Existence results for coupled differential equations of non-integer order with Riemann-Liouville, Erd{\'e}lyi-Kober integral conditions

1 Department of Mathematics, Cankaya University, Ankara, Turkey 2 Institute of Space Science, Magurele-Bucharest, Romania 3 Department of Medical Research, China Medical University, Taichung, Taiwan 4 Department of Mathematics, Sasurie College of Arts and Science, Vijayamangalam, India 5 Department of Mathematics, Gobi Arts and Science College, Gobichettipalayam, India 6 Department of Electrical and electronics engineering, KPR Institute of Engineering and Technology, Coimbatore, India 7 Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore, India


Introduction
Fractional calculus is one of the most widely used mathematical analysis which deals with different ways to represent the real and complex number powers of the differentiation or integration operator and creating a calculus for the same operators in the generalized form. This calculus has numerous applications in the fields of science and engineering viz viscoelasticity, engineering mechanics, control systems, biological population models, etc. In specific, this branch of mathematics involves the methods and notion to solve the differential equations concern with a fractional derivative of unknown function which is also called fractional differential equations (FDEs). Moreover, this fractional calculus has been widely employed for modeling the engineering and physical processes which are possibly represented in terms of FDEs. This type of fractional derivative model is utilized in order to provide accurate modeling of those systems which needs to be accurate modeling of damping and also has the capability of modeling the complex engineering problems [10][11][12][14][15][16][17]23]. In recent years, a variety of numerical and analytical modeling approaches with their applications to new problems have been addressed in the research field of mechanics, electrodynamics of complex medium, and aerodynamics, etc. The application of Erdélyi-Kober fractional integrals is discussed in detail with the examples in [7,11,12,25,27,31]. Unlike integer derivatives, fractional derivatives access the system's global evolution rather than just its local characteristics; as a result, when dealing with certain phenomena, they provide more accurate models of real-world behaviour than standard derivatives. In real life, differential equations of fractional order are used to calculate the movement or flow of electricity, the motion of an object back and forth like a pendulum, and to explain thermodynamic concepts, etc. Additionally, in medical terms, they are used to visualize the progression of diseases. They represent real-world behaviour more accurately than standard derivatives. The coupled system consists of a couple of differential equations with pair of dependent variables and a single independent variable. The coupled system of FDEs becomes a more popular research field due to its vast applications in real-time problems namely anomalous diffusion, ecological models, chaotic systems, and disease models [1,2,8,20,24,26]. Boundary value problems (BVPs) applied to a coupled system with nonlinear differential equations attracting researchers because of its applications in plasma physics and heat conduction; see [3-6, 18, 19, 21, 22, 28, 29, 32], and the references cited therein. The nonlinear coupled system of Riemann-Liouville FDEs x(t), y(t)), RL D p y(t) = g(t, x(t), y(t)), for 0 < t < T and 1 < q, p ≤ 2, was investigated in [30], where RL D q , RL D p denote the Riemann-Liouville fractional derivatives (RLFDs) of order q, p, f , g : [0, T ] × R × R → R are given continuous functions, and α i , β i ∈ R, i = 1, 2, · · · , n are positive real constants. Fixed-point theorems were also employed to prove the main results. The Caputo type FDEs nonlinear coupled system for 0 < t < 1, n − 1 < α 1 ≤ n, m − 1 < α 2 ≤ m, and n, m ≥ 2, were examined in [34], where λ i > 0 is a parameter, D α i 0+ is the standard Caputo derivative; µ i > 0 is a constant, 1 0 a(s)v(s)dA 1 (s), 1 0 b(s)u(s)dA 2 (s) denote the Riemann-Stieltjes integrals. Leray-Schauder's alternative and the contraction mapping principle proved the existence and uniqueness of solutions. In this study, a coupled system with non-linear FDEs is considered and which is represented as in where c D j represents the Caputo derivatives of order j, { j = ς, , 1 , ς 1 }, J p and J q are the Riemann-Liouville integrals of order p, q > 0 and I i ,θ i σ i (i = 1, 2) is the Erdélyi-Kober integrals of order σ i > 0, θ i > 0, i ∈ R(i = 1, 2), f, g : H × R × R → R are continuous functions and µ i , τ i (i = 1, 2) are real constants. The structure of this proposed work is as follows: Section 2 deals with some facts and definitions related to this study. Section 3 gives a solution for the system described in Eq (2) and (3). The examples of the proposed problem are drawn to validate the applications in Section 4. Finally, the discussion is presented.

Preliminaries
This section recollects the definitions and some basics facts related to the proposed study are presented [9,12,23,33].
provided that the right hand side is point wise defined on [0, ∞).
provided the right hand side is point wise defined on R + . Remark 2.4. For η = 1, the above operator is reduced to the Kober operator that was introduced for the first time by Kober in [13]. For 1 = 0, the kober operator is reduced to the Riemann-Liouville integral with a power weight: is equivalent to the fractional integral equations Here the non zero constants Λ 1 and Λ 2 are Proof. The general solution for the Eq (2.1) can be expressed as where c 0 , c 1 , d 0 , d 1 are arbitrary constants. Substituting the (1.4) in Eqs (2.9) and (2.10), the following equations will be obtained. (1) , Substituting the values of c 0 , c 1 , d 0 , d 1 in (2.9) and (2.10) respectively, we get the solution for (2.1).

Main results
Let us introduce the space U = {u : u ∈ C(H, R) and c D ς 1 u ∈ C(H, R)} with the norm defined by Then U, . U is a Banach space and also let us introduce the space Clearly, the product space U × V, . U×V is a Banach space with the norm defined by In view of Lemma 2.5, we define an operator F : Let us present the following assumptions that are used afterward here: are continuous functions and there exists real constants l i , λ i ≥ 0 (i = 1, 2) and l 0 , λ 0 > 0 such that for all u i ∈ R (i = 1, 2). We have For making a simplified expression, the following terms are introduced throughout this study: Proof. The F : U × V → U × V operator is shown to be completely continuous. It follows that the F operator is continuous by the continuity of the f and g functions. Let Θ ⊂ U × V be bounded. Then there exists positive constants N 1 and N 2 such that Step 1: To show that F is uniformly bounded. For each τ ∈ H, we have Thus, we have .
Similarly, we get and which implies that As a result, the following expression is obtained, .
Therefore, the above equation follows the inequalities in which operator F is uniformly bounded.
Step 3: To prove that the set = Hence we have We can have in a similar way, which implies that Thus, we find that The above equation proves that the set is bounded. Therefore, the F operator consists of at least a single fixed point according to the (see [33] Theorem 1.9). As a result, the boundary value problem is represented in Eqs (1.3) and (1.4) also, consist of at least a single solution on H. |g (τ, 0, 0) | < ∞ and we definê .

Consider the set
which implies that Hence, In this same way, we have which implies that In consequence, we get Hence, we get Hence, F Bρ ⊂ Bρ.
Next to prove that F is a contraction mapping on Bρ.
For u i , v i ∈ Bρ, i = 1, 2 and for each τ ∈ H, we have Thus, we obtain Therefore, In a similar way, we can find (|b 3 |)J +q |g(θ, u 1 (θ), c D ς 1 u 1 (θ)) − g(θ, u 2 (θ), c D ς 1 u 2 (θ))| (ζ) which implies that In consequence, we get Consequently, we obtain Thus, the F operator is referred to as a contraction operator (see [33] Theorem 1.4) and produced a unique fixed point that generates a unique solution for the BVP of (1.3) and (1.4) on H.
All of the hypotheses of the theorem 3.1 are satisfied. Therefore, there is a solution for the problem (4.1) on H.
Example 4.2. Consider the following coupled system of non-integer order differential equations subject to the Riemann-Liouville, Erdélyi-Kober integral conditions:
All of the hypotheses of the theorem 3.2 are satisfied. Therefore, there is a unique solution for the problem (4.2) on H.