The existence and stability results of multi-order boundary value problems involving Riemann-Liouville fractional operators

: In this paper, a general framework for the fractional boundary value problems is presented. The problem is created by Riemann-Liouville type two-term fractional di ﬀ erential equations with a fractional bi-order setup. Moreover, the boundary conditions of the suggested system are considered as mixed Riemann-Liouville integro-derivative conditions with four di ﬀ erent orders, which it cover a variety of speciﬁc instances previously researched. Further, the provided problem’s Hyers-Ulam stability and the possibility of a ﬁxed-point approach solution are both investigated. Finally, to support our theoretical ﬁndings, an example is developed.


Introduction and building system
The classical derivatives are local in nature, i.e., using classical derivatives we can describe changes in the neighborhood of a point, but using fractional derivatives we can describe changes in an interval.
Namely, a fractional derivative is nonlocal in nature.This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, polymers, etc.
On the other hand, in the case of the heat conduction equation, the fractional order parameter α means the level of thermal conductivity.If α = 1, the medium's thermal conductivity is normal; if α < 1, the medium has weak conductivity; and if α ≥ 1, the medium has strong conductivity.
Further, in modeling various memory phenomena, it is observed that a memory process usually consists of two stages.One is short with permanent retention, and the other is governed by a simple model of fractional derivative.With the numerical least squares method, the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics but also in biology and psychology.Based on this model, it is found that the physical meaning of the fractional order is an index of memory.For more details, see [1,2].
Fractional calculus and its applications have acquired a lot of interest in several disciplines of engineering and science such as biology, chemistry, physics, economics, control theory, signal and image processing, etc, see [3][4][5] and the references therein.Variant definitions for the fractional derivative have emerged over the years.The most famous ones are the Riemann-Liouville and Caputo fractional derivatives.In recent years, many nonlinear phenomena in numerous fields have been modeled by fractional differential equations.Due to the evolution of fractional calculus, these equations have emerged as a new branch of applied mathematics.Several works on the existence and multiplicity of solutions to fractional boundary value problems (FBVPs) have appeared in view of the qualitative properties of fractional differential equations.
Among the used methods to solve a FBVP, there are the variational methods used by Fix and Roop in [6] and Erwin and Roop in [7].Also, some fixed point techniques have been applied successfully to ensure the existence of solutions of some FBVPs.Here, we may cite the works of Agarwal et al. [8], Benchohra et al. [9], Zhang [10], Ahmad and Nieto [11], etc. Going in the same direction, the critical point theory has been used to investigate the solutions for some FBVPs.For instance, see the works Jiao and Zhou [12] and Tang and Wu [13].On the other hand, stability analysis of fractional differential equations with different types of initial and boundary conditions have attracted many researchers who discussed the analysis of stability in the setting of Ulam-Hyers (UH) and generalized UH theory.For more details, see [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30].
In 2016, the boundary value problem (BVP) with 4-order Riemann-Liouville fractional (RLF) derivatives is studied by Niyom et al. [31]: under appropriate conditions.Also, Niyom et al. [32], modified the above problem under multiple orders of fractional integrals and derivatives as follows: In 2018, Xu et al. [33] examined the existence of solutions and UH stability for the FDEs They focused on the RLF derivative and integral issues of the two-term class of three-point BVPs, where the notions and parameters in (1.1) and (1.2) are defined below the system (1.4).Now, utilizing the concepts from the works described above and combining them, we investigate a new category of coupled boundary value problems (CBVPs) that includes a multi-order RLF equation plus various linear integro-derivative boundary stipulations as follows: where 2 R are continuous functions.As many scholars are interested in exploring the idea of stability for various CBVPs, this can serve as inspiration for us to research the stability of complicated systems with added broad boundary stipulations.Consequently, to be more precise, the main objective of the current manuscript is to find some existing criteria for the solutions to a new general CBVP that includes a two-term fractional differential equation (FDE) (1.4) and multi-order RLF derivatives and integrals.The well-known standard fixed point (FP) theorems are employed in order to achieve this goal.Furthermore, in the follow-up, we examine the HU stability of the suggested problem (1.4) in the unique scenario when 1 = 2 = 1 and * 1 = * 2 = 1.Ultimately, to demonstrate the applicability of our theoretical results, two examples are provided.We think that the BVP that has been proposed is a generic one that incorporates a lot of fractional dynamical systems as special examples in the fields of physics and other applied disciplines.

Basic concepts
Let G > 0 and U = [0, G].Assume that the piecewise continuous function space PC(U, R + ) equipped with the norms z = max{|z(υ)| : υ ∈ U} and r = max{|r(υ)| : υ ∈ U} is a Banach space (BS), then the products of these norms are also a BS under the norm (z, r) = z + r .
Assume also 1 and 2 represent the piecewise continuous function spaces described as Definition 2.1.[34] For a real valued function z : (0, ∞) → R, the RLF integral operator of order ρ is described as where Γ(.) is the Euler gamma function.Definition 2.2.[34] The RLF derivative of order ρ of a function z : (0, ∞) → R takes the form where [ρ] refers to the integer part of real number ρ.
Theorem 2.2.(Banach FP theorem [37]) Every contraction self-mapping defined on a complete metric space admits a unique FP.

Existence results
We begin this section with the lemma below.
Lemma 3.1.The mappings z 0 , r 0 are a solution for CBVP (1.4) if z 0 , r 0 are solutions to the following integral equations: and where Proof.Let (z 0 , r 0 ) be a solution for the Eq (1.4), then, we get Taking the RLF integration of order ρ from both sides of the first equation in (3.4), we have where O 1 , O 2 and O 3 are real constants.From the first boundary stipulation of (1.4), for ρ ∈ (2, 3), we have O 3 = 0.By Lemma 2.3, we can write Using the RLF integral and derivative of order η and s, respectively with η ∈ {η 1 , η 2 }, s ∈ {s 1 , s 2 }, 0 < η < ρ − θ and 2 < θ < ρ, we obtain and Replacing η = η 1 , η = η 2 , s = s 1 , s = s 2 and using the boundary stipulations and and Substituting O 1 and O 2 in (1.4), we have the first part of the solution (3.1).With the same scenario followed above, the second part of the solution (3.2) can easily be obtained.Now, we convert the problem to the FP problem.Based on Lemma 3.1, define an operator where and Remember that the solution to CBVP (1.4) is (z 0 , r 0 ) iff (z 0 , r 0 ) is a FP of Ω.We employ the following notation to streamline calculations: Now, our main theorem is as follows: Theorem 3.1.Assume that the mappings Ξ, Ξ * : U × R 2 → R are continuous and there are constants and then the considered problem (1.4) has a unique solution (US), where T = max{T, T * }, where ∇ i and ∇ * i , i ∈ {1, 2, 3, 4}, Φ and Φ * are defined by (3.3) and N = max{N, N * }.As a first step, we show that ΩQ y ⊂ Q y , where Q y = (z, r) ∈ : (z, r) ≤ y .For any (z, r) ∈ Q y , we have From (3.6) and (3.7), we get which implies that In the same scenario, we can write Applying (3.14) and (3.15) in (3.13) and using (3.12), we have Hence, Ω(z, r) ≤ y and so ΩQ y ⊂ Q y .For each υ ∈ U and for z, r, z, r ∈ , we get Similarly, one can obtain Hence, Since T Λ 4 + Λ 3 < 1, then Ω is a contraction mapping.Using the contraction principle, Ω has a unique FP, which is the US for the CBVP (1.4).Now, we present an existence result by applying Krasnoselskii's FP theorem.Theorem 3.2.Suppose that the mappings Ξ, Ξ * : U × R 2 → R are continuous and there are positive constants T Ξ , T Ξ , T Ξ * , T Ξ * so that for all (τ, z, r) ∈ U × R × R and Λ 3 < 1, then, the CBVP (1.4) has at least one solution. Proof.
Similarly, one can obtain under where , and .

Conclusions
Fractional calculus has found numerous miscellaneous applications connected with real-world problems as they appear in many fields of science and engineering, including fluid flow, signal and image processing, fractal theory, control theory, electromagnetic theory, fitting of experimental data, optics, potential theory, biology, chemistry, diffusion, and viscoelasticity.Due to the many applications that have been mentioned, this branch has become of interest to many writers.Therefore, in this paper, the existence of solutions to a system of two-term FDEs with a fractional bi-order involving the Riemann-Liouville derivative has been established.Also, the considered boundaries are mixed Riemann-Liouville integro-derivative conditions with four different orders.Further, HU stability is studied, and an illustrative example has been introduced.Ultimately, we conclude that our results are new and are considered a further development of the qualitative analysis of fractional differential equations.