝑡₁+𝑦₂𝑡₂≥𝛾𝑦₁,𝑦₂. (3.19) Thus, from (3.15) and (3.19), we obtain 𝒮₁𝑦₁,𝑦₂=max_{𝑡1∈[𝜈−1,𝜈+𝑏]ℕ𝜈−1}^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑓𝑦₁(𝑠+𝜈−1),𝑦₂≥𝑠+𝜇−1_𝑏+1𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗∗−𝜀𝑦₁+𝑦₂≥^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗∗−𝜀𝛾𝑦₁,𝑦₂≥𝑟₂=12𝑦₁,𝑦₂. (3.20) Similarly, by (3.16) and (3.19), we have 𝒮₂𝑦₁,𝑦₂≥12𝑦₁,𝑦₂. (3.21) So, from (3.20) and (3.21), we get 𝒮𝑦₁,𝑦₂≥𝑦₁,𝑦₂, (3.22) whenever (𝑦₁,𝑦₂)∈𝒦∩𝜕Ω₂.
Thus, by (3.14) and (3.22), we get that all of the hypotheses of Lemma 3.2 are satisfied. Consequently, we conclude that 𝒮 has a fixed point, say (𝑦^∗₁,𝑦^∗₂)∈𝒦. Then the theorem is proved.

Theorem 3.5. Let 𝑓,𝑔∶ℝ×ℝ→ℝ be continuous functions. Suppose that the following conditions are satisfied:(i)there exist nonnegative functions 𝑎,ℎ,𝑝,𝑞∈𝐶(ℝ,ℝ⁺) such that𝑓𝑦₁,𝑦₂≤𝑎𝑦₁+𝑝𝑦₂,𝑔𝑦₁,𝑦₂≤ℎ𝑦₁+𝑞𝑦₂, (3.23)(ii)Consider the following:lim_{|𝑦|→+∞}𝑝𝑦𝑦<𝐴,lim_{|𝑦|→+∞}𝑞𝑦𝑦<𝐵, (3.24) where 𝐴=(2max_{𝑡∈[𝜈−1,𝜈+𝑏+1]}∑^𝑏+1_𝑠=0𝐺₁(𝑡,𝑠))⁻¹,𝐵=(2max_{𝑡∈[𝜇−1,𝜇+𝑏+1]}∑^𝑏+1_𝑠=0𝐺₂(𝑡,𝑠))⁻¹, then boundary value problem (1.4) has at least one solution (𝑦₁^∗∗,𝑦₂^∗∗).

Proof. Let 𝜀₁=(1/2)(𝐴−lim_{|𝑦|→+∞}𝑝(𝑦)/|𝑦|). By (ii), there exists a 𝑐₁>0, such that 𝑝(𝑦₂)<(𝐴−𝜀₁)|𝑦₂|, for |𝑦₂|≥𝑐₁. Set 𝑀=max{𝑝(𝑦₂)∶|𝑦₂|≤𝑐₁}, then there exists a 𝑐₂>𝑐₁, such that 𝑀/𝑐₂≤𝐴−𝜀₁, so we have 𝑝(𝑦₂)<(𝐴−𝜀₁)𝑐₂, for |𝑦₂|≤𝑐₂.
Let 𝜀₂=(1/2)(𝐵−lim_{|𝑦|→+∞}𝑞(𝑦)/|𝑦|). In the same way, there exists a constant 𝑐₃>0, such that 𝑞(𝑦₂)<(𝐵−𝜀₂)𝑐₃, |𝑦₂|≤𝑐₃. Define 𝑒₁=max_{𝑡1∈[]𝜈−1,𝜈+𝑏+1}^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑎𝑦₁,𝑒₂=max_{𝑡2∈𝜇−1,𝜇+𝑏+1}^𝑏+1_𝑠=0𝐺₂𝑡₂,𝑠ℎ𝑦₁,𝑐=max𝑐₂,𝑐₃,2𝐴𝜀₁𝑒₁,2𝐵𝜀₂𝑒₂,𝑈=𝑦₁,𝑦₂∣𝑦₁,𝑦₂∈𝒳,𝑦₁,𝑦₂≤𝑐,𝑡₁,𝑡₂∈[𝜈−1,𝜈+𝑏+1]_ℕ𝜈−1×𝜇−1,𝜇+𝑏+1_ℕ𝜇−1. (3.25) Then 𝑈 is a bounded closed-convex set of 𝒳×𝒳, and for any (𝑦₁,𝑦₂)∈𝑈, we have 𝑝(𝑦₂)<(𝐴−𝜀₁)𝑐,𝑞(𝑦₂)<(𝐵−𝜀₂)𝑐. Next, we prove 𝒮∶𝑈→𝑈, where 𝒮 is defined as (2.14).
For (𝑦₁,𝑦₂)∈𝑈, from (i) and (sii), we obtain 𝒮₁𝑦₁,𝑦₂𝑡₁=^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑓𝑦₁,𝑦₂≤^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑎𝑦₁+^𝑏+1_𝑠=0𝐺₁𝑡₁,𝑠𝑝𝑦₂≤𝑒₁+𝐴−𝜀₁𝑐^𝑏+1_𝑠=0𝐺₁𝑡₁≤𝜀,𝑠₁12𝐴𝑐+2𝜀1−¹𝐴1𝑐=2𝑐, (3.26) similarly, 𝒮₂𝑦₁,𝑦₂𝑡₂≤𝜀₂12𝐵𝑐+2𝜀1−²𝐵1𝑐=2𝑐. (3.27) Thus, from (3.26) and (3.27), we have ‖𝒮(𝑦₁,𝑦₂)‖≤𝑐. Then, 𝒮∶𝑈→𝑈.
Note that 𝒮 is a summation operator on a discrete finite set. Hence, 𝒮 is trivially completely continuous. By Schauder fixed-point theorem, the boundary problem value problem (1.4) has at least one solution, say (𝑦₁^∗∗,𝑦₂^∗∗). This completes the proof.

4. Examples

In this section, we will present some examples to illustrate the main results.

Example 4.1. Suppose that 𝜈=3/2, 𝜇=5/4, 𝜂₁=1/1000, 𝜂₂=10/𝛾, 𝛾=min{𝛾₁,𝛾₂}, and 𝛾₁, 𝛾₂ are defined as in Lemma 2.7, 𝑏=2. Let 𝑓(𝑦₁,𝑦₂)∶=((1/1000𝑒^−𝑦2)+(1000/𝛾𝑒^1/𝑦1))(𝑦₁+𝑦₂) and 𝑔(𝑦)∶=((1/2000𝑒^−𝑦1)+(1000/𝛾𝑒^1/𝑦2))(𝑦₁+𝑦₂), then (1.4) becomes −Δ^3/2𝑦₁(𝑡)=𝑓𝑦₁1𝑡+2,𝑦₂1𝑡+4,for𝑡∈[0,3]_ℕ0,−Δ^5/4𝑦₂(𝑡)=𝑔𝑦₁1𝑡+2,𝑦₂1𝑡+4,for𝑡∈[0,3]_ℕ0,𝑦₁−12=Δ𝑦₁72𝑦=0,₂−34=Δ𝑦₂134=0. (4.1) In this case, lim_{(𝑦1,𝑦2)→(0+,0+)}𝑓𝑦₁,𝑦₂𝑦₁+𝑦₂=11000=𝑓^∗,lim_{(𝑦1,𝑦2)→(0+,0+)}𝑔𝑦₁,𝑦₂𝑦₁+𝑦₂=12000=𝑔^∗,lim_{(𝑦1,𝑦2)→(+∞,+∞)}𝑓𝑦₁,𝑦₂𝑦₁+𝑦₂=1000𝛾=𝑓^∗∗,lim_{(𝑦1,𝑦2)→(+∞,+∞)}𝑔𝑦₁,𝑦₂𝑦₁+𝑦₂=1000𝛾=𝑔^∗∗, (4.2)^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)=^𝑏+1_𝑠=0Γ(𝑏+3)(𝑠+𝜈−1)^𝜈−1Γ()()𝜈+𝑏+1𝜈+𝑏−𝑠−1^𝜈−2=^𝑏+1_𝑠=0Γ(𝑏+3)Γ(𝑠+𝜈)Γ(𝜈+𝑏−𝑠)Γ()Γ()Γ()=𝜈+𝑏+1𝑠+1𝑏−𝑠+2_𝑏+1𝑠=0(𝑏+2)(𝑏+1)⋯(𝑏−𝑠+2)(𝑠+𝜈−1)⋯(𝑠+1)()⋯()≤𝜈+𝑏)(𝜈+𝑏−1𝜈+𝑏−𝑠_𝑏+1𝑠=0()⋯()≤()7𝑠+𝜈−1𝑠+1𝑏+2)(𝑏+𝜈)(𝑏+2=4×2×4=56. (4.3) Similarly, we get ^𝑏+1_𝑠=0𝐺₂𝑠+𝜇−1,𝑠≤(𝑏+2)𝑏+𝜇(𝑏+2)=52. (4.4) Thus, ^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗+𝜂₁1≤56×+11000≤110002_𝑏+1𝑠=0,𝐺₂(𝑠+𝜈−1,𝑠)𝑔^∗+𝜂₁1≤52×+12000≤110002. (4.5) On the other hand, ^𝑏+1_𝑠=0𝐺₁(𝑠+𝜈−1,𝑠)=^𝑏+1_𝑠=0(𝑏+2)(𝑏+1)⋯(𝑏−𝑠+2)(𝑠+𝜈−1)⋯(𝑠+1)()⋯()≥𝜈+𝑏)(𝜈+𝑏−1𝜈+𝑏−𝑠_𝑏+1𝑠=0()⋯()⋯()≥()1𝑏+2)(𝑏+1𝑏−𝑠+2)(𝑠+𝜈−1𝑠+1𝑏+2)(𝑏+2)(𝜈−1=2×2×2=2, (4.6) similarly, ^𝑏+1_𝑠=0𝐺₂≥()1𝑠+𝜇−1,𝑠𝑏+2)(𝑏+2𝜇−1≥4×4×4=1. (4.7) Then, ^𝑏+1_𝑠=0𝛾𝐺₁(𝑠+𝜈−1,𝑠)𝑓^∗∗−𝜂₂≥2𝛾×1000𝛾−10𝛾1=1980≥2_𝑏+1𝑠=0,𝛾𝐺₂𝑠+𝜇−1,𝑠𝑔^∗∗−𝜂₂≥𝛾×1000𝛾−10𝛾1=990≥2. (4.8)

Thus, the conditions of Theorem 3.4 are satisfied. So, we obtain that the problem (4.1) has at least one solution.

Example 4.2. Suppose that 𝜈=5/3, 𝜇=7/4,and𝑏=3. Let 𝑓(𝑦₁,𝑦₂)∶=𝑦₁+(𝐴/4)𝑦₂𝑒^−|y2|, 𝑔(𝑦₁,𝑦₂)∶=𝑦₁+(𝐵/5)𝑦₂𝑒^−|𝑦2|, 𝑎(𝑦₁)=|𝑦₁|,ℎ(𝑦₁)=2|𝑦₁|,𝑝(𝑦₂)=(𝐴/3)|𝑦₂|(𝑒^−|𝑦2|+1), and 𝑞(𝑦₂)=(𝐵/4)|𝑦₂|(𝑒^−|𝑦2|+1),𝑦₁,𝑦₂∈ℝ⁺, where 𝐴=2max_{𝑡∈[𝜈−1,𝜈+𝑏+1]}^𝑏+1_𝑠=0𝐺₁(𝑡,𝑠)⁻¹,𝐵=2max_{𝑡∈[𝜇−1,𝜇+𝑏+1]}^𝑏+1_𝑠=0𝐺₂(𝑡,𝑠)⁻¹. (4.9) Then problem (1.4) is −Δ^5/3𝑦₁(𝑡)=𝑦₁2𝑡+3+𝐴4𝑦₂3𝑡+4𝑒^{−|𝑦2(𝑡+(3/4))|},for𝑡∈[0,4]_ℕ0,−Δ^7/4𝑦₁(𝑡)=𝑦₁2𝑡+3+𝐵5𝑦₂3𝑡+4𝑒^{−|𝑦2(𝑡+(3/4))|},for𝑡∈[0,4]_ℕ0,𝑦₁−13=Δ𝑦₁143𝑦=0,₂−14=Δ𝑦₂194=0. (4.10)

It is clear that 𝑓,𝑔,𝑎,ℎ,𝑝,and𝑞 satisfy the conditions of Theorem 3.5. Thus, by Theorem 3.5, we deduce that problem (4.10) has at least one solution.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original paper. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003), and the Natural Science Foundation of Educational Department of Shandong Province (J11LA01).