# Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique

Authors: H. R. Marasi; H. Aydi
Journal: Journal of Mathematics
 (2021)

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## Abstract

The work addressed in this paper is to ensure the existence and uniqueness of positive solutions for initial value problems for nonlinear fractional differential equations with two terms of fractional orders. By virtue of recent fixed point theorems on mixed monotone operators, we get some new straightforward results with a wide range of applications.


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## Body

## 1. Introduction

In recent years, we see a great interest to consider fractional differential equations in various kinds of studies. Namely, these equations have striking applications in diverse science. Moreover, during the past decades, new areas for applications of fractional differential and integral equations with initial boundary conditions have been appeared. In the modeling of significant numbers of phenomena occurring in engineering and scientific subjects, we encountered a problem of such kinds. In multiple cases, modeling by fractional operators can also be done better than modeling by ordinary derivative operators. One can see fractional operators in applications of rheology, control, viscoelasticity, porus structures, chemical physics, electrochemistry, and significant number of other branches of science (see [1–3] for more details). For these reasons, there exist many articles about the existence and uniqueness of solutions of problems containing operators of fractional orders and using fixed point techniques [4–27]. But, the number of research studies on the study of existence and uniqueness of solutions for FDEs with two or more terms of fractional operators is limited. Fujita [28] studied the following specific Cauchy problem:(1)∂α1∂twt,x=∂α2∂twt,x,1≤α1,α2≤2,and established some existence and uniqueness results. In [29], the author considered an initial value problem of the following form:(2)Dαyt=gt,Dβyt,0<t≤1,(3)yk0=μk,0≤k≤n−1,k∈Z,where n−1<β<α<n, and studied existence and uniqueness of related solutions. Based on the results of the above paper, the following exacting conditions should be satisfied on g for the existence of solutions:(C1)
g is a continuous and differentiable function on 0,1×R⟶R.(C2)
Fort∈I, gt,0≠0 and g0,0=0, where I is any compact subinterval of 0,1.(C3)
There exists a continuous functionq which satisfies(4)Csupt∈0,1∫0tt−rα−β−1qrdr<1,n−1<β<α<n,whereC=1/Γα−β.To detect a unique solution other than (C1)–(C3), the following restriction should hold.(C4)
Letw1 and w2 be any real numbers and q∈C0,1,0,∞ be the continuous function appeared in (4), then(5)gt,w1−gt,w2≤qtw1−w2,0≤t≤1.The authors in [16, 30] have studied the same problem. In [16], we observe that the necessary conditions to exist at least one solution for (2) and (3) are (C1), (C2), and the following condition.(C5)
For a continuous functiony∈C0,∞, there exist positive numbers c and d such that(6)gt,yt≤cyt+d,0≤t≤1.Also, the additional condition thatgt,xt satisfies a Lipschitz condition is needed to prove the uniqueness of the solution:(7)gt,z1t−gt,z2t≤Mz1t−z2t,0≤t≤1,whereM is the Lipschitz constant. In [31], the following general form is considered:(8)Dαyt=gt,yt,Dβyt,0<t≤1,(9)yk0=μk,0≤k≤n−1,k∈Z,where n−1<β<α<n,n∈N and Dα and Dβ are the fractional derivative orders in the Caputo definition and g∈C0,1×R. We have used an index fixed point theorem and obtained some new results about existence and also multiplicity of solutions. In this paper, we consider the same general form (8) and (9) and try to use new fixed point theorems proved for operators having mixed monotone property to establish new results with a wide range of applications for existence and uniqueness.


## 2. Preliminaries

For convenience of readers, we provide some useful definitions and previous results which we use throughout the paper.Definition 1.
(see [1, 3]). Let α>0. The operator defined by(10)Iαyt=∫0t1t−rα−1Γαyrdr,where y∈C0,∞∩Lloc10,∞, α∈m−1,m and m∈N, is the fractional integral of y of order α in Riemann–Liouville definition.
After introducing the fractional integral operator, it is time to remind the fractional differentiation definitions. There are several ways to define fractional derivative. Here, we focus on the Caputo definition.Definition 2.
(see [1, 3]). Let n=α+1 where α≥0. If yt∈ACn0,1, then the fractional derivative of y in Caputo sense exists on the interval 0,1, almost everywhere, and is obtained by(11)Dcαyt=∫0tynrΓn−αt−rα−n+1dr.
For the sake of convenience, here, we mention some properties of Caputo fractional operator. For the power functionft=tq,q≥0, taking the Caputo derivative yields(12)Dcαtq=Γq+1Γq−α+1tq−α,q≥n−1,0,q≤n−1.
By virtue of this formula, one can deduceDcαk=kDcαt0=0. Therefore, unlike Riemann–Liouville derivative, the Caputo derivative of a constant function is zero. In the following equality:(13)Dcαa0tr+a1tr−1+⋯+a1=0,α∈m−1,m,the polynomial’s degree is less than or equal to m−1, i. e, r≤m−1. From the following relation, it is clear that similar to the case of integer order, the αth fractional integral of the Caputo fractional derivative of order α requires to know the values of the function and its integer order derivatives:(14)IαDαgt=gt−∑l=0m−1glt0+tll!,m−1<α≤m.
The results of the paper about the existence and uniqueness are expressed by the famous beta function given as follows.Definition 3.
The following integral defines the beta function:(15)Bx,y=∫01sx−11−sy−1ds,x,y>0.
It is known that(16)Br,s=ΓrΓsΓr+s,where Γ denotes the Gamma function.
The following lemma transforms problems (8) and (9) to an integral equation [29, 30], at which the nonlinear term g needs to satisfy the conditions (C1) and (C2).Lemma 1 (see [29]).
Letn∈N,β<α and α,β∈n−1,n. If (C1) and (C2) hold, then y∈Cn0,1 is a solution of the following problem:(17)Dαyx=gx,yx,Dβyx,0<x≤1,(18)yk0=μk,0≤k≤n−1,k∈Z,if and only if(19)yx=∑k=0n−1xkk!ηk+1Γβ∫0xx−rβ−1wrdr,0≤x≤1,where w is a continuous function and satisfies the following integral equation:(20)wx=∫01Kx,rgr,wrdr,0≤x≤1,where(21)Kx,r=1Γα−βx−r+α−β−1,with(22)x−r+α−β−1=x−rα−β−1,r≤x,r,x∈0,10,r≥x.
Here and anywhere below, the functionKx,r is the Green function.Lemma 2.
Letn, m∈N and n−1<β≤m−1<α<m. If (C1) and (C2) hold, then y∈Cm0,1 is a solution of (17) and (18) if and only if(23)yx=∑l=0n−1xll!ηl+∫0xx−rn−1wrdrn−1!,0≤x≤1,where w∈C0,1 satisfies the following integral equation:(24)wx=∑l=0m−n−1xll!ηn+l+1Γα−n∫0xx−rα−n−1×fr,wr,1Γn−β∫0rr−hn−β−1whdhdr,0≤x≤1.
Now, we present the following lemma which will be useful in proving an existence and uniqueness theorem of the solution of (8) and (9).Lemma 3.
For the Green functionGt,s, the following relations hold:(25)∫01Kx,rdr≤Bα−β,α−β,0≤x≤1,∫01Kx,rdr≥Γα−βΓ2α−β,0≤x≤1.Proof.
By making use of (14), it is easy and straightforward to prove the properties for Kx,r.
Now, we provide some definitions and a fixed point theorem involving mixed monotone operators. We suggest references [19, 20] for more details.Definition 4.
LetE,. be a Banach space and P⊆E be a nonempty closed convex subset. P is a cone in E if(I)
For anyx∈P and λ≥0, then λx∈P(II)
For anyx∈P, if −x∈P then x=θwhere θ denotes the zero element of the Banach space E.
If there exists a constantC>0 such that, for any x,y∈P with θ≤x≤y, we have x≤Cy, then P is called a normal cone. Then, we give the definition of a partially ordered Banach space. We say that E is partially ordered by P if x≤y iff y−x∈P, for any x,y∈P. For arbitrary x1,x2∈E, the ordered interval is defined by x1,x2=x∈E|x1≤x≤x2. Let T:E⟶E be an operator. If u≤v implies Tu≤Tv, then T is called increasing.
Let us briefly recall that the operatorT:P×P⟶P is increasing in its first variable and decreasing in its second variable, if from xi,yi∈Pi=1,2,x1≤x2,y1≥y2, one can deduce Tx1,y1≤Tx2,y2. If x≥θ implies that Tx≥θ, then the operator T is called positive.
In the following definition,P is a cone on the Banach space E,..Definition 5.
(see [19, 20]). Suppose T:P×P⟶P is increasing in its first variable and decreasing in its second variable, then T is called a mixed monotone operator.
If for an elementx∈P, Tx,x=x, then x is called a fixed point of T.Theorem 1 (see [4]).
SupposeT:P×P⟶P is a mixed monotone operator and let the following conditions hold:(i)
Ifc∈0,1,x,y∈P, then there exists αc,x,y∈1,∞ such that(26)Tcx,y≤cαc,x,yTx,y.(ii)
There exist two elementsu0,v0∈P,r∈0,1 such that(27)u0≤rv0,Tu0,v0≥u0,Tv0,u0≤v0.Then,T has a unique fixed point u∗in u0,rv0 such that Tu∗,u∗=u∗. Moreover, one can construct successively the following iterates:(28)un=Tun−1,vn−1,vn=Tvn−1,un−1,n=1,2,…,starting fromx0,y0∈u0,rv0 to get the fixed point of T. Also, un−u∗⟶0,vn−u∗⟶0 as n⟶∞.


## 3. Main Results

In this section, we prove new existence and uniqueness results for nonlinear fractional differential equations with two terms of fractional derivatives. Compared with the results of the previous studies, our results are simple and straightforward. They are expressed with the values of beta and gamma functions. For these reasons, these results can be applied for a wide range of problems. To make effective our obtained result, some examples provided at the end of section cannot be considered by techniques of [16, 30]. In order to apply the fixed point results concerning the mixed monotone operators for the study of (8) and (9), we first consider E=C0,1 as a suitable Banach space, and let the norm on E be defined by(29)y=maxyt,0≤t≤1.Let us consider(30)P=y∈E|minyt≥0,0≤t≤1,CDαis positive.It is obvious thatP is a normal cone in E.In the following theorem,g is the function appeared in the right side of equation (8).Theorem 2.
Assume that(C6)
gt,u,v is a continuous function which is increasing in its first variable u and decreasing in its second variable v.(C7)
For anyη∈0,1, let uandv be two arbitrary elements of P so that there exists λη,u,v∈1,∞ such that(31)gt,ηu,v≤ηλη,u,vgt,u,v.(C8)
Letu0andv0 be two elements of P and μ∈0,1 such that(32)u0t≤μv0t,gt,u0,Dβv0≥u0Γ2α−βΓα−β,gt,u0,Dβv0≤v0Bα−β,α−β.Then, for equation (8) with initial conditions (9), we have a unique solution u∗∈u0,μv0. Furthermore, by computing the iterates,(33)un+1t=∫0tKt,rgr,unr,∂∂rvnrdr,vn+1t=∫0tKt,rgr,vnr,∂∂runrdr.Forn=0,1,…, one can find the unique solution. In other words, un−u∗⟶0,vn−u∗⟶0 as n⟶∞.Proof.
First, we define the operatorT as follows:(34)Tu,vt=∫0tKt,rgr,ur,Dβvrdr,where Kt,r is the Green function (21). Taking the definition of the operator T into account and Lemma 3, it is clear that a certain function ut is a solution of (2) and (3) iff ut is a fixed point of T, that is, Tu,u=u. Hence, it will be sufficient to show that T satisfies the conditions of Theorem 1. By virtue of (C6), it immediately follows that T is increasing in its first variable u and decreasing in its second variable v. On the other hand, under the hypothesis of (C7), if η∈0,1 and u,v are two elements of P, then there exists λη,u,v∈1,∞ such that(35)Tηut,vt=∫0tKt,rgr,ηur,∂∂rvrdr≤ηα∫0tKt,rgr,ur,∂∂rvrdr=ηαTut,vt.
Also, with the help of (C8), and using Lemma3, we can choose u0 and v0 such that(36)Tu0t,v0t=∫0tKt,rgr,u0r,∂∂rv0rdr≤u0,Tv0t,u0t=∫0tKt,rgr,v0r,∂∂ru0rdr≥v0.
Consequently, it is obvious that all essentials of Theorem1 are provided by (C6)–(C8), and therefore, there is a unique positive solution u∗,u∗ so that Tu∗,u∗=u∗. It is the unique solution of (8) and (9). The proof is complete.Example 1.
Let us consider the following set of nonlinear differential equations of fractional order(37)Dcm+3/4yt=ψt−Dcm+1/4yt2,0<t≤1,subject to(38)yl0=0,l=0,1,2,…,m,where m is a positive integer, ψt is a continuous function such that HTML translation failed, and η1=π≤ψt≤mπ=η2. In connection with Theorem 2, set α=m+3/4 and β=m+1/4. By virtue of this theorem and Lemma 3, the existence of a unique solution for problems (37) and (38) is equivalent to a unique solution for the following integral equation:(39)yt=∫0tKt,rψr−Dcm+1/4yr2dr,where with the help of Lemma 3, one can deduce L1=1/π≤∫01Kt,rdr≤2/π=L2. We note that if u,v∈P, defined by (30), then gt,u,v=ψt−Dcm+1/4vt2 is increasing in u and decreasing in v. Now, let us define Au,v=∫0tKt,rψr−Dcm+1/4vr2dr and choose u0=1,v0=2m. Then, considering that u0,v0 are constant, we can show that(40)u0≤12v0,Au0,v0≥L1η1≥1=u0,Av0,u0≤2πη2≤2m=v0.
Therefore, using Theorem2, we conclude that problems (37) and (38) possess a unique positive solution such that 1≤yt≤m.Example 2.
In this example, we find a unique positive solution for the nonlinear fractional initial value problems of the following form:(41)Dcn+2/3yt=χt−1yt−Dcn+2/3yt,0<t≤1,(42)yl0=0,l=0,1,2,…,n,where n∈Z+, χt is a continuous function such that χ0≠0, and τ1=315/100≤χt≤945/100=τ2. In connection with Theorem 2, set α=n+2/3 and β=n+1/3. By virtue of this theorem and Lemma 3, the existence of a unique solution for problems (41) and (42) is equivalent to a unique solution for the following integral equation:(43)yt=∫0tKt,rχr−1yr−Dcn+1/3yrdr,where with the help of Lemma 3, one deduces 2≤∫01Kt,rdr≤53/10. We note that if u,v∈P are defined by (30), then gt,u,v=χs−1/ut−Dcn+1/3vt is increasing in u and decreasing in v.
Forc∈0,1, consider 1<αc,u,v<2; then, we have(44)gt,cut,vt=χt−1cut−Dcn+1/3vt≤cαχt−1ut−Dcn+1/3vt=cαgt,ut,vt.
Now, let us defineAu,v=∫0tKt,rχr−1/ur−Dcn+1/3vrdr and choose u0=53/10,v0=51. Then, regarding that u0,v0 are constant functions, we get(45)Au0,v0=∫01Kt,rχr−1u0−Dcn+1/3v0dr≥2γ1−1≥5310=u0,Av0,u0=∫01Kt,rχr−1v0−Dcn+1/3u0dr≤5310γ−2v0≤51=v0.
Therefore, using Theorem2, we conclude that for problems (38)–(41), there is a unique positive solution such that yt∈53/10,51 for each m.


## 4. Conclusion

In this research, we considered nonlinear fractional differential equations of arbitrary order with two terms of fractional orders. By virtue of recent fixed point theorems on normal cones for mixed monotone operators, we obtained some new results for existence and uniqueness of solutions of these equations involving initial conditions, which are more applicable compared with previous existing results. The previous results can be proved based on very strict conditions on the functiong. While, in our paper, the function gt,u,Dαu≡gt,u,v needs to be increasing in its first variable u and decreasing in its second variable v. Also, establishing the conditions (C7) and (C8) is not difficult. On the other hand, our results provided a constructive approach, based on an iterative relation, to find the solution. Indeed, the interval 0,1 can be generalized to 0,T. The presented examples, at the end of the paper, cannot be considered by the previous works.



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