On solutions of fractional multi-term sequential problems via some special categories of functions and (AEP)-property

The main intention of this article is that new techniques of existence theory are used to derive some required criteria pertinent to a given fractional multi-term problem and its inclusion version. In such an approach, we do our research on a fractional integral equation corresponding to the mentioned BVPs. In more precise words, by virtue of this integral equation, we construct new operators which belong to a special category of functions named α-admissible and α-ψ-contraction maps coupled with operators having (AEP)-property. Next, by considering some new properties on the existing Banach space having properties (B) and (Cα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(C_{\alpha })$\end{document}, our argument for ensuring the existence of solutions is completed. In addition, we also add two simulative examples to review our findings by a numerical view.


Introduction
As is well known, fractional calculus (FC), for the sake of its higher accuracy than that of the integer one, is an essential topic that is considered as a strong tool in description of natural laws in many branches of science including electrical networks, rheology, biology, dynamical systems, biophysics coupled with many mathematical modelings formulated by a vast diversity of fractional operators; review for details [1][2][3][4][5][6][7][8].
Among theoretical concepts and methods, the theory of the existence of solution on the large domain of different fractional constructions including differential equations and inclusions has gained the attention of many mathematicians and relevant researchers. Most of them have focused on applying Caputo, Riemann-Liouville (RL), Hadamard, and many other derivation operators to illustrate the underlying fractional differential equations. Along with these structures, we observe different published articles recently in which the existence of solutions is derived for interesting categories of fractional local or nonlocal, multi-term, multi-point, multi-strip, multi-order fractional differential equations; see [9][10][11][12][13][14][15][16][17][18][19][20][21].
In 2018, Tariboon et al. [22] discussed a new category of separated sequential BVP in the context of mixed Hadamard and Caputo operators as follows: m stand for two different kinds of derivatives named Caputo and Hadamard of order σ * 1 , σ * 2 ∈ (0, 1], a 1 , a 2 , a 3 , a 4 ∈ R and * : J * × R → R is continuous. In that article, some standard methods pertinent to fixed point theory are used for obtaining the existence results.
With the help of ideas of the above articles, we investigate and discuss new existence techniques and methods to ensure the existence of solutions for a new multi-term problem illustrated as follows: where RL I (·) 0 and C D (·) 0 denote integration and derivation operators in the sense of Riemann-Liouville and Caputo, respectively, such that θ * > 0, 0 < σ * < 1, 1 < δ * < 2. Besides, we have taken a 1 , a 2 , a 3 , a 4 ∈ R + and * : J × R 2 → R as a continuous map.
Also along with the above problem, we derive the existence of solution for the following inclusion version which takes a form as follows: where we consider H as a multifunction on the product space J × R 2 containing different properties which will be introduced later. By studying a wide range of published articles pertinent to the existence and uniqueness notions in the context of fractional boundary value problems, we see that many authors usually utilize some standard methods based on the famous fixed point techniques to derive desired results in relation to the existence of solutions. The novelty of our work in the current research is that we introduce a new construction of two fractional multi-term BVPs having integral conditions and then build new operators which belong to a new class of specific functions. Two main functions belonging to this class are α-admissible maps and α-ψ-contractive maps. Here, by using these functions on a space having properties (B) and (C α ), we derive the existence results for both suggested BVPs (1) and (2). Besides, by virtue of the (AEP)-property for the obtained multifunction in the proof and with the help of the endpoint notion, we derive another criterion of the existence of solutions. As you will see, our techniques used for supposed problems (1) and (2) have been done in limited works, and for the first time, we apply these methods on a multi-term structures simultaneously.
The arrangement of the notions of this research are as follows: Sect. 2 is devoted to reviewing some primitive notions. In Sect. 3, by means of α-ψ-contractive functions, we prove our existence results for (1). Then, in Sect. 4, we derive other criteria stating the solution's existence for the inclusion BVP (2) by considering the extended category of α-ψcontractive functions on multifunctions. Two numerical examples are also given to confirm findings in Sect. 5. At last, we summarize our method and specify new directions for the future works in Sect. 6.

Preliminaries
At this moment, we collect and review several auxiliary and primitive concepts in the context of our methods used in this research. As you know, the concepts of the Riemann-Liouville operator and the Caputo one have an important role in fractional calculus, we recall some properties of them here.
if the integral exists.
if the integral exists.
such that the integral exists.
where d is considered the metric of M and also d( Then H is said to be: (H1) a Lipschitz map if > 0; (H2) a contraction if 0 < < 1.
is measurable for any υ ∈ R and υ → H(a, υ) is u.s.c for a.e. z ∈ J.
for all |υ| ≤ ζ and for almost all z ∈ J. To meet the argumentative purposes of this paper, we shall apply a specific set of functions and properties pertinent to them. For the first time, Samet, Vetro, and Vetro [29] constructed such a category of mappings in 2012.
The second category of special maps is an extension of the previous one which was constructed by Mohammadi et al. one year later [30].

Definition 2.11 ([30]) Consider all nondecreasing maps
Next theorems are considered as the basis of our arguments until the end of this research. (i) * is α-admissible and α-ψ-contraction; (iii) M has the property (B). In that case, * has a fixed point.
For T 1 and T 1 defined on A, the following assertions hold: Then υ * ∈ A exists such that υ * = T 1 υ * + T 2 υ * .

Theorem 2.14 ([30]) Consider the complete metric space
. In that case, H has a fixed point.
Then H has a unique endpoint if and only if H has the (AEP)-property.

Existence criteria for problem (1)
In the present situation of the current research, we start to follow the essential deductions on the existence notion for possible solutions of problem (1) through existing nonlinear techniques in the theory of fixed points of an assumed operator. To begin the de- In the next lemma, the solution of the supposed problem (1) is exhibited in the framework of an integral equation which will be useful for our subsequent arguments. Lemma 3.1 Let θ * > 0, 0 < σ * < 1, 1 < δ * < 2, a 1 , a 2 , a 3 , a 4 ∈ R + , and φ ∈ C(J, R). In that case, the solution of the fractional BVP which takes the structure is illustrated as where , , are nonzero constants.
By using the second boundary condition a C By using the third boundary condition a C 3 D 2 0 υ(0) + a 4 RL I θ * 0 υ(1) = 0, we get that From (7), we have and hence, by the above notations displayed in (5), we get Inserting the obtained value for c 1 into (8), we reach and by (5) this turns into and so again by (5) it becomes On the other hand, putting the value of c 2 in (9), we try to find c 1 as which implies and so Finally, in view of notations (5), we have At last, we put values of c 0 , c 1 , c 2 computed by the above procedures in (6), and we reach This completes our proof about the structure of solutions for BVP (3).
Inspired by the previous lemma, we now assume an operator T : M → M defined by It is to be noted that υ 0 is regarded as a solution for supposed BVP (1) iff υ 0 is a fixed point for the newly-defined map T. Set: * We are at this moment ready to express and verify the first theorem pertinent to the existence conditions of solutions for BVP (1). (i) for all υ 1 , υ 1 , υ 2 , υ 2 ∈ M with χ * ((υ 1 (z), υ 2 (z)), (υ 1 (z), υ 2 (z))) ≥ 0 and λ * = 1 * (ii) there exists υ 0 ∈ M such that, for all z ∈ J, and also for all z ∈ J and υ 1 , υ 2 ∈ M; (iii) for an arbitrary sequence {υ n } n≥1 ⊆ M with υ n → υ and for any n ∈ N and z ∈ J, the following inequality holds: Then the given BVP (1) has at least one solution.
By following the argument, we focus on our intention to find other existence requirements of solutions for BVP (1) by means of another tool based on the fixed point. (iv) A continuous function ξ is formulated on the closed interval J such that * (z, υ 1 , υ 2 ) - * z, υ 1 , υ 2 ≤ ξ (z) 2 k=1 υ kυ k for all z ∈ J and υ 1 , υ 2 , υ 1 , υ 2 ∈ M; (v) There are a continuous function η * : J → R + and a nondecreasing function ψ : for all z ∈ J and υ 1 , υ 2 ∈ M. Then BVP (1) has at least one solution whenever by terms of constants * 3 and * 4 given by (10).
Proof Let η * = sup z∈J |η * (z)| and put m * = sup υ∈M ψ( υ ), and assume that there is > 0 such that where * i s are given in (10). We define a set A = {υ ∈ M : υ ≤ }, and it is easy to verify that A = ∅ belongs to P CL (M), P CV (M), and P BD (M) simultaneously. Now we define two fractional operators T 1 and T 2 on a set A by and For all z ∈ J and for υ 1 , υ 2 ∈ A , we have Hence T 1 υ 1 (z) + T 2 υ 2 (z) ≤ , which implies that T 1 υ 1 (z) + T 2 υ 2 (z) ∈ A . On the other hand, since * is continuous, this ensures that T 1 will be continuous. Moreover, along with these, we compute for all υ ∈ A . Hence, which guarantees that T 1 is uniformly bounded on A . Now, in the next step, we show that the operator T 1 is compact on A , for this we assume that z 1 , z 2 ∈ J with z 2 > z 1 . Thus we have Also, in a similar manner, Thus T 1 is equicontinuous and T 1 is relatively compact on A . Now, by the Arzela-Ascoli theorem, it is compact. Finally, we prove that T 2 is a contraction. For υ 1 , υ 2 ∈ A , we get Therefore, we get where ξ = sup z∈J |ξ (z)|. Hence, T 2 υ 1 (z) -T 2 υ 2 (z) ≤ ( * 3 + * 4 ) υ 1υ 2 , which implies that Thus, T 2 is a contraction on A with constant K * = * 3 + * 4 < 1. So, from Theorem 2.13, BVP (1) has at least one solution and the proof is completed.

Existence criteria for problem (2)
In this part, we derive the existence of solutions to the inclusion BVP (2). The function υ ∈ C(J, M) is called the solution of problem (2) when it satisfies the boundary conditions and there is ϒ ∈ L 1 (J) such that ϒ(z) ∈ H(z, υ(z), C D 1 0 υ(z)) for almost all z ∈ J and where (viii) There is χ * : R 2 × R 2 → R such that χ * ((υ 1 , υ 2 ), (ύ 1 ,ύ 2 )) ≥ 0 for all υ k ,ύ k ∈ M (k = 1, 2); (ix) If {υ n } is a sequence in M with υ n → υ and for all z ∈ J and n ≥ 1, then there exists a subsequence {υ n j } of {υ n } such that for z ∈ J and j ≥ 1; (x) There exist υ 0 ∈ M and p ∈ N(υ 0 ) such that (13); (xi) For any υ ∈ M and p ∈ N(υ) with there exists a member w * ∈ N(υ) such that for all z ∈ J. In that case, the inclusion problem (2) has at least one solution.

Examples
Here, we provide two simulative examples to review our findings by a numerical view.

Conclusion
In the current research, we discussed some new conditions ensuring the existence of solution for given two multi-term BVPs in two different equation and inclusion versions. Indeed, we defined some operators based on the equivalent integral equation which belonged to a special category of α-admissible and α-ψ-contractions. Also, we investigated the (AEP)-property for such operators. At last, in two separate examples, the numerical simulation of given BVPs was done. As a future and next project, one can apply these techniques for the generalized BVPs having multi-strip, multi-point, multi-order integral conditions simultaneously.