On system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order

We investigate a new class of boundary value problems of a nonlinear coupled system of sequential fractional differential equations and inclusions involving Caputo fractional derivatives and boundary conditions. We use standard fixed-point theory tools to deduce sufficient criteria for the existence and uniqueness of solutions to the problems at hand. Examples are discussed to illustrate the validity of the proposed results.


Introduction
Fractional calculus has gotten a lot of attention over the last two decades. Consequently, there has been a burgeoning interest in the theory and applications of fractional differential equations (FDEs) under various types of initial and boundary conditions (BCs); see, for example [1][2][3][4][5][6][7][8][9][10][11] and the references cited therein. The feature of fractional differentiation and integration has significantly improved the consideration of mathematical modellings of many real-life problems within fractional settings. As a result, it has been realized that this subject has applications in a wide range of technical and physical sciences, including electrodynamics of complex media, control theory ecology, viscoelasticity, biomathematics, electrical circuits, electroanalytical chemistry, aerodynamics and blood flow phenomena. For additional information, the reader can consult the papers [12][13][14][15][16][17][18][19][20][21][22][23][24].
In the remarkable monograph [25], the concept of two fractional order operators (i.e. Sequential Fractional Derivative (SFD)) was discussed. It has been recognized that SFDs and non-SFDs are inextricably linked and thus there have appeared some recent work on sequential fractional differential equations (SFDEs) [26][27][28][29][30]. Apart from the study of fractional-order boundary value problems (BVPs) for equations and inclusions [31][32][33], the study of systems of coupled FDEs has accelerated and attracted interested researchers. Examples on applications of coupled systems include disease models, Lorenz system, ecological models, Duffing system, synchronization of chaotic systems, etc. [34][35][36][37][38]. For the theoretical development of coupled systems of FDEs, we refer the readers to [39][40][41] and the references cited therein. In [42,43], the authors investigated SFDEs with different types of BCs. Sufficient conditions are utilized to demonstrate the consequence of existence and uniqueness to the analysed equation. Numerous mathematicians and applied researchers have attempted to use fractional calculus to model real-world processes. It has been deduced in biology that the membranes of biological organism cells have fractional-order electrical conductance [44] and thus are classified in groups of non-integer-order models. Fractional derivatives are the most successful in the field of rheology because they embody essential features of cell rheological behaviour [45]. In most biological systems, such as HIV infection, hepatitis C virus (HCV) infection, and cancer spread, fractional-order ordinary differential equations are naturally related to systems with long-time memory. Additionally, they are related to fractals, which occur frequently in biological systems. Wang and Li [46] analysed the global dynamics of HIV infection of CD4+ cells.
Arafal et al. [47] studied fractional modelling dynamics of HIV and CD4+ T cells during primary infection. As a result, fractional-order differential equations are thought to be a better tool than integer-order differential equations for describing hereditary properties of various materials and processes. Fractional-order models have become more realistic and practical than their classical integer-order counterparts as a result of this advantage, and their dynamics behaviour is also as stable as their integer-order counterparts. Due to the fact that theoretical results can aid in the development of a more complete understanding of the dynamic behaviour of biological processes, the study of abstract fractional dynamic models is becoming increasingly relevant and important in the modern era. On the other hand, the BCs in (1) are referred to as coupled BCs; they are encountered in the study of reaction-diffusion equations, Sturm-Liouville problems and mathematical biology, among other fields [48][49][50]. Recently in [51], Ahmad et al. discussed the existence and uniqueness of coupled system of FDEs with a novel class of coupled boundary conditions specified by (1) where C D χ is the Caputo fractional derivatives (CFD) of order χ ∈ {α, β}, α, β ∈ (0, 1], f , g : [0, T] × R 2 → R are continuous functions and A is nonnegative constant. The main results are established by converting the system (1) to a fixed point equivalent problem and solving it using standard fixed point theorems. As far as we know, the single-valued and multi-valued maps for the solutions of nonlinear coupled SFDEs with coupled boundary conditions have been rarely investigated. Motivated by the HIV infection model and its application background, we investigate the consequences of existence for a nonlinear coupled system of Caputotype SFDEs subject to coupled boundary conditions of the form: where is the collection of non-empty subsets of R, and φ is positive constant. Further existence investigation is carried out for the following nonlinear coupled under the same assumptions. Ahmad et al. [51] reported the research to study the existence of positive solutions for nonlinear coupled system of fractional differential equations complemented with boundary conditions. We should point out that the term "sequential" is used in this context in the sense that the operator C D ϑ + ϕ C D ϑ−1 can be written as the composition of the operators C D ϑ−1 (D + ϕ). In this article, authors have extend the boundary value problem of Ahmad et al. [51] to nonlinear coupled system of sequential Caputo fractional differential equations and inclusions having the value of unknown functions v and w at the interval endpoints [0, T ] being zero, whereas the impact of the sum of the unknown functions on an arbitrary domain ( , δ) of the given interval [0, T ] remaining constant. Furthermore, the authors have emphasized the major results on existence, uniqueness of solutions for sequential fractional differential equations and inclusions compared against [51]. Unlike the paper [51], the main results of this paper are entirely different in the sense that we consider the main problems in frame of sequential fractional derivative, use different techniques based on Schaefer's, Banach's, Covitz-Nadler's, and nonlinear alternative for Kakutani fixed point theorems and investigate the nonlinear coupled differential inclusion (3) which was not considered in [51]. Additionally, to our knowledge, there are no published outcomes relating system (3). Section 2 provides essential preliminaries along with an auxiliary lemma that is critical for deriving the solution to the given problem. Section 3 is devoted to the main results in which we study the existence and uniqueness of solutions for systems (2) and (3), separately. In Section 4, particular examples consistent with the studied systems and the main theorems are provided.

Preliminaries
This section explores several definitions of multi-valued maps and lemmas that are necessary for proving the primary results [13,16,52,53].
Let (W, · ) be a normed space and that U cl

Further a Caratheodory function
Definition 2.1: The Riemann-Liouville fractional integral of order with the lower limit zero for a function f : [0, ∞) → R is defined as

Definition 2.2:
The Riemann-Liouville fractional derivative of order > 0, n − 1 < < n, n ∈ N for a function f : [0, ∞) → R is defined as Notice that the Riemann-Liouville fractional derivative of order ∈ [n − 1, n) exists almost everywhere on

Definition 2.3:
The Caputo derivative of order ∈ [n − 1, n) for a function f : [0, ∞) → R can be written as Note that the Caputo fractional derivative of order

Lemma 2.4:
Then the integral solution for the linear system of SFDEs: Proof: As argued in [43], the general solution of the system (4) can be written as where c 0 , d 0 are arbitrary constants. Using BCs (4) in (7), we obtain The solutions (5) and (6) are obtained by substituting the values of c 0 and d 0 in (7) respectively.

Main results
The main results are stated and proved in this section.
The results are carried out separately for systems (2) and (3).

Existence results for system (2)
Define W = C(S, R) × C(S, R) as the Banach space endowed with norm (v, w) = sup ι∈S |v(ι)| + sup ι∈S |w(ι)|, for (v, w) ∈ W. Using Lemma 2.4, we convert system (2) into a fixed point problem as v = v, the following operator : W → W is defined by where Following that, we initiate the hypotheses that will be used to demonstrate the paper's primary research results. Let H 1 , H 2 : S × R 2 → R be continuous functions.
To facilitate the computation, we introduce the notation: and In this portion, we prove the consequence of the existence of the BVP (2) via Schaefer's fixed point theorem [56].
Theorem 3.1: Suppose that (Q 1 ) holds. Furthermore, the assumption is that where ϒ 1 , ϒ 2 are defined by (13) and (14). Then the problem (2) has at least one solution on S.

Proof:
We begin by demonstrating that the operator : W → W is c.c. Note that is continuous as the functions H 1 and H 2 are continuous. Now let r ⊂ W be bounded. Then ∃ positive constants L H 1 and L H 2 such that Thus Thus the operator is uniformly bounded as a result of the preceding inequality. Let prove that it determines bounded sets into equicontinuous sets of W, let ι 1 , ι 2 ∈ [0, T ], ι 1 < ι 2 , and (v, u ∈ r ). Then Analogously, we can obtain Take note that in the limit ι 1 → ι 2 , the RHS of the preceding inequalities tends to zero independently of (v, w) ∈ r . Thus the operator (v, w) is equicontinuous, and hence by, Arzela-Ascoli theorem, (v, w) is c.c. Next, it will be verified that the set Using ϒ 1 , ϒ 2 defined by (13)-(14), we get In consequence, we get Then, using Equation (15), we conclude that This demonstrates that the set is bounded. Hence, by Schaefer's fixed point theorem, there exists a solution of (2).
Next, we express our second result, which is based on Banach's fixed point theorem and concerns the existence of a unique solution to (2). Theorem 3.2: Suppose that (Q 2 ) hold and that where C = max{C 1 , C 2 }, K = max{K 1 , K 2 } and ϒ i , i = 1, 2 are defined by (13)- (14). Then the problem (2) has a unique solution. .
which leads to 1 (v, w) when the norm for ι ∈ S. Equivalently, for (v, w)) ∈ B r , one can obtain which demonstrates that maps B r into itself. To demonstrate that the operator is a contraction, let p 1 (a), q 1 (a)) − H 1 (a, p 2 (a), q 2 (a))| da dθ − H 2 (a, p 2 (a), q 2 (a))| da dθ , Clearly, the preceding inequalities imply that As a result, in light of the assumption (16), the operator textitPsi is a contraction. As a result of Banach's contraction mapping theorem, has an unique fixed point. This demonstrates that system (2) has a unique solution on S.

Remark 3.1:
There exist positive functions γ 1 , k 1 ∈ C(S, R + ) and H 1 , H 2 : S × R 2 → R which are continuous functions such that Then system (2) has at least one solution on S.

Remark 3.2:
According to the assumptions of (ε i and ω i non-negative constants, and the criteria of the functions H 1 , H 2 have the following form: (Q 1 ) there are real constants ω i , ε i > 0, i = 1, 2, 3, so and (15) becomes

Existence results for system (3)
and is referred to as a coupled solution for system (3). Let for a.e ι ∈ S}, for a.e ι ∈ S}, define the sets of H 1 , H 2 selections for each (v, w) ∈ W × W. Using Lemma 2.4, the following operators where and Following that, we define the operator : where 1 and 2 are defined in (19) and (20), respectively.
In this portion, we prove the existence of solutions for the BVP (3) via nonlinear alternative of Leray-Schauder [55]. Following that, we initiate the assumptions that will be used to demonstrate the paper's primary research results. [0, ∞) → [0, ∞) and functions l 1 , l 2 ∈ C(S, R + ), such that for each (ι, v, w) ∈ S × R 2 , for each (ι, v, w) ∈ S × R 2 .
where ϒ 1 , ϒ 2 are defined by (13) and (14). Proof: Consider 1 , 2 : W × W → U(W × W) the operators which is given by (19) and (20) respectively. Using the assumption (Q 3 ), the sets V H 1 (v,w) and V H 2 (v,w) are non-empty for each (v, w) ∈ W × W. Then, and where g 1 ∈ 1 (v, w), g 2 ∈ 2 (v, w), and so (g 1 , g 2 ) ∈ (v, w). The operator will be shown to satisfy the Leray-Schauder nonlinear alternative hypotheses in several steps. To begin, we demonstrate that (v, w) has a convex value. Let (g i ,ĝ i ) ∈ ( 1 , 2 ), i = 1, 2. and Let 0 ≤ ν ≤ 1. Then, for each ι ∈ S, we have [νh 11 (a) and We may deduce that V H 1 (v,w) and V H 2 (v,w) have convex values since H 1 and H 2 have convex values. Clearly, and Then we have and Thus we get Then we prove that is equicontinuous. Let ι 1 , Analogously, we can obtain As a consequence, the operator (v, w) is equicontinuous, hence the operator (v, w) is c.c according to the Arzela-Ascoli theorem. We know from [53, Proposition 1.2] that a c.c operator has a closed graph if it is upper semicontinuous. As a result, we must demonstrate that has a closed graph. Let (p n , q n ) → (p * , q * ), (g n ,ĝ n ) ∈ (p n , q n ) and (g n ,ĝ n ) → (g * ,ĝ * ), then we must demonstrate (g * ,ĝ * ) ∈ (p * , q * ). Remember that (g n ,ĝ n ) ∈ (p n , q n ) implies that ∃ h 1n and g n (p n , q n )(ι) Consider the 1 , 2 : L 1 (S, W × W → C(S, W × W continuous linear operators provided by We can deduce from [57] that is a closed graph operator. In addition, we have (g n ,ĝ n ) ∈ ( 1 , 2 ) • (V H 1 (p n ,q n ) , V H 2 (p n ,q n ) ) for all n. Since (p n , q n ) → (p * , q * ), (g n ,ĝ n ) → g * ,ĝ * ), it follows that h 1n and For each ι ∈ S, we obtain which implies that v, w According to (Q 5 ), Z exists such that (v, w) = Z. Let us fix It should be noted that operator : for some υ ∈ (0, 1) by E selection. As a result, we can deduce from the Leray-Schauder nonlinear alternative [55] that has a fixed point (v, w) ∈ E, which is a solution of system (3).
Let (W, d) denote a metric space generated from the normed space (W, · ), and let G d : [57]).

Definition 3.2:
A multi-valued K : W → U(W) operator is called for each c, d ∈ W; and (ii) a contraction iff it is δ-Lipschitz with δ < 1.
The following result makes use of Covitz and Nadler's theorem for multi-valued maps [54].

Theorem 3.4:
Assume that (Q 6 ) and (Q 7 ) holds. Then system (3) has at least one solution on S provided that Proof: Assuming (Q 6 ) that the sets V H 1 (v,w) and V H 2 (v,w) are non-empty for each (v, w) ∈ W × W, H 1 and H 2 have measurable selections (see Theorem III.6 in [58]). Next we demonstrate that the operator fulfils the theorem of Covitz and Nadler's [54].

Examples
In consistence with systems (2) and (3) and the main theorems, we provide some examples in this section.

Concluding remarks
In this research work, we studied a new BVP involving a coupled system of nonlinear SFDEs and inclusions of the Caputo type and supplemented with coupled integral boundary conditions. We have studied a new type of coupled boundary condition that determines the sum of unknown functions at the boundary points and along any arbitrary segment of the domain. Under these conditions, we solved a nonlinear Caputo SFDEs and inclusion system. The consequences of existence and uniqueness were examined via single-valued and multi-valued maps. It is also possible to broaden the scope of this investigation to include fractional differential and integral operators of the Riemann-Liouville and Hadamard types. Our results aren't only novel in the context of the problem, but they also lead to some novel cases involving specific parameter choices. The results of this paper are limited to a few intriguing instances with adequate values for the systems parameters. For instance, our results correspond to those for new coupled Stieltjes boundary conditions if we set (2) and (3), and also we can obtained new existence results through special cases as stated in Remark 3.1 and Remark 3.2. We believe that the results discussed in this paper are of great significance for the scientific audience. Future research could focus on different concepts of stability and existence concerning a neutral time-delay system/inclusion and a time-delay system/inclusion with finite delay.

Declarations
Availability of data and material Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Authors' contributions All authors contributed equally and significantly to this paper. All authors have read and approved the final version of the manuscript.