Qualitative analysis of fractional relaxation equation and coupled system with Ψ–Caputo fractional derivative in Banach spaces

1 Laboratory of Mathematics and Applied Sciences University of Ghardaia, 47000, Algeria 2 Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen 3 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 4 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 5 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction
It is widely recognized that fractional differential equations (FDEs) become one of the most important research topics in numerous different disciplines such as mathematics, physics, biology, chemistry, finance, economics, and engineering, etc, see [1][2][3][4]. The elementary knowledge of fractional calculus can be found in the books of senior scholars [5,6]. While some notable developments on this topic can be found in the monographs of several mathematicians [7][8][9].
On the other hand, the monotone iterative technique combined with the method of upper and lower solutions have been used by several researchers, both for the scalar case and the abstract case with the end goal to establish the existence and uniqueness of extremal solutions for a class of nonlinear ordinary and FDEs (see, for instance, [31][32][33][34][35][36][37][38][39][40]) and the references therein. Motivated by the papers mentioned above, our goal is to extend the results of the recent paper [34] to the abstract framework. To our knowledge, no contributions exist, concerning the existence and uniqueness of extremal solution for a class of nonlinear FDEs in the frame of Ψ-Caputo derivative with initial conditions in Banach spaces via the monotone iterative technique. As a results, we aim to fill this gap in the literature and contribute to enriching this academic area. So, in this paper, we study the existence and uniqueness of extremal solution for the following Ψ-Caputo FDE in an ordered Banach space Y: where c D ς;Ψ a + is the Ψ-Caputo fractional derivative such that 0 < ς ≤ 1, f : I × Y −→ Y is a function fulfillments some suppositions that will be mentioned later, r > 0 and u a ∈ Y.
Next, we continue the results obtained in our recently published work in [23] to prove other properties such as the existence and uniqueness of solutions as well as the Ulam-Hyers (UH) stability results for the following Ψ-Caputo fractional relaxation differential system (Ψ-Caputo FRDS): a + u(ξ) + r 1 u(ξ) = G 1 (ξ, u(ξ), v(ξ)), c D ς 2 ;Ψ a + v(ξ) + r 2 v(ξ) = G 2 (ξ, u(ξ), v(ξ)), with the initial conditions where ς i ∈ (0, 1], r i > 0, G i : I × ℵ × ℵ −→ ℵ, i = 1, 2 are functions fulfillments some suppositions that will be mentioned later, ℵ is a Banach space with norm · and µ 1 , µ 2 ∈ ℵ. We organize the present work as follows: In Sect. 2, we recall basic concepts and results that will be called in the proof of our results. In Sect. 3, we apply the the monotone iterative technique in the presence of upper and lower solutions method to establish the existence and uniqueness of extremal solutions for the given problem (1.1). Whereas Sect. 4 is devoted to the existence and uniqueness of solutions to the coupled system (1.2)-(1.3). Moreover, Sect. 5, contains the UH stability of the proposed system (1.2)-(1.3). Also, two examples to illustrate the effectiveness of the feasibility of our abstract results are provided in Sect. 6. the work is terminated by some concluding remarks in Sect. 7.

Background materials
In this portion, we provide some fundamental concepts on the cones in a Banach space and Kuratowski's measure of noncompactness (KMN) as well as some facts about fractional calculus theory.
All over this part, we suppose that (Y, · , ≤) is a partially ordered Banach space whose positive cone K = {y ∈ Y | y ≥ θ} (θ is the zero element of Y ). Note that every cone K in Y defines a partial ordering in Y given by z 1 ≤ z 2 if and only if z 2 − z 1 ∈ K.
where ν is the normal constant of K, which is the smallest positive number fulfilling the above condition.
For any Let now I := [a, d] (0 < a < d < ∞) be a finite interval and Ψ : I → R be an increasing function with Ψ (ξ) 0, for all ξ ∈ I, and let C(I, Y) be the Banach space of all continuous functions u from I into Y with the norm u ∞ = sup ξ∈I u(ξ) .
Plainly, C(I, Y) is an ordered Banach space whose partial ordering ≤ reduced by a positive cone K C = {u ∈ C(I, Y) : u(ξ) ≥ θ, ξ ∈ I} which is also normal with the same normal constant ν. For more details on cone theory, see [44]. A measurable function u : I → Y is Bochner integrable if and only if u is Lebesgue integrable. By L 1 (I, Y) we denote the space of Bochner-integrable functions u : I → Y, with the norm Next, we define the KMN and grant some of its significant properties.
The following properties about the KMN are well known.
A fixed point technique advantageous to our aims is the following. holds for every subset Q ⊂ Λ, then N has a fixed point. Now, we supply some properties and results regarding the Ψ-fractional calculus as follows.
Definition 2.9 ( [5,10]). For ς > 0 and ξ ∈ I, the Ψ-Riemann-Liouville fractional integral of order ς is given by where z : I −→ R is an integrable function and Γ(·) is the Gamma function defined by Definition 2.10 ( [10]). Let n ∈ N and Ψ, z ∈ C n (I, R) be two functions. Then the Ψ-Riemann-Liouville fractional derivative of a function z of order ς is given by Definition 2.11 ( [10]). Let n ∈ N and Ψ, z ∈ C n (I, R) be two functions. The Ψ-Caputo fractional derivative of z of order ς is defined by where n = [ς] + 1 for ν N, n = ς for ς ∈ N. For the sake of brevity, let us take From the definition, it is clear that In fact, since the fractional integrals of a function z with respect to another function Ψ are generated by iterating the local integral I 1;Ψ a + z(ξ) = ξ a z(s)Ψ (s)ds, then the fractional derivative of a function z with respect to another function Ψ in the sense of Riemannn-Liouville and in the case of ς = n is natural number will be reduced to the local fractional operator z [n] Ψ (ξ). Then, the Caputo type local behavior is accordingly follows via Remark 1 in [13]. For example, if ς = 1 then [1] + 1 = 2 and Then, on the light of (16) in Remark 1 in [13], we have Lemma 2.12 ( [10,15]). Let ς, β > 0, and z ∈ C(I, R). Then for each ξ ∈ I we have Some essential properties of the MLFs are listed in the following Lemma.
Remark 2.16. Notice that, for an abstract function z : I −→ Y, the integrals which show in the preceding definitions are taken in Bochner's frame (see [49]).
The next lemma has an important role in demonstrating our main results.
Proof. From equation (2.2), we have Using the change of variables y = Ψ(ξ) − Ψ( ) we get Using now the change of variables v = λy in the above equation we get This completes the proof.

Remark 2.18 ( [27]
). On the space C(I, Y) we define a Bielecki type norm · B as below Consequently, we have the following proprieties 2. The norms · B and · ∞ are equivalent on C(I, Y), where · ∞ denotes the Chebyshev norm on C(I, Y), i.e; 3. Cauchy problem of Ψ-Caputo FDE (1.1) In this section, we apply the well-known MIT together with the method of UP and LO solutions and the theory of measure of noncompactness to investigate the existence and uniqueness of extremal solutions for the Cauchy problem (1.1) in an ordered Banach space Y.
Before we give our main results, let us defining what we mean by a solution of Ψ-Caputo FDE (1.1).
Definition 3.1. A function u ∈ C(I, Y) be a solution of problem (1.1) such that c D ς;Ψ a + u exists and is continuous on I, if u satisfies the equation c D ς;Ψ a + u(ξ) + ru(ξ) = f(ξ, u(ξ)), for each ξ ∈ I and the condition u(a) = u a . Now, we present the definition of lower and upper solutions of Ψ-Caputo FDE (1.1).
If all inequalities of (3.1) are inverted, we say that u is an upper solution of the Ψ-Caputo FDE (1.1).
The following key lemma is substantial to forward in demonstrating the main results.
has a unique solution is given by As a result of Lemma 3.3, the problem (1.1) can be converted to an integral equation which takes the following form Now, we are willing to give and prove our main findings.
Theorem 3.4. Let Y be an ordered Banach space, whose positive cone K is normal with normal constant ν. Let the following assumpitions are fulfilled (H 1 ) There exist u 0 , y 0 ∈ C(I, Y) such that u 0 and y 0 are lower and upper solutions of the Ψ-Caputo for any ξ ∈ I, and z 1 , z 2 ∈ Y with u 0 (ξ) ≤ z 1 ≤ z 2 ≤ y 0 (ξ).
(H 4 ) There exists a constant L > 0, such that for any ξ ∈ I and decreasing or increasing monotone sequence {u n (ξ)} ⊂ [u 0 (ξ), y 0 (ξ)], Then the Ψ-Caputo FDE (1.1) has minimal and maximal solutions that are between u 0 and y 0 which can be acquired by a monotone iterative procedure starting from u 0 and y 0 , respectively.
Proof. Transform the integral representation (3.4) of the problem (1.1) into a fixed point problem as follows: where T : From the continuity of f, T is well defined. On the other side, for any i.e.
Step 1: In this step, we will show the continuity of the operator T on D. To do this, let {u n } be a sequence in D such that u n → u in D as n → ∞. With ease, we find that f( , u n ( )) → f( , u( )), as n → +∞, due to f is a continuous. Also, from 3.6 we get the following inequality: From fact that the function → 2cΨ ( )(Ψ(ξ) − Ψ( )) ς−1 is Lebesgue integrable over [a, ξ] along with the Lebesgue dominated convergence theorem, we attain It follows that T u n − T u → 0 as n → +∞. Hence the operator T is continuous.
Step 2: We shall show that the operator T has the following two properties: (P 1 ) T is an increasing operator in D; (P 2 ) u 0 ≤ T u 0 , T y 0 ≤ y 0 .
To prove (P 1 ), let z 1 , z 2 ∈ D, such that z 1 ≤ z 2 . Then, from (H 3 ) we obtain which implies that T z 1 ≤ T z 2 . Therefore, T is an increasing operator.
Step 3: T(D) is equicontinuous on I. To do this, choosing, ξ 1 , ξ 2 ∈ I, with ξ 1 ≤ ξ 2 . By (3.6) and Lemma 2.14 we have Since the function M ς −r(Ψ(ξ) − Ψ(a)) ς is continuous on I, the right-hand side of the previous inequality approaches to zero when ξ 1 → ξ 2 independently of u ∈ D. This implies that T(D) is equicontinuous on I. Now define two sequences {u n } and {y n } in D, by the iterative scheme u n = T u n−1 , y n = T y n−1 , for n = 1, 2, . . .
Since Ω is equi-continuous, we have from Lemma 2.5 that Υ (Ω) = Υ (Ω 0 ) = max ξ∈I Υ (Ω 0 (ξ)) = 0. Combining this with the monotonicity and the normality of the cone K, without difficulty we can prove that {u n } itself is convergent in C(I, Y), i.e., there exists u ∈ C(I, Y) such that lim n→∞ u n = u. Similarly, it can be proved that there existsȳ ∈ C(I, Y) such that lim n→∞ y n =ȳ.
Using Lebesgue dominated convergence theorem, and letting n → ∞ in, (3.7) we see that Therefore, u,ȳ ∈ C(I, Y) are fixed points of T .
Step 5: We show the minimal and maximal property of u,ȳ. Suppose that z * is a fixed point of T in D, then we have u 0 (ξ) ≤ z * (ξ) ≤ y 0 (ξ), ξ ∈ I.
By the monotonicity of T , it is uncomplicated to find that Repeating the above arguments, we get u n ≤ z * ≤ y n , n = 1, 2, . . . .

(3.10)
Taking n → ∞ in (3.10), we get u ≤ z * ≤ȳ. Thus u,ȳ are the minimal and maximal fixed points of T in D, so, they also are the minimal and maximal solutions of problem (1.1) in D. Moreover, u andȳ can be obtained by the iterative procedure (3.7) beginning from u 0 and y 0 , respectively. This finishes the proof.
Our next theorem to prove the uniqueness of solution for the Ψ-Caputo FDE (1.1) by applying the monotone iterative technique.
Then Ψ-Caputo FDE (1.1) has a unique solution between u 0 and y 0 , which can be acquired by the iterative procedure beginning from u 0 or y 0 .
Proof. Let {z n } ⊂ D be an increasing monotone sequence, and n, m ∈ N with n > m. (H 2 ) and (H 5 From the normality of positive cone K, we obtain So by Lemma 2.4, we get Υ {f(ξ, z n )} ≤ kνΥ({z n }).
Then condition (H 4 ) holds and from the Theorem 3.4, we realize that Ψ-Caputo FDE (1.1) has minimal and maximal solutions u andȳ in D. Next we prove that u(ξ) ≡ȳ(ξ) in I. Thanks to Lemma 2.14 and (H 5 ), for each ξ ∈ I, we obtain By the normality of positive cone K, it follows that  Now by using the basic concepts mentioned in [13,34] we can easily derive the following lemma which is useful to prove our main results. Lemma 4.2. [13,34] Let ς 1 , ς 2 ∈ (0, 1] be fixed, r 1 , r 2 > 0 and G 1 , G 2 ∈ C(I × ℵ × ℵ, ℵ). Then the coupled systems of Ψ-Caputo FRDS (1.2)-(1.3) is equivalent to the following integral equations In order to establish our main results, we introduce the following assumptions.
(H 3 ) There exist real constants K 1 , K 2 > 0 and a continuous non-decreasing function φ i : , for any ξ ∈ I and each u, v ∈ ℵ.
(H 4 ) For each bounded set H ⊂ ℵ × ℵ, and each ξ ∈ I, the following inequality holds Our first theorem on the uniqueness relies on the fixed point theorem of Banach combined with the Bielecki norm. Proof. Let C(I, ℵ) be a Banach space equipped with the Bielecki norm type · B defined in (2.4). Consequently, the product space E := C(I, ℵ) × C(I, ℵ) is a Banach space, endowed with the Bielecki norm We define an operator S = S 1 , S 2 : E → E by: It should be noted that S is well-defined since both G 1 and G 2 are continuous. Now, we make use of the fixed point theorem of Banach to show that S has a unique fixed point. In this moment, we must show that S is a contraction mapping on E with respect to Bielecki's norm · E,B . Note that by definition of operator S, for any (u 1 , v 1 ), (u 2 , v 2 ) ∈ E and ξ ∈ I, using (H 2 ), and Lemmas 2.14, 2.17, we can get Hence This implies that We can choose λ > 0 such that L 1 λ ς 1 + L 2 λ ς 2 < 1, so the operator S is a contraction with respect to Bielecki's norm · E,B . Thus, an application of Banach's fixed point theorem shows that S has a unique fixed point. So the coupled system of Ψ-Caputo FRDS (1.2)-(1.3) has a unique solution in the space E. This completes the proof. Proof. In order to use the Theorem 2.8, we define a subset B δ of E by Notice that B δ is convex, closed and bounded subset of the Banach space E. We shall prove that S, satisfies all conditions of Theorem 2.8 in a two steps.
Step 1: we show that the operator S maps the set B δ into itself. Indeed, for any (u, v) ∈ B δ and for each ξ ∈ I. By Lemma 2.14 together with assumption (H 3 ) we can get Hence This proves that S transforms the ball B δ into itself. Moreover, in view of assumptions (H 1 ), (H 3 ) and by a similar deduction in Theorem 3.4, one can easily verify that S : B δ −→ B δ is continuous and S(B δ ) is equi-continuous on I.
Remark 5.2. A function (ũ,ṽ) ∈ E is a solution of the inequalities (5.1) if and only if there exist a functions g 1 , g 2 ∈ C(I, ℵ) ( which depend uponũ andṽ respectively, such that (ii) and Lemma 5.3. Let (ũ,ṽ) ∈ E be the solution of the inequalities (5.1), then the following of the inequalities will be satisfied: where S 1 and S 2 are defined by (4.3).
Proof. By Remark 5.2 (ii), we have with the following initial conditions Thanks to Lemma 3.3, the integral representation of (5.2)-(5.3) is expressed as

Examples
To illustrate our results, we provide two examples. Let be the Banach space with the norm z = ∞ j=1 |z j |.
It is clear that condition (H 1 ) holds, and as for all ξ ∈ I and each u 1 , v 1 , u 2 , v 2 ∈ 1 . Hence condition (H 2 ) holds with L 1 = L 2 = 1. Moreover, if we choose, λ > 4, it follows that the mapping S is a contraction with respect to Bielecki's norm. Hence by Theorem 4.3 the coupled system (6.1) has a unique solution which belong to the space C(I, 1 ) × C(I, 1 ). Besides, Theorem 5.4 implies that the coupled system (6.1) is Ulam-Hyers stable with respect to the Bielecki's norm.

Conclusion
The existence and uniqueness theorems of solutions to two classes of Ψ-Caputo-type FDEs and FRDS in Banach spaces have been developed. For the mentioned theorems, the obtained results have been derived by different methods of nonlinear analysis like the method of upper and lower solutions along with the monotone iterative technique, Banach contraction principle, and Mönch's fixed point theorem concerted with the measures of noncompactness. Also, some convenient results about UH stability have been established by utilizing some results of nonlinear analysis. The acquired results have been justified by two pertinent examples. To the best of our knowledge, the current results are recent for FDEs and FRDS involving generalized Caputo fractional derivative. Moreover, these results proven in Banach spaces. Apart from this, the FDEs and FRDS for different values of Ψ includes the study of FDEs and FRDS involving the fractional derivative operators: standard Caputo, Caputo-Hadamard, Caputo-Katugampola, and many other operators.
Finally, we would like to point out that the use of fractional operators with different kernels, reflected by the use of the increasing function Ψ used in the power law, is important in modeling certain physical and engineering problems in which we have memory. This confirms the need of the non-locality nature when we deal with such models. Moreover, the dependency of the kernel on the function Ψ provides us with more possibilities or choices in fitting the real data of some models.
In the future, the above results and analysis can be extended to more sophisticated and applicable problems of FDEs and FRDS involving Ψ-Hilfer operator. It will be also of interest to discuss the implementation of certain conditions in such case studies using the monotone iterative technique.