Novel existence techniques on the generalized φ-Caputo fractional inclusion boundary problem

Our basic purpose is to derive several existence aspects of solutions for a novel class of the fractional inclusion problem in terms of the well-defined generalized φ-Caputo and φ-Riemann–Liouville operators. The existing boundary conditions in such an inclusion problem are endowed with mixed generalized φ-Riemann–Liouville conditions. To reach this goal, we utilize the analytical methods on α-ψ -contractive maps and multifunctions involving approximate endpoint specification to derive the required results. In the final part, we formulate an illustrative simulation example to examine obtained theoretical outcomes by computationally and numerically.


Introduction
Discussion of the mathematical modelings of processes is an interesting part of mathematics in which the mathematicians discuss behaviors of existing systems in terms of several mathematical and computational methods. In this regard, there is a vast range of fractional operators which have a vital role in modeling different phenomena. In fact, during the past years, a large number of mathematicians have formulated well-defined operators for their purposes. For the sake of having a high accuracy of the arbitrary order modelings rather than integer order ones, the fractional versions of these new operators have been welcomed nowadays. For instance, the Caputo and the Riemann-Liouville operators in the fractional frames have appeared repeatedly for modeling the complicated mathematical systems; see [1][2][3][4][5][6][7][8][9][10][11][12]. Later, a wide range of modelings were designed utilizing the Hadamard and Caputo-Hadamard fractional operators; see [13][14][15][16][17][18]. In early 2015, Caputo and Fabrizio [19] propounded a novel nonsingular derivative named the Caputo-Fabrizio operator, and immediately after them Nieto and Losada [20] turned to checking some flexible specifications of such newly defined nonsingular operator. The applicability of this operator was such that many newly designed mathematical models were formulated with respect to the mentioned nonsingular operator; see [21][22][23].
In 2017, a generalization of the Caputo fractional operator called the ϕ-Caputo derivative was introduced by Almeida [24] in which the kernel of this operator depends on an increasing function ϕ. By using this newly defined operator, a limited number of papers have been published so far; see [25][26][27][28][29].
In the light of the new operator introduced in [24] and motivated by the aforementioned work, we design the following ϕ-Caputo differential inclusion boundary value problem: supplemented with mixed integro-derivative conditions in the frame of the ϕ-Riemann-Liouville operators so that 1 < ς * < 2, 0 < ς 1 ≤ 1, ς 2 > 0 and C D ς * ;ϕ a stands for the ς * th ϕ-Caputo derivative with respect to an increasing function ϕ and RL D ς 1 ;ϕ a is the ς 1 th ϕ-Riemann-Liouville derivative and RL I ς 2 ;ϕ a indicates the ς 2 th ϕ-Riemann-Liouville integral. Besides, μ * 1 , μ * 2 ∈ R, ξ , σ ∈ (a, M) andȆ : [a, M] × R → P(R) is a multifunction furnished with some specifications which will be indicated later.
Our basic purpose in the current research is to derive existence aspects of solutions for the above general category of the fractional inclusion boundary value problem in terms of the well-defined generalized ϕ-Caputo and ϕ-Riemann-Liouville operators. The existing boundary conditions are considered as mixed generalized ϕ-Riemann-Liouville bound conditions. To obtain the desired existence criteria, we utilize analytical methods on αψ-contractive maps and multifunctions involving the approximate endpoint specification. We emphasize that this structure of the ϕ-Caputo inclusion problem and ϕ-Riemann-Liouville integro-derivative conditions is a general case based on an arbitrary increasing function ϕ and we can even extend our boundary conditions to multi-strip multi-point conditions in future work. So this problem is novel and unique in this respect.
We prepare the contents of this research paper by the following settings. Some applied properties and auxiliary issues are collected in Sect. 2. Next in Sect. 3, we employ two notions of endpoint and fixed point to derive the existence aspects corresponding to the ϕ-inclusion boundary value problem designed by (1)- (2). In the final section, we formulate an illustrative simulation model to examine theoretical findings computationally and numerically.

Auxiliary preliminaries
Here, we collect and review several fundamental notions in the framework of our analytical methods applied in this paper. As is well known, the concept of the ς * th Riemann-Liouville integral of a continuous function φ : R ≥0 → R is when the right hand side integral exists finitely [32,33]. From here onwards, let ς * ∈ (m -1, m) with m = [ς * ] + 1. For a continuous function φ : R ≥0 → R, the ς * th Riemann-Liouville derivative is represented by when the right hand side integral possesses finite values [32,33]. In the next step, for an absolutely continuous function φ ∈ AC (m) when the right hand side integral possesses values finitely [32,33].
provided that the right hand side of equality is finite-valued.

Definition 2.2 ([
provided that the right hand side of equality is finite-valued. In the similar manner, if ϕ(t) = t, then it is obvious that the ς * th ϕ-Riemann-Liouville derivative (7) reduces to the standard ς * th Riemann-Liouville derivative (4). Inspired by these operators, Almeida presented a new ϕ-version of the Caputo derivative in the following formulation.
when the right hand side of equality possesses values finitely.
It should be noted that, if ϕ(s) = s, then it is obvious that the ς * th ϕ-Caputo derivative (8) reduces to the standard ς * th Caputo derivative (5). In the following, some useful specifications of the ς * th ϕ-Caputo and ς * th ϕ-Riemann-Liouville integro-derivative operators can be seen.

Proposition 2.4 ([24, 33-35])
Let ς * , β * , * > 0 and ϕ ∈ C m ([a, b]) is assumed to be an increasing map with ϕ (s) > 0 for any a ≤ s ≤ b. Then the following statements hold: With due attention to the above proposition, it is simply found that the general series solution of ( C D ς * ;ϕ a φ)(s) = 0 is given by . In the subsequent stage, some concepts and the related specifications on the theory of multifunctions are reviewed.

Notation 2.6
Regard an ordered pair ( , · ) which represents a normed space. We mean by the notations P( ), P bnd ( ), P cmp ( ), P cls ( ) and P cvx ( ) a representation of the category of all nonempty, all bounded, all compact, all closed and all convex subsets of , respectively.

Definition 2.13 ([40])
Regard as a normed space. In this case: The next propositions are required logical tools for establishing the desired results in this research.
ThenȆ possesses an endpoint uniquely iffȆ possesses an approximate endpoint specification.

Existence theorems
After presenting some required concepts in the previous two sections, we are going to indicate our desired existence theorems. To arrive at this goal, In this phase, ( , · ) will be a Banach space. In addition to this, keep in mind the following for convenience: In the subsequent stage, we derive an equivalent relation as an integral equation for the generalized ϕ-Caputo inclusion boundary value problem (1)-(2).
Then φ 0 satisfies the linear ϕ-Caputo differential equation if and only if φ 0 is a solution of the ϕ-Riemann-Liouville integral equation so thatB * 1 (s),B * 2 (s) andB * 3 (s) are three functions depending on a variable s endowed with the following rules: and Proof First φ 0 is assumed to satisfy the generalized ϕ-Caputo differential equation (10). Obviously, we have C D ς * ;ϕ a φ 0 (s) =˘ (s). By taking the ς * th ϕ-Riemann-Liouville integral on both sides of the latter relation, we obtain Now, we intend to seek unknown real constants c * 0 and c * 1 . If we take the ς 1 th ϕ-Riemann-Liouville derivative and the ς 2 th ϕ-Riemann-Liouville integral in s on both sides of (14), respectively, we deduce that and By virtue of the mixed generalized ϕ-Riemann-Liouville boundary value conditions, we get and where (9). After doing some straightforward calculations on (15)-(16), we obtain Now we insert c * 0 and c * 1 into (14). In this case, one may write which illustrates that φ 0 satisfies (12). Conversely, we can simply see that φ 0 is a solution for the generalized ϕ-Caputo fractional boundary value problem (10)-(11) whenever φ 0 satisfies (12). This finishes the argument.

Notation 3.2 For simplicity in some required calculations, set
hold and also, for any s ∈ [a, M], is valid.
In the following we address another criterion for the generalized ϕ-Caputo fractional inclusion boundary value problem (1)-(2) under new hypotheses. More precisely, we infer the existence result with respect to a new property due to Amini [40]. We invoke the approximate endpoint specification for K which is demonstrated in (18).
Proof As a main purpose, we intend to check the existence of an endpoint for the multifunction K : → P( ). To fulfill the mentioned demand, we try to verify that K(φ) is closed for any φ ∈ . Due to (C8) and by the measurability of the mapping s →Ȇ(s, φ(s)) and closedness of it for each φ ∈ , it is deduced thatȆ possesses a measurable selection and so SȆ ,φ = ∅ for each φ ∈ . Accordingly, like the proof of Theorem 3.4, it is simple for a.e. s ∈ [a, M]. In conclusion, similar to the proof of Theorem 3.4, we have The above result yields PH d (K(φ), K(φ )) ≤ ψ( φφ ) for each φ, φ ∈ . Along with this, the condition (C10) expresses that K possesses an approximate endpoint specification. In conclusion, by Proposition 2.16, we arrive at the intended purpose which confirms the issue that K possesses an endpoint uniquely; that is, K(φ * ) = {φ * } for some φ * ∈ . Hence, it is followed that φ * is a solution for the generalized ϕ-Caputo fractional inclusion boundary value problem (1)-(2).

Simulation example
Finally, we support our results by proposing an example to demonstrate the applicability of the findings numerically. Indeed, the theoretical results obtained in Theorem 3.5 are guaranteed by a numerical example.

Conclusion
The basic purpose in the current research is to derive several existence aspects of solutions for a novel general class of inclusion problems in terms of the well-defined generalized ϕ-Caputo and ϕ-Riemann-Liouville operators. The existing boundary conditions in such inclusion problem are endowed with mixed generalized ϕ-Riemann-Liouville conditions. To reach this goal, we utilize two notions of endpoint and fixed point to deduce the existence aspects in relation to the existing inclusion boundary value problem (1)- (2). In other words, the analytical methods on α-ψ-contractions and multifunctions involving an approximate endpoint specification are applied to verify the required theoretical findings. Finally, we present a simulation example to examine our theoretical results computationally and numerically.