# An Iterative Algorithm for Solvingn-Order Fractional Differential Equation with Mixed Integral and Multipoint Boundary Conditions

Authors: Jingjing Tan; Xinguang Zhang; Lishan Liu; Yonghong Wu
Journal: Complexity
 (2021)

---

## Abstract

In this paper, we consider the iterative algorithm for a boundary value problem ofn-order fractional differential equation with mixed integral and multipoint boundary conditions. Using an iterative technique, we derive an existence result of the uniqueness of the positive solution, then construct the iterative scheme to approximate the positive solution of the equation, and further establish some numerical results on the estimation of the convergence rate and the approximation error.


---

## Body

## 1. Introduction

In this paper, we focus on the iterative algorithm for the followingn-order fractional equation involving mixed integral and multipoint boundary conditions:(1)−D0+αxt=ft,xt,xt,0<t<1,xi0=0,i=0,1,2,…,n−2,x1=∑i=1m−2βi∫0ηixsds+∑i=1m−2γixηi,where D0+α is the standard fractional derivative of order α satisfying n−1<α≤n with m,n≥3 and m,n∈N+, 0<η1<η2<⋯<ηm−2<1, βi,γi>0, 1≤i≤m−2, and ft,u,v is may be singular at v=0 and t=0,1.Based on the wide range applications of calculus, in recent years, the study for various differential equations has become a frontier issue of nonlinear field and many mathematical methods and techniques, such as iterative techniques [1–17], dual approach and perturbed techniques [18–23], fixed-point theorems [24–50], lower-upper solution method [51–53], variational method [54–68], numerical methods and stability analysis [69–81], which were developed by many researchers to handle various nonlinear problems. In particular, in describing and modeling viscoelasticity and nonlocal problems in complex analysis, environmental issue, chemistry physics, and statistical physics, a large amount of work [18, 30, 64, 82–111] have shown that fractional differential equations possess greater advantage than classical integer differential equations. In recent work [95], Salm, by using the Hahn-Banach fixed point theorem, studied the following multipoint boundary value problem:(2)−D0+αxt=qtft,xt,0<t<1,n−1<α≤n,n≥3,xk0=0,0≤k≤n−2,x1=∑i=1m−2ζixηi.The weakly continuous solution for the above nonlinear boundary value problem of fractional type was derived. And then, Xie et al. [96] studied the following nonlinear fractional differential equation with a three-point nonlinear boundary condition:(3)D0+αxt+ft,xt=0,0≤t≤1,x0=0,gx1,xη=0,0≤η≤1.By using the method of upper and lower solutions as well as the monotone iterative technique, the results about the existence of extremal solutions were obtained, where the iterative process can start from the fixed upper and lower solutions.Among various techniques of dealing with differential equations, upper and lower solutions’ method and fixed-point methods have been verified to be the most efficient approaches. However, the efficiency of those methods depends essentially on the monotonicity and compactness of the operator. So, how to overcome the requirement of the monotonicity and compactness is a huge challenge, since such qualities are not naturally available and difficult to prove which lead to the complexity to solve some BVP, especially for the fractional nonlinear boundary value problems.Different from [96], in this paper, we develop the new iterative algorithm to overcome the requirement of the compactness for the nonlinear operator. Our work has three major features. Firstly, equation (1) possesses a more general nonlinear term which has two space variables and the boundary condition is a mixed integral and multipoint boundary condition. Secondly, the nonlinear term can be singular in the time variables and the second space variables. In the end, our results are more refined, that is, we not only construct a new iterative process which can perform from any initial value but also obtain the uniform convergence of the iterative sequences; at same time, the estimation of the convergence rate and the approximation error are also given, which imply that more strong results are established under the relatively weaker condition than that of [96].The paper is organized as follows. In Section2, we recall some definitions and lemmas. In Section 3, the unique of positive solutions to BVP (1) are obtained. Finally, in Section 4, an illustrative example is also presented.


## 2. Preliminaries

In this section, we first list some notations and recall related definitions and lemmas to be used in our proofs later.Definition 1 (see [5]).
Letp>0, the Riemann–Liouville standard fractional integral derivative of order p>0 of a function f:0,∞⟶R is given by(4)I0+pfx=1Γp∫0xftx−t1−pdt,D0+pft=1Γn−pddtn∫0tfst−sp−n+1ds,where n=p+1, p denotes the integer part of the real number p.Lemma 1 (see [5]).
Suppose thatn−1<α≤n with n≥3 and h∈L10,1; then, the boundary value problem(5)−D0+αxt=ht,0<t<1,xi0=0,i=0,1,2,…,n−2,x1=∑i=1m−2βi∫0ηixsds+∑i=1m−2γixηi,is given by(6)xt=∫01Gt,shsds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,shsds,where ξ=1−1/α∑i=1m−2βiηiα−∑i=1m−2γiηiα−1>0, and(7)Gt,s=1Γαtα−11−sα−1−t−sα−1,0≤s≤t≤1,tα−11−sα−1,0≤t≤s≤1,Ht,s=1Γα+1tα−11−sα−1−t−sα−1,0≤s≤t≤1,tα−11−sα−1,0≤t≤s≤1.Lemma 2 (see [5]).
For allt,s∈0,1, the functions Gt,s and Ht,s in Lemma 1 satisfy the following properties:(1)
Gt,s and Ht,s are continuous and nonnegative(2)
α−1/Γαtα−11−t1−sα−1s≤Gt,s≤1/Γαtα−11−sα−2(3)
α−1/Γα+1tα−11−t1−sα−1s≤Ht,s≤1/Γα+1tα−11−sα−2
In this paper, we will work in the spaceE=C0,1. Define a set P in E and an operator T:E×E⟶E as follows:(8)P=x∈E|there exists a positive constantlx∈0,1,such thatlxtα−1≤xt≤lx−1tα−1,t∈0,1,Tx,yt=∫01Gt,sfs,xs,ysds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,xs,ysds.
Obviously,tα−1,tα−1∈P×P, so P×P is not empty.


## 3. Main Results

Before claiming our main results, we first introduce the following notations:(i)
M=ξα+1+αm−2(ii)
N=∑i=1m−2βi+αγiηiα−11−ηiTheorem 1.
Assume that(H1)
ft,u,v∈C0,1×0,+∞×0,+∞;0,+∞ and for t,u,v∈0,1×0,+∞×0,+∞, f is increasing with respect to u, decreasing with respect to v(H2)
Fort,u,v∈0,1×0,+∞×0,+∞ and k∈0,1, there exists a constant μ∈0,1 such that ft,ku,k−1v≥kμft,u,v(H3)
0<∫011−sα−2fs,sα−1,sα−1ds<∞
Then, the BVP (1) has unique positive solution x∗t∈P, and there exists a constant 0<l<1 satisfying(9)ltα−1≤x∗t≤l−1tα−1,t∈0,1.Proof.
Firstly, it is easy to know thatx∗ is the solution of the BVP (1) if and only if x∗ satisfies Tx∗,x∗=x∗.
Next, it follows fromH1 that the operator T:P×P⟶P is nondecreasing with respect to x and nonincreasing with respect to y; thus, by H1,H2, and H3, for any x,y∈P×P and t∈0,1, there exist two constants 0<lx<1,0<ly<1 such that(10)lxtα−1≤xt≤lx−1tα−1,lytα−1≤yt≤ly−1tα−1.
Denotelx∗=minlx,ly; then, we have(11)lx∗tα−1≤xt≤lx∗−1tα−1,lx∗tα−1≤yt≤lx∗−1tα−1.
Consequently,(12)Tx,yt=∫01Gt,sfs,xs,ysds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,xs,ysds≤tα−1Γα∫011−sα−2fs,lx∗−1sα−1,lx∗sα−1ds+tα−1ξ∑i=1m−2∫01βiηiα−1Γα+1+γiηiα−1Γα1−sα−2fs,lx∗−1sα−1,lx∗sα−1ds≤lx∗−μtα−1Γα∫011−sα−2fs,sα−1,sα−1ds+lx∗−μtα−1ξ∑i=1m−2∫01βiηiα−1Γα+1+γiηiα−1Γα1−sα−2fs,sα−1,sα−1ds≤lx∗−μtα−1ξα+1+αm−2ξαΓα∫011−sα−2fs,sα−1,sα−1ds=tα−1Mlx∗−μξαΓα∫011−sα−2fs,sα−1,sα−1ds<+∞,(13)Tx,yt=∫01Gt,sfs,xs,ysds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,xs,ysds≥α−1tα−1ξ∑i=1m−2∫01βiηiα−11−ηiΓα+1+γiηiα−11−ηiΓαs1−sα−1fs,lx∗sα−1,lx∗−1sα−1ds≥lx∗μtα−1α−1ξαΓα∑i=1m−2βi+αγiηiα−11−ηi∫01s1−sα−1fs,sα−1,sα−1ds=tα−1α−1Nlx∗μξαΓα∫01s1−sα−1fs,sα−1,sα−1ds.
Let(14)lTx=min12,α−1Nlx∗μξαΓα∫01s1−sα−1fs,sα−1,sα−1ds,Mlx∗−μξαΓα∫011−sα−2fs,sα−1,sα−1ds−1.
Then, there exists a constant0<lTx<1 such that(15)lTxtα−1≤Tx,yt≤lTx−1tα−1,t∈0,1,which implies that the operator T:P×P⟶P is well defined. According to the Arzela-Ascoli theorem, it is easy to know that T:P×P⟶P is completely continuous.
Now, takeht=tα−1, then h,h∈P×P, it follows from (12) and (13) that Th,h∈P. Thus, by the definition of P, there exists a constant 0<lTh<1 such that(16)lThtα−1≤Th,ht≤lTh−1tα−1.
Take(17)0<λ≤lTh1/1−μ,and let(18)x0=λht,y0=λ−1ht.
Now, construct the following iterative sequence:(19)xn=Txn−1,yn−1,yn=Tyn−1,xn−1,n=1,2,…
We assert(20)x0≤x1≤⋯≤xn≤⋯≤yn≤⋯≤y1≤y0.
In fact, it follows from0<λ<1 and (18) that x0,y0∈P and x0≤y0. Moreover,(21)x1=Tx0,y0t=∫01Gt,sfs,λht,λ−1htds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,λht,λ−1htds≥λμ∫01Gt,sfs,ht,htds+λμtα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,ht,htds=λμTh,ht≥λμlThht≥λμλ1−μht=x0,y1=Ty0,x0t=∫01Gt,sfs,λ−1ht,λhtds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,λ−1ht,λhtds≤λ−μ∫01Gt,sfs,ht,htds+λ−μtα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,ht,htds=λ−μTh,ht≤λ−μlTh−1ht≤λ−μλμ−1ht=y0.
On the other hand, it follows fromx0≤y0 and the fact of T being nondecreasing with respect to second variable and nonincreasing with respect to third variable that x1≤y1. Thus, according to the above fact, we have (20) holds.
Notice that, for any nature numbern,(22)xn=Txn−1,yn−1=Tny0,x0=Tnλht,λ−1ht=Tnλ2λ−1ht,λ−2λht≥λ2μnTnλ−1ht,λht=cμnyn,where c=λ2. So, for any nature numbers n and n∗, we have(23)0≤xn+n∗−xn≤yn−xn≤1−cμnyn≤1−cμnλ−1ht⟶0,n⟶+∞,which implies that there exists x∗∈P such that(24)xnt⟶x∗t,uniformly on 0,1. By the same method, we can also prove that(25)ynt⟶x∗t,uniformly on 0,1. In view of the continuous of T, take the limits in xn=Txn,yn, we have x∗=Tx∗,x∗. So, x∗ is a positive solution of BVP (1). Since x∗∈P, for any t∈0,1, there exists a constant l∈0,1 such that(26)ltα−1≤x∗t≤l−1tα−1holds.
Finally, we show that the uniqueness of the positive solution. Lety∗t be another positive solution of BVP (1); then, for any t∈0,1, there exists a constant m∈0,1 such that(27)mtα−1≤y∗t≤m−1tα−1.
Takingλ defined in (17) be small enough such that λ<m. So,(28)x0t≤y∗t≤y0t,t∈0,1.
According toTy∗,y∗=y∗, using the nondecreasing of T, we can show that(29)xnt≤y∗t≤ynt,t∈0,1.
Taking limits to the both sides of (29), we have x∗=y∗. It follows that the solution of BVP (1) is unique. The proof of Theorem 1 is completed.
In the following, we consider the error estimation between unique solution and iterative value.Theorem 2.
Let conditionsH1, H2, and H3 be satisfied. Then, for any initial value z0∈P, there exists a sequence znt that uniformly converges to the unique positive solution x∗t with the following error estimation:(30)maxznt−x∗t=o1−cμn,where c∈0,1 is determined by z0.Proof.
By Theorem1, we know that the positive solution x∗ is unique. For any z0∈P, there exists a constant lz0∈0,1, such that(31)lz0tα−1≤z0t≤lz0−1tα−1.
Similar to Theorem1, we can take λ<minlz0,lTh1/1−μ as a fixed number. It follows that(32)x0t≤z0t≤y0t,t∈0,1.
Letzn=Tzn−1,zn−1, n=1,2,…; then, by monotonicity of the operator T, we have(33)x1t≤z1t≤y1t,t∈0,1.
Thus, it follows from mathematical induction that(34)xnt≤znt≤ynt,t∈0,1.
Take limits for the above inequality, we get thatznt uniformly converges to the unique positive solution x∗ of BVP (1). Using (23), we can now derive the error estimation (30), which implies that the error estimation is the same order infinitesimal of 1−cμn, where c=λ2 and determined by z0. This completes the proof of Theorem 2.


## 4. Example

Let us illustrate the main results with an example.Example 1.
Letα=7/2, m=4,η1=1/3, η2=2/3, β1=3/2, β2=4, γ1=5/2, and γ2=2. We consider the following BVP:(35)D0+7/2xt+atx1/8+bty−1/5=0,0<t<1,x′0=x″0=0,x1=32∫01/3xsds+4∫02/3xsds+52x13+2x23,where ft,x,y=atx1/8+bty−1/5. For any k∈0,1, take μ=1/4, and it is easy to verify that(36)ft,kx,k−1y≥kμft,x,y.
According to the expression off and the above inequality, it follows that H1 and H2 are held. In addition,(37)0<∫011−t3/2ft,t5/2,t5/2dt<∞.
So, all of the assumptions of Theorem1 are satisfied. As a result, BVP (35) has a unique positive solution x∗ and for any initial value x0∈P, and the successive iterative sequence xnt is generated by(38)xnt=∫01Gt,sfs,xn−1s,xn−1sds+tα−1ξ∑i=1m−2∫01βiHηi,s+γiGηi,sfs,xn−1s,xn−1sds,n=1,2,…,and uniformly converges to the unique positive solution x∗ on 0,1. We also obtain the error estimation(39)maxxnt−x∗t=o1−c1/4n,where c∈0,1 is a constant and determined by the initial value x0. Moreover, for any t∈0,1, there exists a constant l∈0,1 which satisfies that(40)lt5/2≤x∗t≤l−1t5/2.Remark 1.
According to the definition offt,x,y, let ft,x,x=atx1/8+btx−1/5; then, f does not have monotonicity with respect to x. Thus, the iterative process in some previous work such as [31, 33, 35] cannot be performed, which implies our developed iterative technique in this paper can be suitable for a wider range of functions; in particular, even if ft,x,y is reduced to only have one space variable ft,x, our results is more general than those of [96].


## 5. Result and Discussion

In this paper, we obtain the results of the existence solutions of then-order fractional equation involving mixed integral and multipoint boundary conditions by using a new iterative algorithm. The efficiency of those methods depends essentially on the monotonicity and compactness of the operator. Different from [96], the iterative process does not need to start with the fixed upper and lower solution. We can not only construct a new iterative process which can perform from any initial value but also obtain the uniform convergence of the iterative sequences; at the same time, the estimation of the convergence rate and the approximation error are also given, which imply that the more strong results are established under the relatively weaker condition than that of [96].



---

*Source: 8898859-2021-01-22.xml*