Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation with Periodic Boundary Condition via Initial Value Problem

Abstract

We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order 𝑞, where 0<𝑞<1. We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute.

1. Introduction

The study of fractional differential equations has acquired popularity in the last few decades due to its multiple applications, see [1–5] for more information. However, it was not until recently that a study on the existence of solutions by using upper and lower solutions, which is well established for ordinary differential equations in [6], has been done for fractional differential equations. See [3, 7–16] for recent work.

In this paper we recall a comparison theorem from [3] for a Caputo fractional differential equation of order 𝑞, 0<𝑞<1, with initial condition. We will use coupled lower and upper solutions combined with a generalized monotone method of initial value problems to prove the existence of coupled minimal and maximal periodic solutions. The results developed provide natural sequences and intertwined sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. Instead of the usual approach as in [6, 17] where the iterates are solutions of linear periodic boundary value problems we have used a generalized monotone method of initial value problems. This idea was presented in [18] for integrodifferential equations. The advantage of this method, compared to what was developed in [3, 15], is that it avoids computing the solution of the linear periodic boundary value problem using the Mittag-Leffler function at every step of the iterates. We also modify the comparison theorem which does not require the Hölder continuity condition as in [3].

2. Preliminary Definitions and Comparison Results

In this section we state some definitions and recall some results for a Caputo initial value problem which we need in our main results. Consider the initial value problem of the form: ^𝑐𝐷^𝑞𝑢(𝑡)((𝑡,𝑢(0)=𝑓𝑡,𝑢))=𝑢₀. (2.1)

Here, ^𝑐𝐷^𝑞𝑢(𝑡) is the Caputo derivative of order 𝑛−1<𝑞≤𝑛 for 𝑡∈[𝑎,𝑏], which is defined in [1–3, 5] as ^𝑐𝐷^𝑞_𝑎+𝑢(𝑡)=1Γ𝑛−𝑞^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝑞−1}𝑢^(𝑛)(𝑠)𝑑𝑠,_𝑐𝐷^𝑞_𝑏−𝑢(𝑡)=(−1)^𝑛Γ𝑛−𝑞^𝑏_𝑡(𝑠−𝑡)^{𝑛−𝑞−1}𝑢^(𝑛)(𝑠)𝑑𝑠. (2.2)

In this paper we will denote ^𝑐𝐷^𝑞𝑢(𝑡)=^𝑐𝐷^𝑞_𝑎𝑢(𝑡).

The relation between the Caputo fractional derivative and the Riemann-Liouville fractional derivative, 𝐷^𝑞, is given by ^𝑐𝐷^𝑞𝑢(𝑡)=𝐷^𝑞𝑢(𝑠)−^𝑛−1_𝑘=0𝑢^(𝑘)(𝑎)()𝑘!𝑠−𝑎^𝑘(𝑡). (2.3)

Throughout this paper we consider the Caputo derivative of order 𝑞, for 0<𝑞<1 and 𝑡∈[0,1]. We start by showing some comparison results relative to initial value problems with the Caputo fractional derivative of order 𝑞.

Lemma 2.1. Let 𝑚(𝑡)∈𝐶¹([0,𝑇],ℝ). If there exists 𝑡₁∈[0,𝑇] such that 𝑚(𝑡₁)=0 and 𝑚(𝑡)≤0 on [0,𝑇], then it follows that ^𝑐𝐷^𝑞𝑚𝑡₁≥0. (2.4)

Proof. Let 𝑡₁∈[0,𝑇], the using the relation (2.3) we have that ^𝑐𝐷^𝑞𝑚𝑡₁=𝐷^𝑞𝑚𝑡₁−𝑚(0)Γ𝑡1−𝑞^−𝑞≥𝐷^𝑞𝑚𝑡₁. (2.5)
Since this lemma was proven in [8] for the Riemann Liouville derivative, we have that 𝐷^𝑞𝑚(𝑡₁)≥0 implies ^𝑐𝐷^𝑞𝑚(𝑡₁)≥0, and the proof is complete.

Remark 2.2. In [3] they proved the above result by assuming that 𝑚(𝑡) is Hölder continuous of order 𝜆>𝑞. Although the proof is correct, it is not useful in the monotone method or any iterative method because we will not be able to prove that each of those iterates are Hölder continuous of order 𝜆>𝑞.

Now we are ready to establish the following comparison theorem.

Theorem 2.3. Let 𝐽=[0,𝑇],𝑓∈𝐶[𝐽×ℝ,ℝ],𝑣,𝑤∈𝐶¹[𝐽,ℝ], and for 𝑡∈𝐽 the following inequalities hold true, ^𝑐𝐷^𝑞𝑣(𝑡)≤𝑓(𝑡,𝑣(𝑡)),𝑣(0)≤𝑢₀,_𝑐𝐷^𝑞𝑤(𝑡)≥𝑓(𝑡,𝑤(𝑡)),𝑤(0)≥𝑢₀. (2.6)
Suppose further that 𝑓(𝑡,𝑢) satisfies the following Lipschitz condition, 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝐿𝑥−𝑦,for𝑥≥𝑦and𝐿>0, (2.7) then 𝑣(0)≤𝑤(0) implies that 𝑣(𝑡)≤𝑤(𝑡),for0≤𝑡≤𝑇. (2.8)

Proof. Assume first without loss of generality that one of the inequalities in (2.6) is strict, say ^𝑐𝐷^𝑞𝑣(𝑡)<𝑓(𝑡,𝑣(𝑡)), and 𝑣₀<𝑤₀, where 𝑣(0)=𝑣₀ and 𝑤(0)=𝑤₀. We will show that 𝑣(𝑡)<𝑤(𝑡) for 𝑡∈𝐽.
Suppose, to the contrary, that there exists 𝑡₁ such that 0<𝑡₁≤𝑇 for which 𝑣𝑡₁=𝑤𝑡₁,𝑣(𝑡)≤𝑤(𝑡),for𝑡<𝑡₁. (2.9)
Setting 𝑚(𝑡)=𝑣(𝑡)−𝑤(𝑡) it follows that 𝑚(𝑡₁)=0 and 𝑚(𝑡)≤0 for 𝑡<𝑡₁. Then by hypothesis and Lemma 2.1 we have that ^𝑐𝐷^𝑞𝑚(𝑡₁)≥0. Thus 𝑓𝑡₁,𝑣𝑡₁>^𝑐𝐷𝑣𝑡₁≥^𝑐𝐷𝑤𝑡₁≥𝑓𝑡₁,𝑤𝑡₁, (2.10) which is a contradiction to the assumption 𝑣(𝑡₁)=𝑤(𝑡₁). Therefore 𝑣(𝑡)<𝑤(𝑡).
Now assume that the inequalities (2.6) are nonstrict. We will show that 𝑣(𝑡)≤𝑤(𝑡).
Set 𝑤_𝜖(𝑡)=𝑤(𝑡)+𝜖𝜆(𝑡), where 𝜖>0 and 𝜆(𝑡)=𝐸_𝑞[2𝐿𝑡^𝑞], where 𝐸_𝑞 is the one parameter Mittag-Leffler function. This implies that 𝑤_𝜖(0)=𝑤₀+𝜖>𝑤₀ and 𝑤_𝜖(𝑡)>𝑤(𝑡) for 𝑡∈(0,𝑇].
Using (2.6) and the Lipschitz condition (2.7), we find that ^𝑐𝐷^𝑞𝑤_𝜖(𝑡)=^𝑐𝐷^𝑞𝑤(𝑡)+𝜀^𝑐𝐷^𝑞𝜆(𝑡)≥𝑓(𝑡,𝑤(𝑡))+2𝜖𝐿𝜆(𝑡)≥𝑓𝑡,𝑤_𝜖(𝑡)−𝜖𝐿𝜆(𝑡)+2𝜖𝐿𝜆(𝑡)=𝑓𝑡,𝑤_𝜖(𝑡)+𝜖𝐿𝜆(𝑡)>𝑓𝑡,𝑤_𝜖(𝑡),0<𝑡≤𝑇. (2.11)
Here we have utilized the fact that 𝜆(𝑡) is the solution of the initial value problem ^𝑐𝐷^𝑞𝜆(𝑡)=2𝐿𝜆(𝑡),𝜆(0)=1>0. (2.12)
Clearly there is no assumption on the growth of 𝐿>0. Applying now the result for strict inequalities to 𝑣(𝑡),𝑤_𝜖(𝑡), we get that 𝑣(𝑡)<𝑤_𝜖(𝑡) for 𝑡∈𝐽, for every 𝜖>0 and consequently making 𝜖→0, we get that 𝑣(𝑡)≤𝑤(𝑡) for 𝑡∈𝐽.

The following corollary will be useful in our main results.

Corollary 2.4. Let 𝑚∈𝐶¹[𝐽,ℝ] be such that ^𝑐𝐷^𝑞𝑚(𝑡)(𝑡),𝑚(0)≤𝐿𝑚=𝑚₀. (2.13)
Then we have from the previous theorem the estimate 𝑚(𝑡)≤𝑚₀𝐸_𝑞𝐿𝑡^𝑞,for0≤𝑡≤𝑇and𝐿>0. (2.14)

The result of the above corollary is still true even if 𝐿=0, which we state separately and prove it.

Corollary 2.5. Let ^𝑐𝐷^𝑞𝑚(𝑡)≤0 on [0,𝑇]. Then 𝑚(𝑡)≤0, if 𝑚(0)≤0.

Proof. By definition of ^𝑐𝐷^𝑞𝑚(𝑡) and by hypothesis, ^𝑐𝐷^𝑞𝑚(𝑡)=1Γ1−𝑞^𝑡₀(𝑡−𝑠)^−𝑞𝑚^′(𝑠)𝑑𝑠≤0, (2.15) which implies that 𝑚′(𝑡)≤0 on [0,𝑇]. Therefore, 𝑚(𝑡)≤𝑚(0)≤0 on [0,𝑇].

Note that the above result may not be true for the Riemann Liouville derivative.

We recall a comparison result from [3] for periodic boundary conditions. As in Theorem 2.3, the proof does not require Hölder continuity.

Theorem 2.6. Let 𝐽=[0,𝑇], 𝑓∈𝐶[𝐽×ℝ,ℝ], 𝑣,𝑤∈𝐶¹[𝐽,ℝ], and for 0<𝑡≤𝑇, ^𝑐𝐷^𝑞𝑣(𝑡)≤𝑓(𝑡,𝑣(𝑡)),𝑣(0)≤𝑣_𝑐(𝑇),𝐷^𝑞𝑤(𝑡)≥𝑓(𝑡,𝑤(𝑡)),𝑤(0)≥𝑤(𝑇). (2.16)
Suppose further that 𝑓(𝑡,𝑢) is strictly decreasing in 𝑢 for each 𝑡, then 𝑣(𝑡)≤𝑤(𝑡),for0≤𝑡≤𝑇. (2.17)

Proof. If the conclusion is not true; that is, if there exists 𝑡₀∈[0,𝑇] such that 𝑣(𝑡₀)>𝑤(𝑡₀), then there exists an 𝜀>0 such that 𝑣𝑡₀=𝑤𝑡₀+𝜀,𝑣(𝑡)≤𝑤(𝑡)+𝜀,for0≤𝑡≤𝑡₀≤𝑇. (2.18)
Setting 𝑚(𝑡)=𝑣(𝑡)−𝑤(𝑡)−𝜀 we find that if 𝑡₀∈(0,𝑇], then 𝑚𝑡₀=0,𝑚(𝑡)≤0,for0<𝑡≤𝑡₀≤𝑇. (2.19)
By Lemma 2.1 we have that ^𝑐𝐷^𝑞𝑚(𝑡₀)≥0. Thus, ^𝑐𝐷^𝑞𝑣(𝑡₀)≥^𝑐𝐷^𝑞𝑤(𝑡₀). We now obtain by hypothesis and by the strictly decreasing nature of 𝑓(𝑡,𝑢) in 𝑢, 𝑓𝑡₀,𝑣𝑡₀≥^𝑐𝐷^𝑞𝑣𝑡₀≥^𝑐𝐷^𝑞𝑤𝑡₀≥𝑓𝑡₀,𝑤𝑡₀>𝑓𝑡₀,𝑣𝑡₀, (2.20) which is a contradiction.
If 𝑡₀=0, then 𝑣(𝑇)≥𝑣(0)=𝑤(0)+𝜀≥𝑤(𝑇)+𝜀, (2.21) so 𝑣(𝑇)>𝑤(𝑇), and by the above argument we also get a contradiction.
Hence 𝑣(𝑡)≤𝑤(𝑡) for 0≤𝑡≤𝑇 and the proof is complete.

Two important cases of this theorem are the following which are useful to prove the uniqueness of the solution of a Caputo fractional differential equation with periodic boundary conditions.

Corollary 2.7. Let 𝑚∈𝐶¹[𝐽,ℝ] be such that ^𝑐𝐷^𝑞𝑚(𝑡)(𝑡),𝑚(0)(𝑇),≤−𝑀𝑚≤𝑚 (2.22) for 0≤𝑡≤𝑇 and 𝑀>0. Then 𝑚(𝑡)≤0 for 0≤𝑡≤𝑇.
Similarly, if ^𝑐𝐷^𝑞𝑚(𝑡)(𝑡),𝑚(0)(𝑇),≥−𝑀𝑚≥𝑚 (2.23) for 0≤𝑡≤𝑇 and 𝑀>0. Then 𝑚(𝑡)≥0 for 0≤𝑡≤𝑇.

Remark 2.8. It is to be noted that in the proof of these equivalent results from [3] we use Lemma 2.1 which does not require the Hölder continuity assumption.

A generelized monotone method for periodic boundary value problems was recently developed in [3, 15]. However it uses the approach established in [6] for ordinary differential equations where the iterates are solutions of the linear periodic boundary value problem, ^𝑐𝐷^𝑞𝑢(𝑡)(𝑡)(𝑡),𝑢(0)(𝑇),=−𝑀𝑢+ℎ=𝑢 (2.24) where 𝑀>0 and ℎ∈𝐶[𝐽,ℝ].

The explicit solution of this equation is given by 𝑢(𝑡)=𝐸_𝑞𝑀𝑡^𝑞1−𝐸_𝑞𝑀(𝑇)^𝑞𝑇₀(𝑇−𝑠)^𝑞−1𝐸_𝑞,𝑞𝑀(𝑇−𝑠)^𝑞ℎ(𝑠)𝑑𝑠+^𝑡₀(𝑡−𝑠)^𝑞−1𝐸_𝑞,𝑞𝑀(𝑡−𝑠)^𝑞ℎ(𝑠)𝑑𝑠, (2.25) where 𝐸_𝑞 and 𝐸_𝑞,𝑞 are Mittag-Leffler functions with one and two parameters, respectively.

This poses a problem to compute the linear iterates since it involves Mittag-Leffler functions. The advantage of our method is that it does not require the Mittag-Leffler function in our computations.

Pandit et al. used the initial value problem to obtain a generalized monotone method in [18] for nonlinear integrodifferential equations with periodic boundary conditions. In the next section we develop a monotone method using this idea.

3. Generalized Monotone Method for the Nonlinear Periodic Boundary Value Problem via Initial Value Problem

In this section, we will develop a generalized monotone method for the nonlinear periodic boundary value problem (3.2), given below, by using coupled upper and lower solutions and the corresponding initial value problem (2.1), where 𝑓 does not depend on 𝑢, 𝑢(𝑡)=𝑢₀+1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠. (3.1)

For that purpose consider the nonlinear periodic boundary value problem of the form: ^𝑐𝐷^𝑞𝑢(𝑡)((𝑡((𝑡,𝑢(0)(𝑇),=𝑓𝑡,𝑢))+𝑔𝑡,𝑢))=𝑢 (3.2) where 𝑓,𝑔∈𝐶[𝐽×ℝ,ℝ] and 𝑢∈𝐶¹[𝐽×ℝ].

If 𝑢∈𝐶¹[0,𝑇] satisfies the fractional differential equation ^𝑐𝐷^𝑞𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡))+𝑔(𝑡,𝑢(𝑡)), (3.3) and 𝑢 is such that 𝑢(0)=𝑢(𝑇) for 𝑡∈𝐽, then 𝑢 is a periodic solution of (3.2).

Furthermore, throughout this paper, we will assume that 𝑓 is increasing in 𝑢 and 𝑔 is decreasing in 𝑢 for 𝑡∈𝐽.

Here below we provide the definition of coupled lower and upper solutions of (3.2).

Definition 3.1. Let 𝑣₀,𝑤₀∈𝐶¹[𝐽,ℝ]. Then 𝑣₀ and 𝑤₀ are said to be, (i)natural lower and upper solutions of (3.2) if ^𝑐𝐷^𝑞𝑣₀(𝑡)≤𝑓𝑡,𝑣₀(𝑡)+𝑔𝑡,𝑣₀(𝑡),𝑣₀(0)≤𝑣₀(𝑇),_𝑐𝐷^𝑞𝑤₀(𝑡)≥𝑓𝑡,𝑤₀(𝑡)+𝑔𝑡,𝑤₀(𝑡),𝑤₀(0)≥𝑤₀(𝑇); (3.4)(ii)coupled lower and upper solutions of Type I of (3.2) if ^𝑐𝐷^𝑞𝑣₀(𝑡)≤𝑓𝑡,𝑣₀(𝑡)+𝑔𝑡,𝑤₀(𝑡),𝑣₀(0)≤𝑣₀(𝑇)_𝑐𝐷^𝑞𝑤₀(𝑡)≥𝑓𝑡,𝑤₀(𝑡)+𝑔𝑡,𝑣₀(𝑡),𝑤₀(0)≥𝑤₀(𝑇); (3.5)(iii)coupled lower and upper solutions of Type II of (3.2) if ^𝑐𝐷^𝑞𝑣₀(𝑡)≤𝑓𝑡,𝑤₀(𝑡)+𝑔𝑡,𝑣₀(𝑡),𝑣₀(0)≤𝑣₀(𝑇),_𝑐𝐷^𝑞𝑤₀(𝑡)≥𝑓𝑡,𝑣₀(𝑡)+𝑔𝑡,𝑤₀(𝑡),𝑤₀(0)≥𝑤₀(𝑇); (3.6)(iv)coupled lower and upper solutions of Type III of (3.2) if, ^𝑐𝐷^𝑞𝑣₀(𝑡)≤𝑓𝑡,𝑤₀(𝑡)+𝑔𝑡,𝑤₀(𝑡),𝑣₀(0)≤𝑣₀(𝑇),_𝑐𝐷^𝑞𝑤₀(𝑡)≥𝑓𝑡,𝑣₀(𝑡)+𝑔𝑡,𝑣₀(𝑡),𝑤₀(0)≥𝑤₀(𝑇). (3.7)

We will state the following four theorems related to coupled lower and upper solutions of Type I and Type II, respectively. We develop the generalized monotone method for the periodic boundary value problem via the initial value problem approach. We obtain natural sequences and intertwined sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions of (3.2).

In the next theorem, we use coupled lower and upper solutions of Type I and obtain natural sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions of (3.2).

Theorem 3.2. Assume that(A1)𝑣₀,𝑤₀ are coupled lower and upper solutions of Type I for (3.2) with 𝑣₀(𝑡)≤𝑢(𝑡)≤𝑤₀(𝑡) on 𝐽; (A2)𝑓,𝑔∈𝐶[𝐽×ℝ,ℝ], 𝑓(𝑡,𝑢(𝑡)) is increasing in 𝑢 and 𝑔(𝑡,𝑢(𝑡)) is decreasing in 𝑢.
Then the sequences defined by ^𝑐𝐷^𝑞𝑣_𝑛+1=𝑓𝑡,𝑣_𝑛+𝑔𝑡,𝑤_𝑛,𝑣_𝑛+1(0)=𝑣_𝑛(𝑇),_𝑐𝐷^𝑞𝑤_𝑛+1=𝑓𝑡,𝑤_𝑛+𝑔𝑡,𝑣_𝑛,𝑤_𝑛+1(0)=𝑤_𝑛(𝑇), (3.8)(3.9) are such that 𝑣_𝑛(𝑡)→𝜌(𝑡) and 𝑤_𝑛(𝑡)→𝑟(𝑡) in 𝐶¹[𝐽,ℝ] uniformly and monotonically, such that 𝜌 and 𝑟 are coupled minimal and maximal solutions of (3.2), respectively, provided that 𝑣₀≤𝑢≤𝑤₀, where 𝑢 is any periodic solution of (3.2). That is, 𝜌 and 𝑟 satisfy the coupled system ^𝑐𝐷^𝑞()𝜌(0)(𝑇),𝜌=𝑓𝑡,𝜌+𝑔𝑡,𝑟on𝐽,=𝜌_𝑐𝐷^𝑞()𝑟(0)(𝑇),𝑟=𝑓𝑡,𝑟+𝑔𝑡,𝜌on𝐽,=𝑟 (3.10) such that 𝜌≤𝑢≤𝑟.

Proof. By hypothesis, 𝑣₀≤𝑢≤𝑤₀. We will show that 𝑣₀≤𝑣₁≤𝑢≤𝑤₁≤𝑤₀.
It follows from (3.5) that ^𝑐𝐷^𝑞𝑣₀(𝑡)≤𝑓𝑡,𝑣₀(𝑡)+𝑔𝑡,𝑤₀(𝑡),𝑣₀(0)≤𝑣₀(𝑇),_𝑐𝐷^𝑞𝑤₀(𝑡)≥𝑓𝑡,𝑤₀(𝑡)+𝑔𝑡,𝑣₀(𝑡),𝑤₀(0)≥𝑤₀(𝑇), (3.11) and by (3.8), we get that ^𝑐𝐷^𝑞𝑣₁=𝑓𝑡,𝑣₀+𝑔𝑡,𝑤₀,𝑣₁(0)=𝑣₀(𝑇). (3.12)
Therefore, 𝑣₀(0)≤𝑣₀(𝑇)=𝑣₁(0). If we let 𝑝=𝑣₀−𝑣₁, then 𝑝(0)≤0 and, ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝑣₀−^𝑐𝐷^𝑞𝑣₁≤𝑓𝑡,𝑣₀+𝑔𝑡,𝑤₀−𝑓𝑡,𝑣₀−𝑔𝑡,𝑤₀=0. (3.13)
Since ^𝑐𝐷^𝑞𝑝≤0 and 𝑝(0)≤0, by Corollary 2.5 we have that 𝑝(𝑡)≤0 and, consequently, 𝑣₀(𝑡)≤𝑣₁(𝑡) on 𝐽. By a similar argument we can show that 𝑣₁(𝑡)≤𝑢, 𝑢≤𝑤₁(𝑡), and 𝑤₁(𝑡)≤𝑤₀(𝑡). Thus, 𝑣₀≤𝑣₁≤𝑢≤𝑤₁≤𝑤₀.
Now we will show that 𝑣_𝑘≤𝑣_𝑘+1 for 𝑘≥1.
Assume that 𝑣_𝑘−1≤𝑣_𝑘≤𝑢≤𝑤_𝑘≤𝑤_𝑘−1, (3.14) for 𝑘>1.
Let 𝑝=𝑣_𝑘−𝑣_𝑘+1. Then 𝑣_𝑘(0)=𝑣_𝑘−1(𝑇)≤𝑣_𝑘(𝑇)=𝑣_𝑘+1(0), (3.15) so 𝑝(0)≤0. By the increasing nature of 𝑓 and the decreasing nature of 𝑔 it follows that ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝑣_𝑘−^𝑐𝐷^𝑞𝑣_𝑘+1=𝑓𝑡,𝑣_𝑘−1+𝑔𝑡,𝑤_𝑘−1−𝑓𝑡,𝑣_𝑘−𝑔𝑡,𝑤_𝑘≤0. (3.16)
Similarly, by Corollary 2.5 we have that 𝑝(𝑡)≤0 and consequently 𝑣_𝑘(𝑡)≤𝑣_𝑘+1(𝑡).
By a similar argument we can show that 𝑤_𝑘+1≤𝑤_𝑘. Using the hypothesis that 𝑣₀(𝑡)≤𝑢(𝑡)≤𝑤₀(𝑡) on 𝐽, the above argument and induction we can show that 𝑣_𝑘+1≤𝑢≤𝑤_𝑘+1. Therefore for 𝑛>1, 𝑣₀≤𝑣₁≤𝑣₂≤⋯≤𝑣_𝑛≤𝑢≤𝑤_𝑛≤⋯≤𝑤₂≤𝑤₁≤𝑤₀. (3.17)
Now we have to show that the sequences converge uniformly. We will use Arzela-Ascoli theorem by showing that the sequences are uniformly bounded and equicontinuous.
First we show uniform boundedness. By hypothesis both 𝑣₀(𝑡) and 𝑤₀(𝑡) are bounded on [0,𝑇], then there exists 𝑀>0 such that for any 𝑡∈[0,𝑇], |𝑣₀(𝑡)|≤𝑀, and |𝑤₀(𝑡)|≤𝑀. Since 𝑣₀(𝑡)≤𝑣_𝑛(𝑡)≤𝑤₀(𝑡) for each 𝑛>0, it follows that 0≤𝑣_𝑛(𝑡)−𝑣₀(𝑡)≤𝑤₀(𝑡)−𝑣₀(𝑡), (3.18) and consequently {𝑣_𝑛(𝑡)} is uniformly bounded. By a similar argument {𝑤_𝑛(𝑡)} is also uniformly bounded.
To prove that {𝑣_𝑛(𝑡)} is equicontinuous, let 0≤𝑡₁≤𝑡₂≤𝑇. Then for 𝑛>0, 𝑣_𝑛𝑡₁−𝑣_𝑛𝑡₂=𝑣_𝑛(𝑇)+1Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑠,𝑤_𝑛−1(𝑠)𝑑𝑠−𝑣_𝑛(𝑇)−1Γ𝑞^𝑡2₀𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑠,𝑤_𝑛−1(𝑠)=1𝑑𝑠Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑠,𝑤_𝑛−1(𝑠)−1𝑑𝑠Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑡,𝑤_𝑛−1(𝑡)≤1𝑑𝑠Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑠,𝑤_𝑛−1(𝑠)+1𝑑𝑠Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑡,𝑤_𝑛−1(𝑡)𝑑𝑠. (3.19)
Since {𝑣_𝑛(𝑡)} and {𝑤_𝑛(𝑡)} are uniformly bounded and 𝑓(𝑡,𝑢(𝑡)) and 𝑔(𝑡,𝑢(𝑡)) are continuous on [0,𝑇], there exists 𝑀 independent of 𝑛 such that 1Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑠,𝑤_𝑛−1(𝑠)+1𝑑𝑠Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑡,𝑤_𝑛−1(𝑡)≤𝑑𝑠𝑀Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1𝑑𝑠+𝑀Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑑𝑠=−𝑀𝑞𝑡𝑞Γ¹−𝑠^𝑞𝑡1₀+𝑀𝑞𝑡𝑞Γ²−𝑠^𝑞𝑡1₀−𝑀𝑞𝑡𝑞Γ²−𝑠^𝑞𝑡2_𝑡1=𝑀Γ𝑡𝑞+1^𝑞1+𝑀Γ𝑡𝑞+1²−𝑡₁^𝑞−𝑀Γ𝑡𝑞+1^𝑞2+𝑀Γ𝑡𝑞+1²−𝑡₁^𝑞≤2𝑀Γ𝑡𝑞+1²−𝑡₁^𝑞=2𝑀Γ𝑡𝑞+1¹−𝑡₂^𝑞. (3.20)
Thus, for any 𝜀>0 there exists 𝛿>0 independent of 𝑛 such that for each 𝑛, 𝑣_𝑛𝑡₁−𝑣_𝑛𝑡₂<𝜀, (3.21) provided that |𝑡₁−𝑡₂|<𝛿.
Similarly we can prove that {𝑤_𝑛(𝑡)} is equicontinuous and uniformly bounded.
This proves that {𝑣_𝑛(𝑡)} and {𝑤_𝑛(𝑡)} are uniformly bounded and equicontinuous on [0,𝑇]. Hence by Arzela-Ascoli’s theorem there exist subsequences {𝑣_𝑛𝑘(𝑡)} and {𝑤_𝑛𝑘(𝑡)} which converge uniformly to 𝜌(𝑡) and 𝑟(𝑡), respectively. Since the sequences are monotone, the entire sequences converge uniformly.
We have shown that the sequences converge in 𝐶[0,𝑇]. In order to show that they converge in 𝐶¹[0,𝑇], observe that since each 𝑣_𝑛 is constructed as follows ^𝑐𝐷^𝑞𝑣_𝑛=𝑓𝑡,𝑣_𝑛−1+𝑔𝑡,𝑤_𝑛−1,𝑣_𝑛(0)=𝑣_𝑛−1(𝑇), (3.22) we get that 𝑣_𝑛(𝑡)=𝑣_𝑛−1(𝑇)+1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑣_𝑛−1(𝑠)+𝑔𝑠,𝑤_𝑛−1(𝑠)𝑑𝑠. (3.23)
Taking limits when 𝑛→∞, we obtain by the Lebesgue dominated convergence theorem that 𝜌(𝑡)(𝑇)+1=𝜌Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝜌(𝑠)+𝑔(𝑠,𝑟(𝑠))𝑑𝑠. (3.24)
Hence 𝑣_𝑛(𝑡)→𝜌(𝑡) in 𝐶¹[0,𝑇]. Furthermore, the above expression is equivalent to ^𝑐𝐷^𝑞()𝜌(0)(𝑇).𝜌=𝑓𝑡,𝜌+𝑔𝑡,𝑟on𝐽,=𝜌 (3.25)
By a similar argument 𝑤_𝑛(𝑡)→𝑟(𝑡) in 𝐶¹[0,𝑇] and it can be shown that ^𝑐𝐷^𝑞()𝑟(0)(𝑇).𝑟=𝑓𝑡,𝑟+𝑔𝑡,𝜌on𝐽,=𝑟 (3.26)
Since 𝑣_𝑛≤𝑢≤𝑤_𝑛 on [0,𝑇] for all 𝑛, we get that 𝜌≤𝑢≤𝑟 on [0,𝑇] which shows that 𝜌 and 𝑟 are minimal and maximal periodic solutions of (3.2), respectively. This completes the proof.

Remark 3.3. In [3] the uniqueness is shown by making additional assumptions to Theorem 3.2 and using Corollary 2.7. However the iterates are solutions of the form (2.25). In our result, we have proved the existence of coupled minimal and maximal periodic solutions of (3.2), in particular if 𝑔≡0 we get minimal and maximal periodic solutions of (3.2).

The next result also uses coupled upper and lower solutions of Type I. However, we obtain intertwined sequences which converge to coupled minimal and maximal periodic solutions of (3.2). The proof is similar to Theorem 3.2, hence we do not provide the proof.

Theorem 3.4. Assume that conditions (A1) and (A2) of Theorem 3.2 are true. Then the iterative scheme given by ^𝑐𝐷^𝑞𝑣_𝑛+1=𝑓𝑡,𝑤_𝑛+𝑔𝑡,𝑣_𝑛,𝑣_𝑛+1(0)=𝑤_𝑛(𝑇),_𝑐𝐷^𝑞𝑤_𝑛+1=𝑓𝑡,𝑣_𝑛+𝑔𝑡,𝑤_𝑛,𝑤_𝑛+1(0)=𝑣_𝑛(𝑇), (3.27) give alternating monotone sequences {𝑣_2𝑛,𝑤_2𝑛+1} and {𝑣_2𝑛+1,𝑤_2𝑛} satisfying 𝑣₀(𝑡)≤𝑤₁(𝑡)≤⋯≤𝑣_2𝑛(𝑡)≤𝑤_2𝑛+1(𝑡)≤𝑢(𝑡)≤𝑣_2𝑛+1(𝑡)≤𝑤_2𝑛(𝑡)⋯𝑣₁(𝑡)≤𝑤₀(𝑡), (3.28) for each 𝑛≥1 on 𝐽, provided that 𝑣₀≤𝑢≤𝑤₀.
Furthermore {𝑣_2𝑛,𝑤_2𝑛+1}→𝜌 and {𝑣_2𝑛+1,𝑤_2𝑛}→𝑟 in 𝐶¹[𝐽,ℝ], where 𝜌 and 𝑟 are coupled minimal and maximal periodic solutions of (3.2), respectively; that is, if 𝑣₀≤𝑢≤𝑤₀ then 𝜌≤𝑢≤𝑟, and 𝜌 and 𝑟 satisfy the coupled system ^𝑐𝐷^𝑞()𝜌(0)(𝑇),𝜌=𝑓𝑡,𝜌+𝑔𝑡,𝑟on𝐽,=𝜌_𝑐𝐷^𝑞()𝑟(0)(𝑇).𝑟=𝑓𝑡,𝑟+𝑔𝑡,𝜌on𝐽,=𝑟 (3.29)

In the previous two theorems we assumed the existence of coupled lower and upper solutions of type I. We can state two more results involving coupled lower and upper solutions of Type II, however they require an additional assumption in order to obtain natural or intertwined sequences converging to coupled minimal and maximal periodic solutions of problem (3.2).

Theorem 3.5. Assume that(B1)𝑣₀ and 𝑤₀ are coupled lower and upper solutions of Type II for (3.2) with 𝑣₀(𝑡)≤𝑤₀(𝑡) on 𝐽,(B2)𝑓,𝑔∈𝐶[𝐽×ℝ,ℝ], 𝑓(𝑡,𝑢(𝑡)) is increasing in 𝑢, and 𝑔(𝑡,𝑢(𝑡)) is decreasing in 𝑢.
If 𝑢(𝑡) is a solution of (3.2) such that 𝑣₀(𝑡)≤𝑢(𝑡)≤𝑤₀(𝑡). Then the sequences defined by (3.27) give alternating monotone sequences {𝑣_2𝑛,𝑤_2𝑛+1} and {𝑣_2𝑛+1,𝑤_2𝑛} satisfying 𝑣₀(𝑡)≤𝑤₁(𝑡)≤⋯≤𝑣_2𝑛(𝑡)≤𝑤_2𝑛+1(𝑡)≤𝑢(𝑡)≤𝑣_2𝑛+1(𝑡)≤𝑤_2𝑛(𝑡)⋯𝑣₁(𝑡)≤𝑤₀(𝑡), (3.30) for each 𝑛≥1 on 𝐽, provided that 𝑣₀≤𝑤₁≤𝑢≤𝑣₁≤𝑤₀.
Furthermore {𝑣_2𝑛,𝑤_2𝑛+1}→𝜌 and {𝑣_2𝑛+1,𝑤_2𝑛}→𝑟 in 𝐶¹[𝐽,ℝ], where 𝜌 and 𝑟 are coupled minimal and maximal solutions of (3.2), respectively; that is, if 𝑣₀≤𝑢≤𝑤₀ then 𝜌≤𝑢≤𝑟, and 𝜌 and 𝑟 satisfy the coupled system ^𝑐𝐷^𝑞()𝜌(0)(𝑇),𝜌=𝑓𝑡,𝜌+𝑔𝑡,𝑟on𝐽,=𝜌_𝑐𝐷^𝑞()𝑟(0)(𝑇).𝑟=𝑓𝑡,𝑟+𝑔𝑡,𝜌on𝐽,=𝑟 (3.31)

Theorem 3.6. Assume that conditions (B1) and (B2) of Theorem 3.5 are true. Then the sequences defined by (3.8) and (3.9) are such that 𝑣_𝑛(𝑡)→𝜌(𝑡) and 𝑤_𝑛(𝑡)→𝑟(𝑡) in 𝐶¹[𝐽,ℝ] uniformly and monotonically, provided that 𝑣₀≤𝑣₁≤𝑢≤𝑤₁≤𝑤₀, where 𝜌 and 𝑟 are coupled minimal and maximal solutions of (3.2), respectively; that is, 𝜌≤𝑢≤𝑟 and 𝜌 and 𝑟 satisfy the coupled system ^𝑐𝐷^𝑞()𝜌(0)(𝑇),𝜌=𝑓𝑡,𝜌+𝑔𝑡,𝑟on𝐽,=𝜌_𝑐𝐷^𝑞()𝑟(0)(𝑇).𝑟=𝑓𝑡,𝑟+𝑔𝑡,𝜌on𝐽,=𝑟 (3.32)

Remark 3.7. The proof of Theorems 3.5 and 3.6 follow on the same lines as Theorem 3.2. However, it is easy to compute coupled lower and upper solutions of Type II.

4. Numerical Examples

In this section we present some numerical examples which are application of Theorem 3.5.

Example 4.1. Consider the problem ^𝑐𝐷^1/2𝑢(𝑡)=𝑢3−𝑢²3,𝑢(0)=𝑢(1). (4.1)
Clearly 𝑢≡0 and 𝑢≡1 are solutions of the equation.
Since 𝐻(𝑢)=𝑢−𝑢² is increasing in 𝑢 for 𝑢≤0.5 and decreasing for 𝑢≥0.5, we let 𝑔()=𝑢𝑡,𝑢3−𝑢²3, (4.2) and 𝑣₀=1/2 and 𝑤₀=3/2 be lower and upper solutions of type II, respectively, because 0=^𝑐𝐷^1/2𝑣₀≤𝑔𝑡,𝑣₀=𝑣₀3−𝑣²₀3=1,120=^𝑐𝐷^1/2𝑤₀≥𝑔𝑡,𝑤₀=𝑤₀3−𝑤²₀31=−4. (4.3)
We construct our sequences according to Theorem 3.5 and show graphically in Figure 1 that {𝑤_2𝑛,𝑣_2𝑛+1}→1.

Figure 1 

Dashed: {𝑣2𝑛,𝑤2𝑛+1}. Solid: {𝑤2𝑛,𝑣2𝑛+1}.

Example 4.2. Consider the problem ^𝑐𝐷^1/2𝑢(𝑡)=𝑢8−𝑢²2+𝑡2,𝑢(0)=𝑢(0.5). (4.4)
Since 𝐻(𝑡,𝑢)=(𝑢/8)−(𝑢²/2)+(𝑡/2) is increasing in 𝑢 for 𝑢≤0.125 and decreasing for 𝑢≥0.125, we let 𝑔()=𝑢𝑡,𝑢8−𝑢²2+𝑡2, (4.5) and 𝑣₀=1/8 and 𝑤₀=1 be lower and upper solutions of type II, respectively, because 0=^𝑐𝐷^1/2𝑣₀≤𝑔𝑡,𝑣₀=𝑣₀8−𝑣²₀2+𝑡2≤1+𝑡1282,0=^𝑐𝐷^1/2𝑤₀≥𝑔𝑡,𝑤₀=𝑤₀8−𝑤²₀2+𝑡23=−8+𝑡2, (4.6) where 0≤𝑡≤1/2.
Using Theorem 3.5 we show in Figure 2 four steps of {𝑣_2𝑛,𝑤_2𝑛+1} and four steps of {𝑤_2𝑛,𝑣_2𝑛+1}. Observe that {𝑤_2𝑛,𝑣_2𝑛+1} converges more quickly to the periodic solution.

Figure 2 

𝑤4(0)≈0.75 and 𝑤4(0.5)≈0.7435.

Example 4.3. Consider ^𝑐𝐷^1/2𝑢(𝑡)𝑢=−²2(),𝑢(0)(1).+𝑡1−𝑡=𝑢 (4.7)
Since 𝑓(𝑡,𝑢)=−(𝑢²/2)+𝑡(1−𝑡) is increasing in 𝑢 for 𝑢≤0 and decreasing for 𝑢≥0, 𝑔()𝑢𝑡,𝑢=−²2+𝑡(1−𝑡), (4.8) and 𝑣₀=0 and 𝑤₀=1 are lower and upper solutions of type II, respectively, because 0=^𝑐𝐷^1/2𝑣₀≤𝑔𝑡,𝑣₀𝑣=−²⁰2+𝑡(1−𝑡)=𝑡(1−𝑡),0=^𝑐𝐷^1/2𝑤₀≥𝑔𝑡,𝑤₀𝑤=−²⁰2()1+𝑡1−𝑡=−2+𝑡(1−𝑡), (4.9) where 0≤𝑡≤1/2, which implies that 0≤𝑡(1−𝑡)≤0.25
Once again using Theorem 3.5 we computed in Figure 3 four steps of {𝑣_2𝑛,𝑤_2𝑛+1} and 4 steps of {𝑤_2𝑛,𝑣_2𝑛+1}. As in the previous examples {𝑤_2𝑛,𝑣_2𝑛+1} converges more quickly to the periodic solution.

Figure 3 

𝑤4(0)≈0.619877 and 𝑤4(1)≈0.610418.

5. Concluding Remarks

In our main results as well as in our numerical results we have not used the assumption needed to obtain unique solution of (3.2). If we make appropriate uniqueness assumption on the nonlinear terms 𝑓(𝑡,𝑢) and 𝑔(𝑡,𝑢) then our linear iterates will require the computation of the Mittag-Leffler function. We plan to take up this work in the near future.