Existence Results and the Monotone Iterative Technique for Nonlinear Fractional Differential Systems with Coupled Four-Point Boundary Value Problems

Abstract

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.

1. Introduction

This paper discusses the coupled four-point boundary value problems 𝐷^𝑝𝑥(𝑡)(𝑡)(𝑡)()𝐷+𝑓𝑡,𝑥,𝑦=0,𝑡∈0,1,1<𝑝≤2,_𝑞𝑦(𝑡)(𝑡)(𝑡)()𝑥(0)(0)(1)(𝜉)(1)𝜂,+𝑔𝑡,𝑥,𝑦=0,𝑡∈0,1,1<𝑞≤2,=𝑦=0,𝑥=𝑎𝑦,𝑦=𝑏𝑥 (1) where 𝑓 and 𝑔∶(0,1)×ℝ×ℝ→ℝ are continuous, 𝜉,𝜂∈(0,1), 𝑎,𝑏>0 with 𝑎𝑏<1, and 𝐷^𝑝𝑥 denotes the Caputo fractional derivative of 𝑥 with 1<𝑝≤2 defined by 𝐷^𝑝𝑥(𝑡)=𝐼^2−𝑝𝑥^′′(𝑡)=1Γ𝑝^𝑡₀(𝑡−𝑠)^1−𝑝𝑥^′′(𝑠)𝑑𝑠. (2)𝐼^2−𝑝 is the Riemann-Liouville fractional integral of order 2−𝑝; see [1–4].

It is well known that 𝐼^2−𝑝𝐷^2−𝑝𝑥(𝑡)=𝑥(𝑡)−¹_𝑘=0𝑥^(𝑘)0⁺𝑡𝑘!^𝑘,𝐷^2−𝑝𝐼^2−𝑝𝑥(𝑡)=𝑥(𝑡). (3)

Fractional differential equation’s modeling capabilities in physics, chemistry, economics, and other fields, over the last few decades, have resulted in the rapid development of the theory of fractional differential equations; we refer the reader to the books [1–4]. On the other hand, the study of systems involving coupled boundary value problems is also important as such systems occur in the study of reaction-diffusion equations and Sturm-Liouville problems, for example, [5–16]. In [17–25], using the upper and lower solutions method and the monotone iterative method, the authors considered the existence of solutions of initial value problems and boundary value problems for fractional differential equations. But, as far as we know, there have been few papers which have considered the existence of solutions of (1) by means of the monotone iterative method.

Motivated by the above papers, in this paper, we will investigate the existence of a solution of problem (1) by means of the upper and lower solutions method and the monotone iterative method. The novelty of this paper is that Caputo-type fractional differential systems involve two different fractional derivatives 𝐷^𝑝 and 𝐷^𝑞 and that the nonlinear terms 𝑓, 𝑔 in the systems (1) involve unknown functions 𝑥(𝑡) and 𝑦(𝑡).

In the following, we denote 𝐸=𝐶²([0,1],ℝ),𝐸₁=𝐶([0,1],(0,+∞)). (4)

2. Preliminaries and Lemmas

In this section, we introduce the definition of the lower and upper solutions and present some existence and uniqueness results for linear problems together with comparison results for differential systems (1) which will be needed in the next section.

Throughout this paper, we always assume that the following condition is satisfied:

(𝐻₁)   0<𝑎𝑏<1.

Definition 1. (𝑢₀,𝑣₀)∈𝐸×𝐸 is called a lower system of solutions of differential system (1) if 𝐷^𝑝𝑢₀(𝑡)+𝑓𝑡,𝑢₀(𝑡),𝑣₀(𝑡)(),𝐷≥0,𝑡∈0,1_𝑞𝑣₀(𝑡)+𝑔𝑡,𝑢₀(𝑡),𝑣₀(𝑡)(),𝑢≥0,𝑡∈0,1₀(0)≤0,𝑣₀(0)𝑢≤0,₀(1)≤𝑎𝑣₀(𝜉),𝑣₀(1)≤𝑏𝑢₀𝜂. (5) Analogously, (𝛼₀,𝛽₀)∈𝐸×𝐸 is called an upper system of solutions if it satisfies the reversed inequalities.

If 𝑢₀(𝑡)≤𝛼₀(𝑡) and 𝑣₀(𝑡)≤𝛽₀(𝑡), 𝑡∈[0,1], we say that (𝑢₀,𝑣₀) and (𝛼₀,𝛽₀) are ordered lower and upper system of solutions of (1). In what follows, we assume that (𝑢₀,𝑣₀) and (𝛼₀,𝛽₀) are ordered lower and upper system of solutions of (1) and define the sector Ω=𝑥,𝑦∈𝐸×𝐸∶𝑢₀(𝑡),𝑣₀(𝑡)≤𝑥(𝑡)(𝑡)≤𝛼,𝑦₀(𝑡),𝛽₀(𝑡),𝑡∈[0,1], (6) where the vectorial inequalities mean that the same inequalities hold between their corresponding components.

Lemma 2 (see [17]). Let 𝑧(𝑡)∈𝐸 and 𝑟(𝑡)∈𝐸₁. If 𝑧(𝑡) satisfies the inequality −𝐷^𝑝𝑧(𝑡)(𝑡)𝑧(𝑡)(](),𝑧(0)(1)≤−𝑟,𝑝∈1,2,𝑡∈0,1≤0,𝑧≤0, (7) then 𝑧(𝑡)≤0, ∀𝑡∈[0,1].

We have the following important result.

Lemma 3 (comparison theorem). Let 𝑀(𝑡),𝑁(𝑡)∈𝐸₁ be given. Assume that 𝑥(𝑡),𝑦(𝑡) satisfy −𝐷^𝑝𝑥(𝑡)≤−𝑀(𝑡)𝑥(𝑡),𝑡∈(0,1),−𝐷^𝑞𝑦(𝑡)(𝑡)𝑦(𝑡)(),𝑥(0)(0)𝑥(1)(𝜉)(1)𝜂.≤−𝑁,𝑡∈0,1≤0,𝑦≤0,≤𝑎𝑦,𝑦≤𝑏𝑥 (8) Then 𝑥(𝑡)≤0, 𝑦(𝑡)≤0,  ∀𝑡∈[0,1].

Proof. Suppose the contrary. By Lemma 2, We consider the following three possible cases.
Case  1. Consider 𝑥(1)≤0 and 𝑦(1)>0. By Lemma 2, 𝑥(𝑡)≤0, ∀𝑡∈[0,1]. Then 𝑦(1)≤𝑏𝑥(𝜂)≤0 which contradicts 𝑦(1)>0.
Case  2. Consider 𝑦(1)≤0 and 𝑥(1)>0. By Lemma 2, 𝑦(𝑡)≤0, ∀𝑡∈[0,1]. Then 𝑥(1)≤𝑎𝑦(𝜉)≤0 which contradicts 𝑥(1)>0.
Case  3. Consider 𝑥(1)>0 and 𝑦(1)>0. By Lemma 2, we have 𝑥(1)=max_𝑡∈[0,1]𝑥(𝑡)>0 and 𝑦(1)=max_𝑡∈[0,1]𝑦(𝑡)>0. We only prove that 𝑥(1)=max_𝑡∈[0,1]𝑥(𝑡)>0. If not, 𝑥(𝑡) has a local positive maximum at some 𝑡₀∈(0,1) such that 𝑥(𝑡₀)=max_𝑡∈[0,1]𝑥(𝑡)>0. Then, by Theorem 2.1 in [21], we have the fact that the Caputo derivative of the function 𝑥 is nonpositive at the point 𝑡₀. Thus, 0≤−𝐷^𝑝𝑥𝑡₀≤−𝑀𝑡₀𝑥𝑡₀<0, (9) which is a contradiction. Furthermore, considering the boundary condition 𝑦(1)≤𝑏𝑥(𝜂), there exists 𝑡₁∈[0,𝜂) such that 𝑥(𝑡)≤0,𝑡∈0,𝑡₁;𝑥(𝑡)≥0,𝑡∈𝑡₁,1. (10) A similar proof, for 𝑦(𝑡), gives us that there exists 𝑡₂∈[0,𝜉) such that 𝑦(𝑡)≤0,𝑡∈0,𝑡₂;𝑦(𝑡)≥0,𝑡∈𝑡₂,1. (11) It follows from (10) and (11) that 𝑥(1)≤𝑎𝑦(𝜉)≤𝑎𝑦(1)≤𝑎𝑏𝑥𝜂≤𝑎𝑏𝑥(1), (12) which implies that 𝑎𝑏≥1, a contradiction. Hence 𝑥(𝑡)≤0,  𝑦(𝑡)≤0,  ∀𝑡∈[0,1].

Corollary 4. Let 𝑀(𝑡),𝑁(𝑡)∈𝐸₁ be given. Assume that 𝑥(𝑡),𝑦(𝑡) satisfy −𝐷^𝑝𝑥(𝑡)=−𝑀(𝑡)𝑥(𝑡),𝑡∈(0,1),−𝐷^𝑞𝑦(𝑡)(𝑡)𝑦(𝑡)(),𝑥(0)(0)𝑥(1)(𝜉)(1)𝜂.=−𝑁,𝑡∈0,1=0,𝑦=0,=𝑎𝑦,𝑦=𝑏𝑥 (13) Then 𝑥(𝑡)=𝑦(𝑡)=0, ∀𝑡∈[0,1].

Lemma 5. Let 𝜌, 𝜎∈𝐶[0,1], then the linear differential system with coupled four-point boundary value problem 𝐷^𝑝𝑥(𝑡)(𝑡)()𝐷+𝜌=0,𝑡∈0,1,1<𝑝≤2,_𝑞𝑦(𝑡)(𝑡)()𝑥(0)(0)(1)(𝜉)(1)𝜂+𝜎=0,𝑡∈0,1,1<𝑞≤2,=𝑦=0,𝑥=𝑎𝑦,𝑦=𝑏𝑥 (14) has integral representation 𝑥(𝑡)=¹₀𝐺₁(𝑡,𝑠)𝜌(𝑠)𝑑𝑠+¹₀𝐻₁()𝜎(𝑠)𝑦(𝑡)=𝑡,𝑠𝑑𝑠,₁₀𝐺₂(𝑡,𝑠)𝜎(𝑠)𝑑𝑠+¹₀𝐻₂(𝑡,𝑠)𝜌(𝑠)𝑑𝑠, (15) where 𝐺₁(𝑡,𝑠)=𝐺_𝑝(𝑡,𝑠)+𝑎𝑏𝜉𝑡𝐺1−𝑎𝑏𝜉𝜂^𝑝,𝐻𝜂,𝑠₁(𝑡,𝑠)=𝑎𝑡𝐺1−𝑎𝑏𝜉𝜂^𝑞(),𝐺𝜉,𝑠₂(𝑡,𝑠)=𝐺_𝑞(𝑡,𝑠)+𝑎𝑏𝜂𝑡𝐺1−𝑎𝑏𝜉𝜂^𝑞(),𝐻𝜉,𝑠₂(𝑡,𝑠)=𝑏𝑡𝐺1−𝑎𝑏𝜉𝜂^𝑝,𝐺𝜂,𝑠_𝑝()=𝑡()𝑡,𝑠1−𝑠^𝑝−1−(𝑡−𝑠)^𝑝−1Γ𝑝𝑡(),0≤𝑠≤𝑡≤1,1−𝑠^𝑝−1Γ𝑝,0≤𝑡≤𝑠≤1. (16)

Proof. It follows from [21] that (14) is equivalent to the system of integral equations 𝑥(𝑡)=𝑥(1)𝑡+¹₀𝐺_𝑝()𝜌(𝑠)[],𝑦(𝑡)(1)𝑡,𝑠𝑑𝑠,𝑡∈0,1=𝑦𝑡+¹₀𝐺_𝑞(𝑡,𝑠)𝜎(𝑠)𝑑𝑠,𝑡∈[0,1]. (17) By coupled four-point boundary value conditions of problem (14), we have 𝑦(1)=𝑏𝑥𝜂=𝑏𝜂𝑥(1)+𝑏¹₀𝐺_𝑝𝜂,𝑠𝜌(𝑠)𝑑𝑠,𝑥(1)=𝑎𝑦(𝜉)=𝑎𝜉𝑦(1)+𝑎¹₀𝐺_𝑞(𝜉,𝑠)𝜎(𝑠)𝑑𝑠. (18)(19) After simple computation, we get 𝑥(1)=𝑎𝑏𝜉1−𝑎𝑏𝜉𝜂¹₀𝐺_𝑝𝜌(𝑠)+𝑎𝜂,𝑠𝑑𝑠1−𝑎𝑏𝜉𝜂¹₀𝐺_𝑞(𝜉,𝑠)𝜎(𝑠)𝑑𝑠,𝑦(1)=𝑎𝑏𝜂1−𝑎𝑏𝜉𝜂¹₀𝐺_𝑞()𝜎(𝑠)+𝑏𝜉,𝑠𝑑𝑠1−𝑎𝑏𝜉𝜂¹₀𝐺_𝑝𝜂,𝑠𝜌(𝑠)𝑑𝑠. (20)(21) Substituting (20) into (18) and (21) into (19), respectively, we obtain the desired results.

Now we enunciate the following existence and uniqueness results for differential system: 𝐷^𝑝𝑥(𝑡)(𝑡)𝑥(𝑡)(𝑡)()𝐷−𝑀+𝜌=0,𝑡∈0,1,1<𝑝≤2,_𝑞𝑦(𝑡)(𝑡)𝑦(𝑡)(𝑡)()𝑥(0)(0)(1)(𝜉)(1)𝜂,−𝑁+𝜎=0,𝑡∈0,1,1<𝑞≤2,=𝑦=0,𝑥=𝑎𝑦,𝑦=𝑏𝑥 (22) where 𝑀, 𝑁∈𝐸₁.

Lemma 6. Let 𝑀, 𝑁∈𝐸₁. Then differential system (22) has a unique solution.

Proof. Indeed, by Lemma 5, differential system (22) is equivalent to the operator equation 𝑥,𝑦=𝑇𝑥,𝑦+𝜌,𝜎, (23) where 𝑇(𝑡)=−𝑥,𝑦₁₀𝐺₁(𝑡,𝑠)𝑀(𝑠)𝑥(𝑠)𝑑𝑠−¹₀𝐻₁()𝑁(𝑠)𝑦(𝑠)−𝑡,𝑠𝑑𝑠,₁₀𝐺₂()𝑁(𝑠)𝑦(𝑠𝑠)−𝑡,𝑠)(𝑑𝑠₁₀𝐻₂()𝑀(𝑠)𝑥(𝑠),𝜌(𝑡)=𝑡,𝑠𝑑𝑠₁₀𝐺₁(𝑡,𝑠)𝜌(𝑠)𝑑𝑠+¹₀𝐻₁()𝜎(𝑠)𝜎(𝑡)=𝑡,𝑠𝑑𝑠,₁₀𝐺₂(𝑡,𝑠)𝜎(𝑠)𝑑𝑠+¹₀𝐻₂(𝑡,𝑠)𝜌(𝑠)𝑑𝑠. (24) We apply the Fredholm theorem to find the unique solution of differential system (22). By using standard arguments, we can easily show that 𝑇∶𝐶[0,1]×𝐶[0,1]→𝐶[0,1]×𝐶[0,1] is linear completely continuous. Also, by Corollary 4, the operator equation (𝑥,𝑦)=𝑇(𝑥,𝑦) has only the zero solution. Thus, for given (𝜌,𝜎)∈𝐶[0,1]×𝐶[0,1], operator equation (23) has a unique solution in 𝐶[0,1]×𝐶[0,1], by the Fredholm theorem. This ends the proof.

3. Main Results

In this section, we prove the existence of extremal solutions of differential system (1).

Theorem 7. Assume that 𝑓∈𝐶([0,1]×ℝ×ℝ,ℝ), 𝑔∈𝐶([0,1]×ℝ×ℝ,ℝ). Let (𝑢₀,𝑣₀) and (𝛼₀,𝛽₀) be ordered lower and upper system of solutions of (1). In addition, we assume that
(𝐻₂)  𝑓(𝑡,𝑥,𝑦) is nondecreasing in 𝑦 and there exists 𝑀(𝑡)∈𝐸₁ such that 𝑓𝑡,𝑥₁,𝑦−𝑓𝑡,𝑥₂,𝑦≥−𝑀(𝑡)𝑥₁−𝑥₂, (25) where 𝑢₀(𝑡)≤𝑥₂≤𝑥₁≤𝛼₀(𝑡),  𝑣₀(𝑡)≤𝑦≤𝛽₀(𝑡);
(𝐻₃)  𝑔(𝑡,𝑥,𝑦) is nondecreasing in 𝑥 and there exists 𝑁(𝑡)∈𝐸₁ such that 𝑔𝑡,𝑥,𝑦₁−𝑔𝑡,𝑥,𝑦₂≥−𝑁(𝑡)𝑦₁−𝑦₂, (26) where 𝑣₀(𝑡)≤𝑦₂≤𝑦₁≤𝛽₀(𝑡), 𝑢₀(𝑡)≤𝑥≤𝛼₀(𝑡).

Then differential system (1) has extremal solutions in the section Ω.

Proof. Let us define two sequences {(𝑢_𝑛,𝑣_𝑛),(𝛼_𝑛,𝛽_𝑛)} by relations 𝐷^𝑝𝑢_𝑛+1(𝑡)−𝑀(𝑡)𝑢_𝑛+1(𝑡)+𝑓𝑡,𝑢_𝑛(𝑡),𝑣_𝑛(𝑡)+𝑀(𝑡)𝑢_𝑛(𝑡)(),𝐷=0,𝑡∈0,1_𝑞𝑣_𝑛+1(𝑡)−𝑁(𝑡)𝑣_𝑛+1(𝑡)+𝑔𝑡,𝑢_𝑛(𝑡),𝑣_𝑛(𝑡)+𝑁(𝑡)𝑣_𝑛(𝑡)(),𝑢=0,𝑡∈0,1_𝑛+1(0)=𝑣_𝑛+1(0)=0,𝑢_𝑛+1(1)=𝑎𝑣_𝑛+1(𝜉),𝑣_𝑛+1(1)=𝑏𝑢_𝑛+1𝜂,𝐷^𝑝𝛼_𝑛+1(𝑡)−𝑀(𝑡)𝛼_𝑛+1(𝑡)+𝑓𝑡,𝛼_𝑛(𝑡),𝛽_𝑛(𝑡)+𝑀(𝑡)𝛼_𝑛(𝑡)(),𝐷=0,𝑡∈0,1_𝑞𝛽_𝑛+1(𝑡)−𝑁(𝑡)𝛽_𝑛+1(𝑡)+𝑔𝑡,𝛼_𝑛(𝑡),𝛽_𝑛(𝑡)+𝑁(𝑡)𝛽_𝑛(𝑡)(),𝛼=0,𝑡∈0,1_𝑛+1(0)=𝛽_𝑛+1(0)=0,𝛼_𝑛+1(1)=𝑎𝛽_𝑛+1(𝜉),𝛽_𝑛+1(1)=𝑏𝛼_𝑛+1𝜂, (27) for 𝑛=1,2,…. Note that {(𝑢₁,𝑣₁),(𝛼₁,𝛽₁)} are well defined, by Lemma 6. First, we show that 𝑢₀,𝑣₀≤𝑢₁,𝑣₁≤𝛼₁,𝛽₁≤𝛼₀,𝛽₀. (28) Let 𝑧=𝑢₀−𝑢₁, 𝑤=𝑣₀−𝑣₁. This and the assumption that (𝑢₀,𝑣₀) is a lower system of solutions of (1) yield −𝐷^𝑝𝑧(𝑡)≤−𝑀(𝑡)𝑧(𝑡),𝑡∈(0,1),−𝐷^𝑞𝑤(𝑡)(𝑡)𝑤(𝑡)(),𝑧(0)(0)(1)(𝜉),𝑤(1)𝜂.≤−𝑁,𝑡∈0,1≤0,𝑤≤0,𝑧≤𝑎𝑤≤𝑏𝑧 (29) Hence, 𝑢₀(𝑡)≤𝑢₁(𝑡) and 𝑣₀(𝑡)≤𝑣₁(𝑡), 𝑡∈[0,1], by Lemma 3. By a similar way, we can show that 𝛼₁(𝑡)≤𝛼₀(𝑡) and 𝛽₁(𝑡)≤𝛽₀(𝑡), 𝑡∈[0,1]. Now we put 𝑧=𝑢₁−𝛼₁, 𝑤=𝑣₁−𝛽₁. Hence, in view of assumptions (𝐻₂), (𝐻₃), we have −𝐷^𝑝𝑧(𝑡)=−𝑀(𝑡)𝑧(𝑡)−𝑓𝑡,𝛼₀(𝑡),𝛽₀(𝑡)−𝑀(𝑡)𝛼₀(𝑡)+𝑓𝑡,𝑢₀(𝑡),𝑣₀(𝑡)+𝑀(𝑡)𝑢₀(𝑡)≤−𝑀(𝑡)𝑧(𝑡)−𝑓𝑡,𝛼₀(𝑡),𝛽₀(𝑡(𝑡)𝛼)−𝑀₀(𝑡)+𝑓𝑡,𝑢₀(𝑡),𝛽₀(𝑡)+𝑀(𝑡)𝑢₀(𝑡)≤−𝑀(𝑡)𝑧(𝑡),𝑡∈(0,1),−𝐷^𝑞𝑤(𝑡)(𝑡)𝑤(𝑡)(),𝑧(0)(0)(1)(𝜉),𝑤(1)𝜂.≤−𝑁,𝑡∈0,1=0,𝑤=0,𝑧=𝑎𝑤=𝑏𝑧 (30) This and Lemma 3 prove that (𝑢₁,𝑣₁)≤(𝛼₁,𝛽₁), 𝑡∈[0,1], so, relation (28) holds.
Now we show that (𝑢₁,𝑣₁) is a lower system of solution of problem (1). Note that 𝐷^𝑝𝑢₁(𝑡)+𝑓𝑡,𝑢₁(𝑡),𝑣₁(𝑡)≥𝑀(𝑡)𝑢₁(𝑡)−𝑓𝑡,𝑢₀(𝑡),𝑣₀(𝑡)−𝑀(𝑡)𝑢₀(𝑡)+𝑓𝑡,𝑢₁(𝑡),𝑣₀(𝑡(),𝐷)≥0,𝑡∈0,1_𝑞𝑣₁(𝑡)+𝑔𝑡,𝑢₁(𝑡),𝑣₁(𝑡)(),𝑢≥0𝑡∈0,1₁(0)=𝑣₁(0)=0,𝑢₁(1)=𝑎𝑣₁(𝜉),𝑣₁(1)=𝑏𝑢₁𝜂, (31) by assumptions (𝐻₂), (𝐻₃). It proves that (𝑢₁,𝑣₁) is a lower system of solution of (1). Similarly, we can prove that (𝛼₁,𝛽₁) is an upper system of solution of problem (1).
By mathematical induction we can show that 𝑢₀,𝑣₀≤𝑢₁,𝑣₁≤⋯≤𝑢_𝑛,𝑣_𝑛≤𝛼_𝑛,𝛽_𝑛≤⋯≤𝛼₁,𝛽₁≤𝛼₀,𝛽₀ (32) for 𝑡∈[0,1] and 𝑛=1,2,…. Employing standard arguments we see that the sequences {(𝑢_𝑛,𝑣_𝑛),(𝛼_𝑛,𝛽_𝑛)} converge to their limit functions (𝑢_∗,𝑣_∗), (𝛼^∗,𝛽^∗), respectively. Indeed, (𝑢_∗,𝑣_∗) and (𝛼^∗,𝛽^∗) are solutions of problem (1) and (𝑢₀(𝑡),𝑣₀(𝑡))≤(𝑢_∗,𝑣_∗)≤(𝛼^∗,𝛽^∗)≤(𝛼₀,𝛽₀) on [0,1].
We need to show now that (𝑢_∗,𝑣_∗) and (𝛼^∗,𝛽^∗) are extremal solutions of problem (1) in the segment Ω. To prove it, we assume that (𝑥,𝑦) is another solution of problem (1) and (𝑢_𝑛,𝑣_𝑛)≤(𝑥(𝑡),𝑦(𝑡))≤(𝛼_𝑛(𝑡),𝛽_𝑛(𝑡)), 𝑡∈[0,1] for some positive integer 𝑛. Put 𝑧=𝑢_𝑛+1−𝑥, 𝑤=𝑣_𝑛+1−𝑦. Hence, in view of assumptions (𝐻₂), (𝐻₃), we have −𝐷^𝑝𝑧(𝑡)(𝑡)𝑧(𝑡)(𝑡)(𝑡)(𝑡)𝑥(𝑡)=−𝑀−𝑓𝑡,𝑥,𝑦−𝑀+𝑓𝑡,𝑢_𝑛(𝑡),𝑣_𝑛(𝑡)+𝑀(𝑡)𝑢_𝑛(𝑡)≤−𝑀(𝑡)𝑧(𝑡)−𝑓𝑡,𝑥(𝑡),𝑦(𝑡)−𝑀(𝑡)𝑥(𝑡)+𝑓𝑡,𝑢_𝑛(𝑡),𝑦(𝑡)+𝑀(𝑡)𝑢_𝑛(𝑡)≤−𝑀(𝑡)𝑧(𝑡),𝑡∈(0,1),−𝐷^𝑞𝑤(𝑡)(𝑡)𝑤(𝑡)(),𝑧(0)(0)(1)(𝜉),𝑤(1)𝜂.≤−𝑁,𝑡∈0,1=0,𝑤=0,𝑧=𝑎𝑤=𝑏𝑧 (33) Hence, (𝑢_𝑛+1(𝑡),𝑣_𝑛+1(𝑡))≤(𝑥(𝑡),𝑦(𝑡)), 𝑡∈[0,1], by Lemma 3. By a similar way, we can show that (𝑥(𝑡),𝑦(𝑡))≤(𝛼_𝑛+1(𝑡),𝛽_𝑛+1(𝑡)), 𝑡∈[0,1]. So by induction, (𝑢_𝑛(𝑡),𝑣_𝑛(𝑡))≤(𝑥(𝑡),𝑦(𝑡))≤(𝛼_𝑛(𝑡),𝛽_𝑛(𝑡)), 𝑡∈[0,1] on [0,1] for all 𝑛. Taking the limit as 𝑛→+∞, we conclude (𝑢_∗(𝑡),𝑣_∗(𝑡))≤(𝑥(𝑡),𝑦(𝑡))≤(𝛼^∗(𝑡),𝛽^∗(𝑡)), 𝑡∈[0,1]. That is, (𝑢_∗(𝑡),𝑣_∗(𝑡)) and (𝛼^∗(𝑡),𝛽^∗(𝑡)) are extremal systems of solutions of (1) in Ω.

4. Example

Consider the following problems: 𝐷^5/4𝑥(𝑡)(𝑡)+1+sin𝑡−2𝑥8𝑦³(𝑡)𝑡³(),𝐷=0,𝑡∈0,1_7/4𝑦(𝑡)−𝑦³(𝑡)+𝑥²(𝑡)(),𝑥(0)(0)(1)=1+1=0,𝑡∈0,1=𝑦=0,𝑥4𝑦12,𝑦(1)3=2𝑥4. (34) Obviously, 𝑓1𝑡,𝑥,𝑦=sin𝑡−2𝑥+8𝑦³𝑡³,𝑔𝑡,𝑥,𝑦=−𝑦³+𝑥²+1. (35) Take (𝑢₀(𝑡),𝑣₀(𝑡))=(0,0), (𝛼₀(𝑡),𝛽₀(𝑡))=(𝑡,2); then 𝐷^5/4𝑢₀(𝑡)+sin𝑡−2𝑢₀(𝑡)+18𝑣³₀(𝑡)𝑡³(),𝐷=sin𝑡≥0,𝑡∈0,1_7/4𝑣₀(𝑡)−𝑣³₀(𝑡)+𝑢²₀(𝑡)(),𝑢+1=1≥0,𝑡∈0,1₀(0)=𝑣₀(0)=0,𝑢₀(1)=14𝑣₀12,𝑣₀(1)=2𝑢₀34,𝐷^5/4𝛼₀(𝑡)+sin𝑡−2𝛼₀(𝑡)+18𝛽³₀(𝑡)𝑡³=sin𝑡−2𝑡+𝑡³(),𝐷≤0,𝑡∈0,1_7/4𝛽₀(𝑡)−𝛽³₀(𝑡)+𝛼²₀(𝑡)+1=−7+𝑡²(),𝛼≤0,𝑡∈0,1₀(0)=0,𝛽₀(0)𝛼≥0,₀(1)≥14𝛽₀12,𝛽₀(1)≥2𝛼₀34. (36) It shows that (𝑢₀(𝑡),𝑣₀(𝑡)) and (𝛼₀(𝑡),𝛽₀(𝑡)) are lower and upper systems of solutions of (34).

On the other hand, it is easy to verify that conditions (𝐻₂), (𝐻₃) hold for 𝑀(𝑡)=2 and 𝑁(𝑡)=12.

By Theorem 7, problem (34) has an extremal system of solutions (𝑢_∗(𝑡),𝑣_∗(𝑡)) and (𝛼^∗(𝑡),𝛽^∗(𝑡)), which can be obtained by taking limits from some iterative sequences.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The Project is supported by the National Natural Science Foundation of China (11371221, 61304074), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province (201303074), and Foundation of SDUST.