Tripled fixed point techniques for solving system of tripled-fractional differential equations

Abstract: The intended goal of this manuscript is to discuss the existence of the solution to the below system of tripled-fractional differential equations (TFDEs, for short):  Θ [k(α) − ג(α, k(α))] = a (α, r(α), I(r(α))) + a (α, l(α), I(l(α))) , Θ [l(α) − ג(α, l(α))] = a (α, k(α), I(k(α))) + a (α, r(α), I(r(α))), Θ [r(α) − ג(α, r(α))] = a (α, l(α), I(l(α))) + a (α, k(α), I(k(α))), k(0) = 0, l(0) = 0, r(0) = 0, a.e. α ∈ Ω, τ > 0, μ ∈ (0, 1),


Nonlinear analysis and fractional differential equations
Fractional calculus has been given proper attention in the last few decades by researchers. This subject gained new structures on an unlimited scale and are mainly applied in all branches of basic sciences, especially engineering sciences.
Because of fractional differential equations (FDEs) frequent appearance, it was particularly important, in many applications such as fluid mechanics, viscoelasticity, biology, physics and engineering. Recently, the related literature has been developed for application in FDEs in nonlinear dynamics [2][3][4][5][6]. Another reason why these equations are widespread is most FDEs do not have exact analytic solutions, approximation and numerical techniques. Consequently, it is used to give the solution of fractional ordinary differential equations, integral equations and fractional partial differential equations of physical interest.
Fixed point theory (FPT) speaks about two variants of arguments, FPT on metric spaces and topological problems under FPT. Topological problems under FPT is of particular interest to topologists and theoretical computer scientists, while FPT on metric spaces is of great importance in computing, computational biology, bio-informatics. This is another reason why the strong relationship between the FPT and the rest of the disciplines is very strong, Which leads to widespread. The main advantage of using FDEs is related to the fact that we can describe the dynamics of complex non-local systems with memory.
Another direction, nonlinear analysis used in the study of dynamical systems represented by nonlinear differential and integral equations. Since some of these equations that represent a dynamical system do not have an analytical solution, therefore studying the turmoil of these problems is very beneficial. There are different types of turmoil differential equations and the important type here is called a hybrid differential equation (HDE) [16]. From this moment, this branch has become very important for many researchers see [17][18][19]. As well as, hybrid FPT can be used to improve the existence theory for the hybrid equations.
The below first-order hybrid DE with linear turmoils of second type introduced by Dhage and Jadhav [20]: where Ω ∈ [α 0 , α 0 + ρ), ρ > 0, for some fixed α 0 , ρ ∈ R, and ‫,ג‬ ∈ C(Ω × R, R). Via this notions they discussed the existence of the minimal and maximal solution for it and obtained exciting results about the strict and nonstrict differential inequalities. The problem (1.1) developed in a fractional version under the title FHDE involving the Riemann-Liouville (RL) differential operators of order 0 < µ < 1 by Lu et al. [21] as follows:

2)
‫,ג‬ ∈ C(Ω × R, R). They showed the existence theorem for FHDEs by applying mixed Lipschitz and Carathéodory conditions. From this standpoint, the concept has become widely used in the field of fractional analysis and has become a huge turning point, see [22][23][24][25][26][27][28][29][30]. The problem (1.2) generalized to two-point boundary value problem, so-called a coupled system of FDEs and some massive results to find a solutions of coupled nonlinear fractional reaction-diffusion equations are presented, see [31,32].

Basic tools
We shall agree in this part on (Ω × R, R) refers to the class of continuous functions ‫ג‬ : Ω × R → R, and C(Ω × R × R, R) the class of functions : Ω × R × R → R such that, the mapping Hence, the class C(Ω × R × R, R) is called Carathéodory class of functions on Ω × R × R, and if it bounded by a Lebesgue integrable function on Ω, then it called Lebesgue integrable. Now we shall present some previous results that are used in the next section.
Definition 2.1. [33] The usual form of the RL-fractional integral operator of order τ is where τ > 0, and the function defined on L 1 (R + ).
Definition 2.2. [33] The usual form of the Caputo fractional derivative of the function is where τ ∈ R + (a positive real number) such that ξ − 1 < τ ≤ ξ, ξ ∈ N and (ξ) ( ) is exists, and function of class C.
The below result will be generalized in this paper as previously presented by Burton [1].
Lemma 2.5. [1] Suppose that ∇ is a Banach space, ℘ ∅ is a closed convex bounded subset of it. Let : ∇ → ∇ and : ℘ → ∇ be two operators such that (i) for all k, l ∈ ∇, < 1, we get k − l ≤ k − l , (ii) the completely continuous property hold for the operators , Then the the operator equation k = k + l has a solution in ℘.
In 2011, Coupled fixed point notion is generalized to TFP concept by Berinde and Borcut [35] in the setting of partially ordered metric spaces. Via the mentioned spaces they presented pivotal results about TFP theorems. For the authors contributions in this direction, see [36][37][38][39][40][41][42].
Here, consider Ψ refers to the family of all functions ψ :

Main theorem
In the beginning of this part, we know that ∇ = C(Ω, R) is a Banach space with respect to the supremum norm and the pointwise operations, if it defined on the supremum norm.
The two operations defined here are scalar multiplication and a sum on ∇ × ∇ × ∇ = ∇ 3 as follows: for all k, l, r ∈ ∇, ∈ R. Then ∇ 3 is a vector space. The below Lemma are very important in the sequel and his proof is clear: Then with respect to this norm, ∇ is a Banach space.

Now our main theorem in this section is valid for viewing.
Theorem 3.2. Assume that ∇ is a Banach space, ℘ ∅ is a closed, convex, and bounded subset of it and ℘ = ℘ 3 . Let : ∇ → ∇ and , Υ : ℘ → ∇ be three operators such that ( † i ) there is ψ R ∈ Ψ such that for all k, l ∈ ∇, and for some > 0, we get ( † ii ) the completely continuous property hold for the oberators and Υ; Then there exists at least a tripled fixed point (tfp) of the operator Z(k, l, r) = k + l + Υr in ℘ * , whenever ∈ (0, 1).
this leads to the operator Z(k, l, r) has at least one TFP. Now we prove that the operators , and Υ satisfy the conditions of Theorem 3.2 as follows: • Prove that is a contraction. Apply assumption ( † i ) for all k = (k 1 , k 2 , k 3 ), l = (l 1, l 2 , l 3 ), r = (r 1 , r 2 , r 3 ) ∈ ℘ , one can werite which leads to is Lipschitzian, hence it is a contraction with a constant . • Show that and Υ are compact and continuous operators on ℘ . Assume the sequence (k n ) = (k 1n , k 2n , k 3n ) ∈ ℘ converging to a point k = (k 1 , k 2 , k 3 ) ∈ ℘ , it follows by the continuity of and Υ that lim n→∞ k n = lim n→∞ k 2n , lim n→∞ k 1n , lim Hence, the operators and Υ are continuous. Also, we have similarly, or all k ∈ ℘ , where ℘ = sup{ k : k ∈ ℘} and Υ℘ = sup{ Υk : k ∈ ℘}. This shows that and Υ are uniformly bounded on ℘ .

Solve a system of tripled-fractional differential equations
In this section, we discuss the existence solution of the system (1.3) under the below hypotheses: For all α ∈ Ω, and k, l ∈ R, there is a constant ≥ γ > 0 such that .
The below lemma is very important in the existence results.
Now we are ready to present our basic theory for this part. Proof. Put ∇ = C(Ω, R) and ℘ ⊆ ∇ defined by It's obvious that ℘ is a closed, convex, and bounded subset of Banach space ∇.
By assumption (t ii ), for k, l ∈ ∇, α ∈ Ω, we have Passing the the supremum over α, one can write .
It follows from (4.2) that is a nonlinear contraction on ∇ with a control function 1 4 ψ, where ψ(v) = γr +r .
Next, we prove that and Υ are compact and continuous on ℘. Assume the sequence {k n } ∈ ℘ converging to a point k ∈ ℘, then for all α ∈ Ω and by Lebesgue dominated convergence theorem, we have Likewise, we can clarify that lim n→∞ Υk n (α) = Υk(α), for all α ∈ Ω. Hence and Υ are continuous.
The non-trivial below example support Theorem 4.2.

Conclusion
Undoubtedly, the theory of FDEs attracted many scientists and mathematicians to work on. The results have been obtained by using FPTs. FP technique play an important role in solutions of nonlinear initial-value problems of FDEs. From this point, in this manuscript, a tfp theorem and some lemmas to discuss the theoretical results are obtained. Also as an application, system of TFDEs has been created and a solution was obtained for it. Lastly, non-trivial example are presented here to support our application.