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

We investigate a nonlinear system of pantograph-type fractional di ﬀ erential 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 ﬁ xed 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.


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][2][3][4][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][7][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][15][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][22][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][25][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][28][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 with t ∈ ½1, e = H , ð ∈ ð3, 4, ‫ג‬ ∈ ð0, 1Þ 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 , k,k under the norm kyk = max t∈H jyj: Consequently, P is a Banach space such that P = Y × Y with norms kðv, YÞk = kvk + kYk or kðv, YÞk = max fkvk, kYkg. 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.

Preliminaries
Here, we recall some needful preliminary results. Definition 1. For a function v : ðJ Þ = ð1, eÞ ⟶ R, the fractional Hadamard integral is expressed as [30]: if the above integral exists.
Journal of Function Spaces ðM 6 Þ For simplicity, we introduce the notation as follows:

Theorem 7.
Let v ∈ C½1, e and x ∈ AC k σ ½1, e, the solution for linear problem converts to the following form: Proof. Thanks to Lemma (4), Equation (18) by making use of the considered boundary conditions vð1Þ = v ′ ð1Þ = 0, we get a 0 = a 1 = 0 also by from this, we can say that a 2 = 3ΦðvÞ and a 3 = −2ΦðvÞ. By making use of a 0 , a 1 , a 2 , and a 3 in (20), we obtain the solution as follows: Also, for Y ∈ C½1, e, and z ∈ AC k σ ½1, e, the solution of may be expressed as Corollary 8. The solution of the concerned problem (1) is expressed as follows: Theorem 9. Consider two functions f , g possesses continuity and then the solution of (25) is ðv, YÞ ∈ P, if f ðv, Y Þ is the solution of (1).

Journal of Function Spaces
Proof. Let ðv, YÞ, ðv, YÞ ∈ P and for all t ∈ H, we have where In a similar way, we obtain where Hence, from (28) and (30), one has where Δ = max t∈H fΔ 1 , Δ 2 g. Hence, it is obvious that F is contraction; therefore, (1) has a unique result.
Proof. Let We define a subset B of P which is closed. That is, Let us define the following operators as It is obvious thatT 1 = ℵ 1 + S 1 ,T 2 = ℵ 2 + S 2 . Further, we prove that For any ðv, Y Þ ∈ B, we have In a similar way, we obtain The preceding calculations imply that kTðv, Y Þk ≤ ρ, which clarify thatTðBÞ ⊆ B: For ðv, YÞ, ðv, YÞ ∈ B, we can write it as We can also prove that Journal of Function Spaces 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 ðv, YÞ ∈ B, we have In the same way, one can get Therefore, from (42) and (43), it implies Hence, from (44), the boundedness of ℵ can also be deduced on B. Take any ðv, Y Þ ∈ B. Subsequently, for t 1 , t 2 ∈ H with t 1 ≤ t 2 ∈ ½1, e, one has From the previous inequality, we can claim that (45) approaches to zero on t 1 ⟶ t 2 . As ℵ 1 possesses the properties of continuity and boundedness, it clearly means that ℵ 1 possesses uniform boundedness. Therefore, kℵ 1 ðvðt 2 Þ, Yðt 2 ÞÞ − ℵ 1 ðvðt 1 Þ, Yðt 1 ÞÞk ⟶ 0 as t 1 tends to t 2 . Similarly, kℵ 2 ðvðt 2 Þ, Yðt 2 ÞÞ − ℵ 2 ðvðt 1 Þ, Yðt 1 ÞÞk ⟶ 0 as t 1 tends to t 2 . Hence, all the assumptions of at least one solution for system (1) are achieved.
Proof. Let for arbitrary solutions ðv, YÞ, ðv, YÞ ∈ P, and for all t ∈ H , we have where Similarly, one has So, from (46) and (48), we get Using (50), we have Since M converges to zero, hence the result of (1) is UH stable.

Applications
Example 13. Taking a coupled system as.

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.

Data Availability
The data used in this research work is contained in paper.

Conflicts of Interest
There are no conflict of interest that exist.