# Applications of Fixed Point Theory to Investigate a System of Fractional Order Differential Equations

Authors: Zareen A. Khan; Israr Ahmad; Kamal Shah
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We investigate a nonlinear system of pantograph-type fractional differential equations (FDEs) via Caputo-Hadamard derivative (CHD). We establish the conditions for existence theory and Ulam-Hyers-type stability for the underlying boundary value system (BVS) of FDE. We use Krasnoselskii’s and Banach’s fixed point theorems to obtain the desired results for the existence of solution. Stability is an important aspect from a numerical point of view we investigate here. To justify the main work, relevant examples are provided.


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## Body

## 1. Introduction

The generalized form of ordinary calculus is called fractional calculus. This newly developed branch of mathematics has numerous applications in many scientific fields including the study of nonlinear oscillations of earthquakes, nanotechnology, and other engineering disciplines. Also, fractional derivatives and integrals have the ability to explore the dynamics of many real-world problems more comprehensively and extensively. To these characteristics of the said area, researchers in the past several decades have taken great interest to investigate FDEs for a different kind of analysis. For applications and usefulness, see [1–5]. The concerned study includes optimization, stability and numerical results, and theoretical analysis. In this regard, existence theory for different kinds of problems of FDEs has been investigated and plenty of research work has been done (see [6–8]).One of the new emerging classes of FDEs is known as pantograph differential equations (PDEs). The work related to this new research field has been published in large numbers. Initially, pantograph differential equations (PDEs) were studied with delay terms [9, 10], material modeling [11], and modeling lasers, especially quantum dot lasers [12]. Basically, PDEs give change in terms of a dependent variable at a previous time [13]. Some beneficial research has been performed in this area [14–16]. Further, these types of FDEs occur in traffic models, control systems, population dynamics, and many natural phenomena.In the last few decades, the stability analysis for FDEs has been established very well. Therefore. different kinds of stability notions have been constructed in literature including exponential, Mittag-Leffler, and Lyapunov. The mentioned stability concepts have been very well investigated for FDEs. Among these, UH stability analysis is an important tool that has gained the attention of researchers [17, 18]. The aforesaid UH stability has extended to other forms in large many articles [19, 20]. The UH stability analysis method has been developed for ordinary and FDEs over the last twenty years [21–23].It is remarkable that great interest has been observed to derive various kinds of results including qualitative and numerical for higher-order problems under BCs [24–26]. Since fractional derivative has various definitions, each and every definition has its own uncharacteristic features. One of the well-known definitions is called the Caputo-Hadamard derivative. The said area has been initiated in the last few years (for detail, see [27–29]). After that, the said definition has been used in large numbers of articles. Motivated from aforesaid work, the qualitative study of a coupled system of FDEs under BCs with fractional CHD has not been investigated properly involving proportional delay term. Therefore, using the results from fixed point theory, we studied the qualitative aspects of the system of FDEs under BCs with CHD given as
(1)CDð1+vt+ft,vגt,Yt=0,CDð1+Yt+gt,vt,Yגt=0,v1=v′1=0=v′e,ve=φv,Y1=Y′1=0=Y′e,Ye=ΨY,with t∈1,e=H,ð∈3,4,ג∈0,1 also the functions f,g:H×R×R⟶R and Φ,Ψ:Y⟶R are continuous functions. The complete norm space is defined by Y,, under the norm y=maxt∈Hy.Consequently,P is a Banach space such that P=Y×Y with norms v,Y=v+Y or v,Y=maxv,Y. We established sufficient conditions under which the problem under our investigation has at least one solution. Further, some adequate results are studied to check the stability of the UH type for the corresponding solution. These results are derived by using fixed point theory and nonlinear analysis. The analysis is justified by pertinent examples.


## 2. Preliminaries

Here, we recall some needful preliminary results.Definition 1.
For a functionv:J=1,e⟶R, the fractional Hadamard integral is expressed as [30]:
(2)I1+ðvt=1Γð∫1tlntθð−1vθdθθ,if the above integral exists.Definition 2.
For a functionv:J⟶R, the fractional Hadamard derivative is denoted as [30]:
(3)CD1+ðvt=σkI1+k−ð=1Γk−ðtddtk∫1tlntθk−ð−1vθdθθ,where k=ð+1 and σ=td/dt.Lemma 3 (see [30]).
Letvt∈ACσk1,e, then for fractional differential equation (FDE)
(4)CD1+ðvt=0,ð∈k−1,k,the solution is given as follows:
(5)vt=∑j=0k−1ajlntj,j=1,2,3,⋯,k−1,whereaj∈R.Lemma 4 (see [30]).
The FDE holds the result in the following:(6)Ia+ðCDað+vt=vt+∑j=0k−1ajlntj,j=1,2,3,⋯,k−1,where k=ð+1.Definition 5 (see [31]).
Let for operatorsV1,V2∋V1,V2:P⟶Y, denoted by
(7)vt=V1v,Yt,Yt=V2v,Ytis called UH stable if for real positive constants aii=1,2,3,4,δii=1,2 and for each solution v∧,Y∧∈P, we have
(8)v∧−V1v∧,Y∧≤δ1,Y∧−V2v∧,Y∧≤δ2,there exist a solution v̄,Ȳ∈P of (7),∋(9)V1v̄,Ȳ−V1v∧,Y∧≤b1v̄−v∧+b2Ȳ−Y∧,V2v̄,Ȳ−V2v∧,Y∧≤b3v̄−v∧+b4Ȳ−Y∧.
Furthermore, if the matrix(10)M=b1b2b3b4converges to zero, then the solution of (7) is UH stable.Theorem 6 (see [32–34]).
LetE≠∅ be closed convex subset of the Banach space P, and there exist two operators F,ℵ such that aFx+ℵy∈E whenever x,y∈E,bF is continuous and compact, and cℵ is contraction. So one has z=v,Y∈E such that Fz+ℵz=z.
M1 For all v,Y∈CH,R,∃RΦ∗,Rψ∗>0∍(11)Φv−Φv̄≤RΦ∗v−v̄,ψY−ψȲ≤Rψ∗Y−Ȳ.
M2 For all v,v̄,Y,Ȳ∈CH,R∀t∈H∃Lf∗>0∋(12)∣ft,vגt,Yt−ft,v̄גt,Ȳt∣≤Lf∗∣v−v̄+Y−Ȳ∣.
M3 For all v,v̄,Y,Ȳ∈CH,R∀t∈H∃L∗g>0∋(13)ℵt,vt,Yגt−ℵt,v̄t,Ȳגt≤Lg∗v−v̄+Y−Ȳ.
M4 There exist positive real numbers Cf∗,Df∗, and Mf∗,Mf∗∋(14)ft,vגt,Yt≤Cf∗v+Df∗Y+Mf∗,∣ℵt,vt,Yגt∣≤Cg∗∣v∣+Dg∗Y+Mg∗.
M5 There exist positive real numbers κii=1,2,βΦ,βΨ∋(15)Φv≤κ1v+βΦ,ΨY≤κ2Y+βΨ.
M6 For simplicity, we introduce the notation as follows:
(16)ℏt=3lnt2−2lnt3.


## 3. Main Results

Theorem 7.
Letv∈C1,e and x∈ACσk1,e, the solution for linear problem
(17)CD1+ðvt=xt,t∈H,ð∈3,4,(18)v1=v′1=v′e=0,ve=Φvconverts to the following form:
(19)vt=Φvℏt+lΓð∫1tlntθð−1xθdθθ.Proof.
Thanks to Lemma (4), Equation (18) obtained the form
(20)vt=I1+ðxt+a0+a1lnt+a2lnt2+a3lnt3,by making use of the considered boundary conditions v1=v′1=0, we get a0=a1=0 also by
(21)ve=Φv,v′e=0⇒,Φv=a2+a3,0=2a2+3a3,from this, we can say that a2=3Φv and a3=−2Φv. By making use of a0,a1,a2, and a3 in (20), we obtain the solution as follows:
(22)vt=Φvℏt+lΓð∫1tlntθð−1xθdθθ.
Also, forY∈C1,e, and z∈ACσk1,e, the solution of
(23)CD1+ðYt=zt,t∈H,ð∈3,4,Y1=Y′1=Y′e=0,Ye=ΨYmay be expressed as
(24)yt=Ψyℏt+lΓð∫1tlntθð−1zθdθθ.Corollary 8.
The solution of the concerned problem (1) is expressed as follows:
(25)vt=Φvℏt+lΓð∫1tlntθð−1fθ,vגθ,Yθ,dθθ,yt=Ψyℏt+lΓð∫1tlntθð−1gθ,vθ,Yגθ,dθθ.Theorem 9.
Consider two functionsf, g possesses continuity and then the solution of (25) is v,Y∈P, if fv,Y is the solution of (1).Proof.
Ifv,Y is the solution of (25), then by the differentiation of both sides of (25), we have (1). However, if v,Y is a solution of (1), then v,Y is the solution of (25).LetF1,F2:P⟶P∋(26)F1v,Y=Φvℏt+lΓð∫1tlntθð−1fθ,vגθ,Yθdθθ,F2v,Y=Ψyℏt+lΓð∫1tlntθð−1gθ,vθ,Yגθdθθ,and Fv,Y=F1v,YF2v,Y. Hence, solution of (25) is a fixed point of F.Theorem 10.
IfΔ<1, with the help of assumptions M1−M3, system (1) has at most one solution, where
(27)Δ=maxRΦ∗+Lf∗Γð+1,RΨ∗+Lg∗Γð+1.Proof.
Letv,Y,v̄,Ȳ∈P and for all t∈H, we have
(28)F1v,Y−F1v̄,Ȳ≤maxt∈H∣Φv−Φv̄∣ℏt+maxt∈HlΓð∫1tlntθð−1fθ,vגθ,Yt−fθ,v̄גθ,Ȳtdθθ≤maxt∈HRΦ∗v−v̄ℏt+maxt∈HlntðLf∗Γð+1v−v̄+Y−Ȳ≤RΦ∗+Lf∗Γð+1v−v̄+Lf∗Γð+1Y−Ȳ≤Δ1Y−Ȳ+Y−Ȳ,where
(29)Δ1=RΦ∗+Lf∗Γð+1.
In a similar way, we obtain(30)F2v,Y−F2v̄,Ȳ≤Δ2v−v̄+Y−Ȳ,where
(31)Δ2=RΨ∗+Lg∗Γð+1.
Hence, from (28) and (30), one has
(32)Fv,Y−Fv̄,Ȳ∣≤maxΔ1,Δ2v−v̄+Y−Ȳ=Δv−v̄+Y−Ȳ,where Δ=maxt∈HΔ1,Δ2. Hence, it is obvious that F is contraction; therefore, (1) has a unique result.Theorem 11.
In the light of hypothesesM1,M4,andM5 together with condition maxY1,Y2<1, system (1) has a minimum of one solution.Proof.
Let(33)J1=κ1+κ2+Cf∗+Df∗+Cg∗+Dg∗Γð+1,J2=βΦ+βΨ+Mf∗+Mg∗Γð+1.
We define a subsetB of P which is closed. That is,
(34)B=v,Y∈P:v,Y≤ρ,forρ≥maxJ21−J1.
Let us define the following operators as(35)ℵ1v,Y=1Γð∫1tlntθð−1fθ,vגθ,Yθdθθ,ℵ2v,Y=1Γð∫1tlntθð−1gθ,vθ,Yגθdθθ,S1vt=Φvℏt,S2Yt=ΨYℏt.
It is obvious thatT~1=ℵ1+S1,T~2=ℵ2+S2. Further, we prove that
(36)T~v,Y=ℵv,Y+Sv,Y∈B,forallv,Y∈B.
For anyv,Y∈B, we have
(37)T1v,Y=Φvℏt+lΓð∫1tlntθð−1fθ,vגθ,Yθdθθ≤maxt∈H+Φvℏt+maxt∈H1Γð∫1tlntθð−1fθ,vגθ,Yθdθθ≤maxt∈Hℏtκ1v+βΦ+maxt∈HCf∗v+Df∗Y+Mf∗1Γð+1lntað≤κ1ρ+βΦ+Cf∗ρ+Df∗ρ+Mf∗Γð+1≤ρ2.
In a similar way, we obtain(38)∣T~2v,Y∣≤κ2ρ+βΨ+Cg∗ρ+Dg∗ρ+Mg∗Γð+1≤ρ2.
The preceding calculations imply thatT~v,Y≤ρ, which clarify that T~B⊆B. For v,Y,v̄,Ȳ∈B, we can write it as
(39)S1v−S1v̄≤maxt∈HΦv−Φv̄≤maxt∈HRΦ∗ℏtv−v̄≤Y1v−v̄.
We can also prove that(40)S2Y−S2Ȳ≤Y2Y−Ȳ,where
(41)Y1=RΦ∗,Y2=RΨ∗.
Clearly, (39) and (40) assure the contraction of S. Now, we need to show the relative compactness of ℵ. Now, as f and g are continuous, hence ℵ is continuous too. For v,Y∈B, we have
(42)ℵ1v,Y≤maxt∈HlΓð∫1tlntθð−1fθ,vגθ,Yθdθθ≤Cf∗ρ+Df∗ρ+Mf∗Γð+1.
In the same way, one can get(43)ℵ2v,Y≤Cg∗ρ+Dg∗ρ+Mg∗Γð+1.
Therefore, from (42) and (43), it implies
(44)ρ≥ℵv,Y.
Hence, from (44), the boundedness of ℵ can also be deduced on B. Take any v,Y∈B. Subsequently, for t1,t2∈H with t1≤t2∈1,e, one has
(45)ℵ1vt1,Yt1−ℵ1vt2,Yt2≤1Γð∫1t1lnt1θð−1−lnt2θð−1fθ,vגθ,Yθdθθ+1Γð∫t1t2lnt2θð−1fθ,vגθ,Yθdθθ≤1Γð+1Cf∗v+Df∗Y+Mf∗×lnt2ð+2lnt2t1ð−lnt1ð.
From the previous inequality, we can claim that (45) approaches to zero on t1⟶t2. As ℵ1 possesses the properties of continuity and boundedness, it clearly means that ℵ1 possesses uniform boundedness. Therefore, ℵ1vt2,Yt2−ℵ1vt1,Yt1⟶0 as t1 tends to t2. Similarly, ℵ2vt2,Yt2−ℵ2vt1,Yt1⟶0 as t1 tends to t2. Hence, all the assumptions of at least one solution for system (1) are achieved.


## 4. Stability Results

Theorem 12.
Under the hypothesisM1−M3 together with condition Δ<1, the considered system has UH stable solution.Proof.
Let for arbitrary solutionsv,Y,v̄,Ȳ∈P, and for all t∈H, we have
(46)F1v,Y−F1v̄,Ȳ≤maxt∈H∣Φv−Φv̄∣ℏt+maxt∈H1Γð∫1tlntθð−1fθ,vגθ,Yt−fθ,v̄גθ,Ȳtdθθ≤maxt∈HℏtRΦ∗v−v̄+maxt∈HlntðLf∗Γð+1v−v̄+Y−Ȳ≤RΦ∗+Lf∗Γð+1v−v̄+Lf∗Γð+1Y−Ȳ≤b1Y−Ȳ+b2Y−Ȳ,where
(47)b1=RΦ∗+Lf∗Γð+1,b2=Lf∗Γð+1.
Similarly, one has(48)F2v,Y−F2v̄,Ȳ≤b3Y−Ȳ+b4Y−Ȳ,where
(49)b3=RΨ∗+Lg∗Γð+1,b4=Lg∗Γð+1.
So, from (46) and (48), we get
(50)F1v,Y−F1v̄,Ȳ≤b1Y−Ȳ+b2Y−Ȳ,F2v,Y−F2v̄,Ȳ≤b3Y−Ȳ+b4Y−Ȳ.
Using (50), we have
(51)M=b1b2b3b4.
SinceM converges to zero, hence the result of (1) is UH stable.


## 5. Applications

Example 13.
Taking a coupled system as.(52)CD1+3.8vt+sin∣v0.3t∣+cos∣Yt∣+et+4t2+103=0,t∈H,CD1+3.8Yt+t3+∣vt−Y0.3t∣et+50=0,t∈H,v1=v′1=0=v′e,ve=sin∣v∣30,Y1=Y′1=0=Y′e,Ye=cosY29.
From above,ð=3.8,ג=0.3 after calculation, we have Lf∗=0.0001,Lg∗=0.02,RΦ∗=0.033,RΨ∗=0.034,
(53)Δ1=0.0331,Δ2=0.0351.
It is obvious thatmaxΔ1,Δ2=0.0351<1. So (52) has a unique solution by Theorem 10. Moreover, from the values of bi,i=1,2,3,4, we have
(54)M=0.03310.00010.03510.0011,after calculation, the eigenvalues are ð1=0.0332,ð2=0.0010. Therefore ΛM=0.0332<1. Thus, the given system is HU stable by using Theorem 12.Example 14.
Consider the following problem:(55)CD1+3.7vt+arctant10+v0.4t=0,t∈H,CD1+3.7Yt+lnt8+Y0.4t=0,t∈H,v1=v′1=0=v′e,ve=v+t260,Y1=Y′1=0=Y′e,Ye=sinY25.
From above,ð=3.7,ג=0.4 after calculation, we have Lf∗=0.1218,Lg∗=0.125,RΦ∗=0.016,RΨ∗=0.04,
(56)Δ1=0.0239,Δ2=0.0481.
It is obvious thatmaxΔ1,Δ2=0.0481<1. So (55) has a unique solution by Theorem 10. Moreover, from the values of bii=1,2,3,4, we have
(57)M=0.02390.00790.04810.0081,after calculation, the eigenvalues are ð1=0.0351,ð2=−0.0102. Therefore, ΛM=0.0351<1. Thus, the given system is HU stable by using Theorem 12.


## 6. Conclusion

In this research work, nonlinear BVPs of FDEs containing proportional delay with CHD operator have been successfully investigated. We have utilized the techniques of fixed point theory and nonlinear analysis, to develop the existence and stability results for the proposed system. Through some examples, the main results have been justified. In the future, one can investigate the aforementioned system of FDEs for more complicated boundary conditions.



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