# Rapid Convergence of Approximate Solutions for Fractional Differential Equations

Authors: Xiran Wu; Junyan Bao; Yufeng Sun
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we develop a generalized quasilinearization technique for a class of Caputo’s fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions of orderm (m≥0), and we obtain the convergence of the sequences of approximate solutions by establishing the convergence of order k (k≥2).


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## Body

## 1. Introduction

Fractional differential equations have received attention from some researchers because they have extensive application in mechanics, biochemistry, electrical engineering, medicine, and many other fields (see [1–6]). For more information about the basic theory of fractional differential equations, we can refer to the monographs [7–9] and references cited therein. It is well known [10] that the monotone iterative technique offers an approach for obtaining approximate solutions to a wide variety of nonlinear differential equations. Recently, there are some results on the monotone sequences of approximate solutions converging uniformly to a solution of fractional differential equations by employing monotone iterative technique and generalized monotone iterative method coupled with the method of upper and lower solutions, which can be found in [11–21].In view of applications, it is very significant to study the rate of convergence of solutions. The quasilinearization method [22] is one of the effective methods to obtain a sequence of approximate solutions with quadratic convergence, and it is extremely useful in scientific computations due to its accelerated rate of convergence as in [23, 24]. A few results of quadratic convergence for fractional differential equations were also obtained by applying quasilinearization, such as the initial value problem of Caputo’s fractional differential equations [13, 25, 26], fractional differential equations via initial time different lower and upper solutions [27], and the system of fractional differential equations [28]. However, to the best of our knowledge, there are few results of rapid convergence of fractional differential equations. Recently, Wang and others obtained the results on rapid convergence of solutions for various differential equations [29–33]. Inspired and motivated by [34, 35], in the present paper, we will discuss the rapid convergence of approximate solutions of fractional differential equations when the forcing function is the sum of hyperconvex and hyperconcave functions with coupled lower and upper solutions, and construct sequences of approximate solutions that converge rapidly to the extremal solutions of (1) by using an improved quasilinearization method (rate of convergence k≥2).


## 2. Preliminaries

Consider the initial value problem of Caputo’s fractional differential equations (IVP):(1)Dcqx=ft,x+gt,x,xt0=x0,where f,g:J×R⟶R are continuous functions, J=t0,T and 0<q<1.A functionxt is called a solution of IVP (1) if it satisfies (1).Firstly, we give the following definitions and lemmas.Definition 1.
The locally Hӧlder continuous functionsα0,β0:J⟶R are coupled lower and upper solutions of type I of IVP (1) if the following inequalities hold:
(2)Dcqα0≤ft,α0+gt,β0,α0t0≤x0,Dcqβ0≥ft,β0+gt,α0,β0t0≥x0.Definition 2.
The locally Hölder continuous functionsα0,β0:J⟶R are coupled lower and upper solutions of type II of IVP (1) if the following inequalities hold:
(3)Dcqα0≤ft,β0+gt,α0,α0t0≤x0,Dcqβ0≥ft,α0+gt,β0,β0t0≥x0.Letfkt,x=∂kft,x/∂xk denote the kth partial of f with respect to x,k∈N. x=supt∈Jxt.Definition 3.
A functionf is called m-hyperconvex, m∈N, if fm+1t,x≥0; f is called m-hyperconcave if the inequality is reversed.The linear IVP of Caputo’s fractional differential equation is given by(4)Dcqx=λx+θt,xt0=x0,where θ:J⟶R is a locally Hölder continuous function, t∈J,λ∈R. The unique solution of (4) can be expressed in the following form [7]:
(5)xt=x0Eqλt−t0q+∫t0tt−sq−1Eq,qλt−sqθsds,where
(6)Eqt=∑k=0∞tkΓqk+1,Eq,qt=∑k=0∞tkΓqk+q,are Mittag-Leffler’s functions, and Γ denotes the Gamma function.Remark 4.
We note that (5) remains valid if x and θt are functions mapping from J⟶Rn, x0∈Rn, and λ is an n×n matrix.We need the following lemmas to prove our main results, which proofs can be found in literature [26].Lemma 5.
Assume thatx:J⟶R is a locally Hӧlder continuous function such that for t1∈t0,T, xt1=0 and xt<0 for t∈t0,t1. Then cDqxt1≥0.Lemma 6.
Assume thatxnt is a family of continuous functions on J. For each n>0 and the Caputo’s fractional differential equations, we have the following:
(7)Dcqxnt=ft,xnt,xnt0=x0,the functions ft,xnt satisfy ft,xnt≤M for t∈J. Then, the family xnt is equicontinuous on J.In our further investigations, we need the following comparison results.Lemma 7.
Assume that one of the following conditions holds:
H1α0,β0 are coupled lower and upper solutions of type I of (1) and
(8)ft,x1−ft,x2≤Lx1−x2,gt,x1−gt,x2≥−Lx1−x2,where x1≥x2,L≥0 is a constant
H2α0,β0 are coupled lower and upper solutions of type II of (1) and
(9)ft,x1−ft,x2≥−Lx1−x2,gt,x1−gt,x2≤Lx1−x2,where x1≥x2,L≥0 is a constant
Then,α0t0≤β0t0 implies α0t≤β0t on J.Proof.
Firstly, we prove that the conclusion is valid when H1 holds. To do this, let β~0t=β0t+εEq3Lt−t0q and α~0t=α0t−εEq3Lt−t0q for any small ε>0 so that β~0t>β0t, α~0t<α0t, and β~0t0>β0t0≥α0t0>α~0t0. Then, in view of H1, we have
(10)Dcqα~0≤ft,α0+gt,β0−3LεEq3Lt−t0q<ft,α~0+gt,β~0.
Similarly, we havecDqβ~0>ft,β~0+gt,α~0.
We next show thatα~0t<β~0t on J, which proves the conclusion as ε⟶0. Letting Qt=α~0t−β~0t, suppose that Qt<0 is not true on J. Then, there exists a t1∈t0,T such that Qt1=0 and Qt<0 for t∈t0,t1. In view of Lemma 5, cDqQt1≥0, that is, cDqα~0t1≥cDqβ~0t1. Thus, we arrive at the following contradiction:
(11)ft1,α~0t1+gt1,β~0t1>cDqα~0t1≥cDqβ~0t1>ft1,β~0t1+gt1,α~0t1.Similar to the proof process above, we can obtain the result of Lemma7 when H2 holds.Lemma 8.
Assume thatf,g∈CΩ,R, where Ω=t,x:α0t≤x≤β0t,t∈J, and one of the following conditions holds:
H3α0,β0 are coupled lower and upper solutions of type I of (1) such that α0t≤β0t on J, and gt,x is monotone nonincreasing in x for t∈J
H4α0,β0 are coupled lower and upper solutions of type II of (1) such that α0t≤β0t on J, and ft,x is monotone nonincreasing in x for t∈J
Then, there exists a solutionxt of (1) satisfying α0t≤xt≤β0t on J.Proof.
Suppose that H3 holds. Consider the mapping P:J×R⟶R defined by
(12)Pt,x=maxα0t,minxt,β0t.
Then(13)cDqx=ft,Pt,x+gt,Pt,x,and it has a solution xt on J with xt0=x0.
Firstly, we prove thatα0t≤xt on J. Letting α¯0t=α0t−εEqLt−t0q, for any small ε>0 such that α¯0t<α0t on J. We can prove that α¯0t<xt on J, which shows that α0t≤xt as ε⟶0. Setting mt=α¯0t−xt, suppose mt<0 is not true on J, then there exists a t2∈t0,T such that mt2=0 and mt<0 for t∈t0,t2. From Lemma 5, it follows that cDqmt2≥0, that is, cDqα¯0t2≥cDqxt2. Therefore,
(14)Dcqxt2≤Dcqα¯0t2<Dcqα0t2≤ft2,α0t2+gt2,β0t2.
On the other hand, we have(15)cDqxt2=ft2,Pt2,x+gt2,Pt2,x≥ft2,α0t2+gt2,β0t2,and it contradicts with (14). This contradiction proves the claim.Similarly, lettingβ¯0t=β0t+εEqLt−t0q, we can find that xt≤β0t.It is easy to construct the proofs of the results relative to H4. We omit the details.To obtain the results of this paper, we need to consider two-dimensional Caputo’s fractional differential systems:(16)cDqX=Ht,X,Xt0=X0,where H∈CJ×R2,R2 and X:J⟶R2 is a locally Hölder continuous function.Lemma 9.
Assume thatv,w:J⟶R2 are locally Hölder continuous functions satisfying the following:
(17)cDqv≤Ht,v,vt0≤X0,cDqw≥Ht,w,wt0≥X0,and whenever X≥Y,
(18)Hit,x1,x2−Hit,y1,y2≤Lx1−y1+x2−y2,where L≥0 is a constant, i=1,2. Then vt0≤wt0 implies vt≤wt on J.Proof.
Letw¯t=wt+εEq2Lt−t0q and v¯t=vt−εEq2Lt−t0q for any small ε>0 and ε=ε1,ε2 so that w¯t>wt, v¯t<vt, and w¯t0>wt0≥vt0>v¯t0. Consequently, we obtain, for each i
(19)cDqv¯i=cDqvi−2LεiEq2Lt−t0q≤Hit,v¯1,v¯2−LεiEq3Lt−t0q<Hit,v¯1,v¯2.
Similarly, we have(20)cDqw¯i=cDqwi+2LεiEq2Lt−t0q>Hit,w¯1,w¯2.
We next prove thatv¯t<w¯t on J, which shows the required conclusion as ε⟶0. Suppose that v¯t<w¯t is not true on J, then there exists an indexj, and a t3∈t0,T such that v¯jt3=w¯jt3 and v¯jt<w¯jt for t∈t0,t3. Set mt=v¯jt−w¯jt, it then follows from Lemma 5 that cDqmt3≥0, that is cDqv¯jt3≥cDqw¯jt3. Furthermore,
(21)Hjt3,v¯1t3,v¯2t3>cDqv¯jt3>Hjt3,w¯1t3,w¯2t3,which leads to a contradiction. This completes the proof.


## 3. Main Results

In this section, we consider thatft,x and gt,x are hyperconvex and hyperconcave in x of order m−1, respectively. We first give some inequalities depending on whether m is even or odd [34]:
(i)
m=2k
(22)ft,η≥∑i=02k−1fit,ξη−ξii!,(23)ft,η≤∑i=02k−2fit,ξη−ξii!+f2k−1t,ηη−ξ2k−12k−1!,(24)gt,η≤∑i=02k−1git,ξη−ξii!,(25)gt,η≥∑i=02k−2git,ξη−ξii!+g2k−1t,ηη−ξ2k−12k−1!.
(ii)
m=2k+1
(26)ft,η≤∑i=02k−1fit,ξη−ξii!+f2kt,ηη−ξ2k2k!,η≥ξ,(27)ft,η≥∑i=02k−1fit,ξη−ξii!+f2kt,ηη−ξ2k2k!,η≤ξ,(28)gt,η≥∑i=02k−1git,ξη−ξii!+g2kt,ηη−ξ2k2k!,η≥ξ,(29)gt,η≤∑i=02k−1git,ξη−ξii!+g2kt,ηη−ξ2k2k!,η≤ξ,Based on the above inequalities, we have the following result which is relative to the coupled lower and upper solutions of type I in Definition1 when m is even.Theorem 10.
Consider the following assumptions:
A1α0,β0 are coupled lower and upper solutions of type I of (1) with α0≤β0 on J
A2f,g∈C2kΩ,R such that ft,x and gt,x are the hyperconvex and hyperconcave in x of the order 2k−1, respectively
A3gxt,x satisfies
(30)gxt,x≤ming2kt,xβ0−α02k−12k−2!≤0onΩ
Then, there exist monotone sequencesαnt and βnt converging uniformly to the solution xt of (1) on J and the convergence is of the order 2k.Proof.
It follows from assumption A2 that the inequalities (22), (23), (24), and (25) hold. Consider the following fractional differential equations:
(31)cDqvt=Ft,α,β;v,w≜∑i=02k−1fit,αv−αii!+∑i=02k−2git,βw−βii!+g2k−1t,αw−β2k−12k−1!,vt0=x0,(32)cDqwt=Gt,α,β;w,v≜∑i=02k−2fit,βw‐βii!+∑i=02k−1git,αv−αii!+f2k−1t,αw−β2k−12k−1!,wt0=x0.
Firstly, applying (31) and (32) and taking α=α0,β=β0, we obtain
(33)cDqvt=Ft,α0,β0;v,w,vt0=x0,(34)cDqwt=Gt,α0,β0;w,v,wt0=x0.
Condition A1 and inequalities (22), (23), (24), and (25) imply
(35)cDqα0≤ft,α0+gt,β0=Ft,α0,β0;α0,β0,α0t0≤x0,Dcqβ0≥∑i=02k−1fit,α0β0−α0ii!+∑i=02k−2git,β0α0−β0ii!+g2k−1t,α0α0−β02k−12k−1!=Ft,α0,β0;β0,α0,β0t0≥x0,Dcqα0≤∑i=02k−2fit,β0α0−β0ii!+f2k−1t,α0α0−β02k−12k−1!+∑i=02k−1git,α0α0−β0ii!=Gt,α0,β0;α0,β0,α0t0≤x0,cDqβ0≥ft,β0+gt,α0=Gt,α0,β0;β0,α0,β0t0≥x0.
Employing assumption A3 and the Taylor series expansion with the Lagrange remainder, we get
(36)Fwt,α0,β0;v,w=gwt,w−g2kt,ξ2w−β02k−2ξ1−α02k−2!≤0,Gvt,α0,β0;w,v=gvt,v−g2kt,η1v−α02k−12k−1!≤0,where α0≤ξ2≤ξ1≤β0 and α0≤η1≤β0. Therefore, Ft,α0,β0;v,w and Gt,α0,β0;w,v are nonincreasing in w and v, respectively. By Lemma 8, there exist solutions α1t and β1t of (33), (34) on J such that α0t≤α1t≤β0t and α0t≤β1t≤β0t. Furthermore, in view of the inequalities (22), (23), (24), and (25) and condition A2, we have
(37)cDqα1=Ft,α0,β0;α1,β1≤ft,α1+gt,β1,α1t0=x0,cDqβ1=Gt,α0,β0;β1,α1≥ft,β1+gt,α1,β1t0=x0,and in view of H1, we have α1t≤β1t for t∈J. Hence, α0t≤α1t≤β1t≤β0t.
By induction, for alln, we can obtain that
(38)α0t≤α1t≤⋯≤αnt≤βnt≤⋯≤β1t≤β0t,where αn and βn are solutions of
(39)cDqvt=Ft,αn−1,βn−1;v,w,vt0=x0,cDqwt=Gt,αn−1,βn−1;w,v,wt0=x0,and
(40)cDqαn=Ft,αn−1,βn−1;αn,βn≤ft,αn+gt,βn,αnt0=x0,(41)cDqβn=Gt,αn−1,βn−1;βn,αn≥ft,βn+gt,αn,βnt0=x0.
According to (40) and (41), αn and βn are coupled lower and upper solutions of type I of (1). We have gx≤0 on Ω from assumption A3. It then follows from Lemma 8 that xt is a solution of (1) on J satisfying αnt≤xt≤βnt. Hence, we have
(42)α0t≤α1t≤⋯≤αnt≤x≤βnt≤⋯≤β1t≤β0t.
By (42), the sequences αn and βn are uniformly bounded on J. From Lemma 6, the sequences αn and βn are equicontinuous on J. Consequently, by employing the Ascoli-Arzela Theorem, sequences αn and βn are uniformly convergent on J.
Finally, we prove that the convergence ofαn and βn is of the order 2k.
SetWnt=xt−αnt≥0 and Vnt=βnt−xt≥0 for t∈J with Wnt0=Vnt0=0. Using the known conditions and the mean value theorem, we obtain
(43)DcqWn+1t=ft,x+gt,x−∑i=02k−1fit,αnαn+1−αnii!+∑i=02k−2git,βnβn+1−βnii!+g2k−1t,αnβn+1−βn2k−12k−1!=ft,x+gt,x−ft,αn+1−f2kt,δ1αn+1−αn2k2k!+gt,βn+1−g2k−1t,δ2βn+1−βn2k−12k−1!+g2k−1t,αnβn+1−βn2k−12k−1!=fxt,δ3x−αn+1−gxt,δ4βn+1−x+f2kt,δ1αn+1−αn2k2k!−g2kt,δ5βn−βn+12k−1δ2−αn2k−1!≤fxt,δ3Wn+1t−gxt,δ4Vn+1t+f2kt,δ1x−αn2k2k!−g2kt,δ5βn−x2k−1βn−x+x−αn2k−1!=fxt,δ3Wn+1−gxt,δ4Vn+1+f2kt,δ1Wn2k2k!−g2kt,δ5Vn2k−1Vn+Wn2k−1!≤k1Wn+1+k2Vn+1+k3Wn2k+k4Vn2k−1Vn+Wn,where αn≤δ1≤αn+1≤δ3≤x≤δ4≤βn+1≤δ2≤βn, αn≤δ5≤δ2, 0≤fxt,x≤k1, and 0≤−gxt,x≤k2 and it follows from assumption A2 that f2k and g2k are bounded on Ω, that is, 0≤f2kt,x/2k!≤k3,0≤−g2kt,x/2k−1!≤k4.
Similarly, we have(44)DcqVn+1t=∑i=02k−2fit,βnβn+1−βnii!+f2k−1t,αnβn+1−βn2k−12k−1!+∑i=02k−1git,αnαn+1−αnii!−ft,x−gt,x=ft,βn+1−f2k−1t,σ1βn+1−βn2k−12k−1!+f2k−1t,αnβn+1−βn2k−12k−1!+gt,αn+1−g2kt,σ2αn+1−αn2k2k!−ft,x−gt,x=fxt,σ3βn+1−x−gxt,σ4x−αn+1+f2kt,σ5βn−βn+12k−1σ1−αn2k−1!−g2kt,σ2αn+1−αn2k2k!≤fxt,σ3Vn+1−gxt,σ4Wn+1−g2kt,σ2x−αn2k2k!+f2kt,σ5βn−x2k−1βn−x+x−αn2k−1!=fxt,σ3Vn+1−gxt,σ4Wn+1+f2kt,σ5Vn2k−1Vn+Wn2k−1!−g2kt,σ2Wn2k2k!≤k1Vn+1+k2Wn+1+k5Vn2k−1Vn+Wn+k6Wn2k,where
(45)αn≤σ2≤αn+1≤σ4≤x≤σ3≤βn+1≤σ1≤βn,αn≤σ5≤σ1,0≤f2kt,x2k−1≤k5,0≤−g2kt,x2k!≤k6.For the following inequalities
(46)cDqWn+1t≤k1Wn+1+k2Vn+1+k3Wn2k+k4Vn2k−1Vn+Wn,Wn+1t0=0,cDqVn+1t≤k1Vn+1+k2Wn+1+k5Vn2k−1Vn+Wn+k6Wn2k,Vn+1t0=0,we can get
(47)cDqΦ≤λΦ+Ψ,Φt0=Ψ0,where
(48)Φ=Wn+1Vn+1,λ=k1k2k2k1,Ψ=Ψ1Ψ2,Ψ0=00, in which
(49)Ψ1=k3Wn2k+k4Vn2k−1Vn+Wn,Ψ2=k5Vn2k−1Vn+Wn+k6Wn2k.
Formulas (4) and (5) and Lemma 9 imply
(50)Φt≤∫t0tt−sq−1Eq,qλt−sqdsΨ≤MΨ,where
(51)M=T−t0qqEq,qk1T−t0qT−t0qqEq,qk2T−t0qT−t0qqEq,qk2T−t0qT−t0qqEq,qk1T−t0q=M1M2M2M1.
Hence, we have(52)Wn+1t≤M1k3Wn2k+k4Vn2k−1Vn+Wn+M2k5Vn2k−1Vn+Wn+k6Wn2k≤M1k3Wn2k+k4Vn2k−1Vn+Wn+M2k5Vn2k−1Vn+Wn+k6Wn2k=c1Wn2k+c2Vn2k−1Vn+Wn,where c1=M1k3+M2k6, c2=M1k4+M2k5. Similarly, we get
(53)Vn+1t≤c3Wn2k+c4Vn2k−1Vn+Wn.
Employing the Binomial Theorem(54)a+b2k=C2k0a2k+C2k1a2k−1b+⋯+C2kra2k−rbr+⋯+C2k2kb2k≥a2k+b2k+a2k−1b,and the inequalities (52) and (53), we obtain
(55)Wn+1+Vn+1t≤c1+c3Wn2k+c2+c4Vn2k−1Vn+Wn=c5Wn2k+c6Vn2k+c6Vn2k−1Wn≤c7Wn+Vn2k,that is,
(56)xt−αn+1t+βn+1t−xt≤c7xt−αnt+βn+1t−xt2k,where c1+c3=c5, c2+c4=c6, c7=maxc5,c6.
The following theorem is relative to the coupled lower and upper solutions of type II in Definition1 when m is odd.Theorem 11.
Consider the following assumptions:
D1α0,β0 are coupled lower and upper solutions of type II of (1) with α0≤β0 on J.
D2f,g∈C2k+1Ω,R such that ft,x and gt,x are hyperconvex and hyperconcave in x of order 2k, respectively.
D3fxt,x satisfies
(57)fxt,x≤−maxf2k+1t,xβ0−α02k2k−1!≤0onΩ.
Then, there exist monotone sequencesαnt and βnt converging uniformly to the solution xt of (1) on J and the convergence is of the order 2k.Proof.
It follows from assumption D2 that the inequalities (26), (27), (28), and (29) hold. In order to construct the sequences αnt and βnt, consider the following fractional differential equations:
(58)cDqαnt=Ft,αn−1,βn−1;βn,αn≜∑i=02k−1fit,βn−1βn−βn−1ii!+f2kt,αn−1βn−βn−12k2k!+∑i=02k−1git,αn−1αn−αn−1ii!+g2kt,βn−1αn−αn−12k2k!,αnt0=x0,Dcqβnt=Gt,αn−1,βn−1;αn,βn≜∑i=02k−1fit,αn−1αn−αn−1ii!+f2kt,βn−1αn−αn−12k2k!+∑i=02k−1git,βn−1βn−βn−1ii!+g2kt,αn−1βn−βn−12k2k!,βnt0=x0.
Similar to the proof of Theorem10, the conclusion can be obtained. We omit the details.



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