# Fixed Point and Endpoint Theories for Two Hybrid Fractional Differential Inclusions with Operators Depending on an Increasing Function

Authors: Sh. Rezapour; M. Q. Iqbal; A. Hussain; A. Zada; S. Etemad
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

The main concentration of the present research is to explore several theoretical criteria for proving the existence results for the suggested boundary problem. In fact, for the first time, we formulate a new hybrid fractional differential inclusion in theφ-Caputo settings depending on an increasing function φ subject to separated mixed φ-hybrid-integro-derivative boundary conditions. In addition to this, we discuss a special case of the proposed φ-inclusion problem in the non-φ-hybrid structure with the help of the endpoint notion. To confirm the consistency of our findings, two specific numerical examples are provided which simulate both φ-hybrid and non-φ-hybrid cases.


---

## Body

## 1. Introduction

Arbitrary order calculus theory is considered as an important topic of research for all mathematicians, researchers, engineers, and scientists due to the applicability of mentioned theory in several contexts in engineering and applied science and its flexibility to model different systems and phenomena having memory effects (see, e.g., [1–3] and reference therein). Several arbitrary order derivatives have been introduced in the past decade, and the most common of them are Riemann-Liouville, Caputo, and Hadamard derivatives. Hence, arbitrary order boundary value problems (BVPs) can be formulated in the framework of different operators. In the meantime, some recent research investigations have been conducted with the aid of these operators to establish the relevant analytical results for proposed BVPs. For instance, Alzabut et al. [4] investigated the oscillatory behavior of a kind of fractional differential equations (FDEs) supplemented with damping and forcing terms by terms of generalized proportional operators. In [5], Baleanu et al. modeled an applied instrument in engineering in the context of a hybrid Caputo FBVP and studied its existence theory. Also, the same authors [6] established similar results by means of Caputo and Riemann-Liouville conformable derivation and integration operators. In 2019, Matar et al. [7] devoted their focus on solvability of nonlinear systems of FDEs via nonlocal initial value problems by terms of fixed point methods and after that, Mohammadi et al. [8] utilized another fractional operator entitled Caputo–Hadamard for modeling a hybrid FBVP with Hadamard integral boundary conditions. Zhou et al. [9] presented a fractional antiperiodic model of Langevin equation and investigated qualitative aspects of its solutions with the aid of techniques appeared in functional analysis. Similarly, one can find some papers on applications of fractional operators [10–13].In 2017, a generalization of the Caputo fractional operator known asφ-Caputo derivative (φ-CF) was presented by Almeida [14] in which its kernel is with respect to a given increasing function φ. One of the most important advantages of the φ-CF derivative operator is its ability to produce all previous fractional derivatives, and also, it involves the semigroup property. As a result, φ-CF derivative is known as an extended structure of arbitrary order derivative operators.To get acquainted with some previous research works done based onφ-CF operators so far, we refer to a paper published by Wahash et al. [15]. In that paper, Wahash et al. designed a generalized φ-fractional differential equation with a simple integral condition as
(1)DC0+σ∗;φwz=ℏ∗z,wz,w0=v+d∫01ξquqdq,where z∈0,1, σ∗∈0,1, ν∈ℝ+, and d∈ℝ≥0, and also, ℏ∗:0,1×ℝ+⟶ℝ+ stands for a continuous function along with ξ∈Lℝ+10,1. The lower-upper solution is a technique implemented in that article by Wahash et al. in which they utilized a fixed point method on cones. Further, lower-upper control maps are provided with respect to the nonlinear term without a certain monotonicity criterion [15]. Similarly, by using the newly introduced φ-CF operator and its generalizations, several articles have been published like as [16] in which Almeida et al. considered a FDE via a Caputo derivative with respect to a kernel function and reviewed some applications of them. Derbazi et al. [17] used such a generalized operator to investigate a nonlinear initial value problem via monotone iterative method. Samet et al. derived some Lyapunov-type inequalities in relation to an antiperiodic FBVP involving φ-Caputo operator [18]. The analysis of the stability to an φ-Hilfer impulsive FDE is another instance of applications of such generalized operators which was studied by Sousa et al. in [19]. In 2020, Tariboon et al. [20] turned to establishment of existence theorems to sequential generalized inclusion FBVP, and then, Thabet et al. [21] achieved to similar findings on a new structure of the pantograph inclusion FBVP. In a higher level, Vivek et al. [22] defined generalized φ-operators in the context of partial operators and analyzed a PDE in the φ-Caputo settings.With regard to ideas of aforesaid research works, we consider the followingφ-hybrid fractional differential inclusion in the sense of Caputo represented as
(2)DCaσ∗;φϖ∗zN∗z,ϖ∗z∈O~z,ϖ∗z,supplemented with separated mixedφ-hybrid-integro-derivative boundary conditions
(3)m~1ϖ∗zN∗z,ϖ∗zz=a=s1∗+m~2ϖ∗zN∗z,ϖ∗zz=a,m~1RLIaμ∗;φDCaμ~;φϖ∗zN∗z,ϖ∗zz=T=s2∗+m~2RLIaμ∗;φDCa1;φϖ∗zN∗z,ϖ∗zz=T,wherez∈a,T with a≥0, σ∗∈1,2, μ~∈0,1, μ∗>0, m~1,m~2∈ℝ≠0, and s1∗,s2∗∈ℝ+. Two notations DCa⋅;φ and IRLa⋅;φ stand for the φ-CF derivative and the φ-Riemann-Liouville integral (φ-RLF), respectively. Also, notice that DCa1;φ=1/φ′zd/dz. Besides, N∗:a,T×ℝ⟶ℝ is assumed to be a nonzero continuous single-valued operator, and O~:0,1×ℝ⟶Pℝ is assumed to be a set-valued operator equipped with some required properties. Notice that by putting N∗z,ϖ∗z=1, the given φ-hybrid Caputo fractional differential inclusion BVP (2) and (3) is transformed into a non-φ-hybrid separated inclusion BVP presented by
(4)DCaσ∗;φϖ∗z∈O~z,ϖ∗z,z∈a,T,m~1ϖ∗a=s1∗+m~2ϖ∗a,m~1IRLaμ∗;φDCaμ~;φϖ∗T=s2∗+m~2IRLaμ∗;φDCa1;φϖ∗T.Note that by taking into account the authors’ knowledge, there are no research manuscripts onφ-CF operators involving mixed φ-hybrid-integro-derivative boundary conditions simultaneously. In addition, this given structure is formulated in a unique and general form in which we can consider some standard special cases studied before. Here, we derive some analytical criteria to prove the existence results for the proposed novel φ-hybrid fractional differential inclusion in the φ-Caputo settings (2) equipped with separated mixed φ-hybrid-integro-derivative boundary conditions (3). The applied approach to achieve desired purposes is based on Dhage’s fixed point result. In addition, we discuss the special case of the proposed φ-inclusion problem in the non-φ-hybrid version with the aid of the endpoint notion. We organize the present manuscript as the following construction. In Section 2, we briefly collect auxiliary preliminaries on the φ-fractional operators and some required notions on the multifunctions and related properties. In Section 3, the existence criteria of solutions for both proposed φ-hybrid and non-φ-hybrid BVPs (2)–(4) are derived by two different analytical methods. To confirm the applicability of our analytical findings, two simulative numerical examples are formulated in Section 4 which cover both φ-hybrid and non-φ-hybrid cases.


## 2. Auxiliary Preliminaries

By continuing the path ahead, we assemble and recall several auxiliary and fundamental notions in the direction of our theoretical methods implemented in this paper. The concept of RLF integral forϖ∗:0,+∞⟶ℝ of order σ∗>0 is defined as
(5)IRL0σ∗ϖ∗z=∫0zz−qσ∗−1Γσ∗ϖ∗qdq,provided that the integral has finite value [23, 24]. In this position, let us take n−1<σ∗<n in which n=σ∗+1. Regarding a continuous function ϖ∗:0,+∞⟶ℝ, the RLF derivative of order ρ∗ is defined as
(6)DRL0σ∗ϖ∗z=ddzn∫0zz−qn−σ∗−1Γn−σ∗ϖ∗qdq,provided that the integral has finite value [23, 24]. In the next step, for an absolutely continuous n-times differentiable real-valued function ϖ∗ on 0,+∞, the derivative in the Caputo settings of order σ∗ is defined as
(7)DC0σ∗ϖ∗z=∫0zz−qn−σ∗−1Γn−σ∗ϖ∗nqdq,such that it is finite-valued [23, 24]. Now, let φ∈Cna,b be increasing with φ′z>0,∀z∈a,b. Then, an integral in the sense of φ-Riemann-Liouville for an integrable ϖ∗:a,b⟶ℝ of order σ∗ depending on increasing function φ is defined as
(8)IRLaσ∗;φϖ∗z=1Γσ∗∫azφ′qφz−φqσ∗−1ϖ∗qdq,provided that the RHS of above equality involves the finite value [25, 26]. It is clear that if we take φz=z, then φ′z=1, and thus by inserting them into (8), we see that the φ-RLF integral is converted to the standard RLF integral given by (5). For a continuous function ϖ∗:0,+∞⟶ℝ, a derivative in the sense of φ-RL of order σ∗ is given by
(9)DRLaσ∗;φϖ∗z=1Γn−σ∗1φ′zddzn∫azφ′qφz−φqn−σ∗−1ϖ∗qdq,provided that the RHS of above equality exists [25, 26]. If φz=z, then the φ-RLF derivative (9) is converted to the standard RLF derivative (6). Motivated by such operators, Almeida gave a φ-version of the CF derivative as follows:
(10)DCaσ∗;φϖ∗z=1Γn−σ∗∫azφ′qφz−φqn−σ∗−11φ′qddqnϖ∗qdq,provided that the RHS of above equality exists [14]. If φz=z, then the φ-CF derivative (10) is converted to the standard CF derivative (7). Some useful properties of the φ-CF and φ-RLF operators can be seen in the following.Lemma 1 [14, 24].
Letσ∗,ϱ∗,β∗>0 and φ∈Cna,b be increasing with φ′z>0,∀z∈a,b. Then,
(i1)IRLaσ∗;φIRLaϱ∗;φϖ∗z=IRLaσ∗+ϱ∗;φϖ∗z
(i2)IRLaσ∗;φφz−φaβ∗y=Γβ∗+1/Γσ∗+β∗+1φy−φaσ∗+β∗
(i3)DCaσ∗;φφz−φaβ∗y=Γβ∗+1/Γβ∗−σ∗+1φy−φaβ∗−σ∗,β∗>σ∗
(i4)DRLaσ∗;φIRLaρ∗;φϖ∗z=IRLaρ∗−σ∗;φϖ∗z,σ∗<ρ∗
For instance, we plot the graph ofφ-RLF integral and φ-CF derivative of ϖz=z−16.5 for φz=2z+3/2 in Figure 1.Lemma 2 [14].
Letn−1<σ∗<n. Then, for each ϖ∗∈Cn−1a,b,
(11)IRLaσ∗;φDCaσ∗;φϖ∗z=ϖ∗z−∑j=0n−1δφjϖ∗aj!φz−φaj,δφ=1φ′zddz.Figure 1
The RLF-integral and CF-derivative ofϖz=z−16.5 for φz=2z+3/2.In accordance with above lemma, the authors proved that the series solution for given homogeneous differential equationDCaσ∗;φϖ∗z=0 has such a form
(12)ϖ∗z=∑j=0n−1k~j∗φz−φaj=k~0∗+k~1∗φz−φa+k~2∗φz−φa2+⋯+k~n−1∗φz−φan−1,where n−1<σ∗<n and k~0∗,k~1∗,⋯,k~n−1∗∈ℝ [14].We consider the normed space by notationW,·W. Also, we introduce the notations PW, PbndW, PclsW, PcmpW, and PcvxW for the category of all nonempty subsets, all bounded subsets, all closed subsets, all compact subsets, and all convex subsets of W, respectively. In the subsequent path, a metric function attributed to Pompeiu-Hausdorff ℙℍdW:PW×PW⟶ℝ∪∞ is defined by
(13)ℙℍdWE1,E2=maxsupe1∈E1dWe1,E2,supe2∈E2dWE1,e2,so that dWE1,e2=infe1∈E1dWe1,e2 and dWe1,E2=infe2∈E2dWe1,e2 [27]. We say that O~:W⟶PclsW is Lipschitzian with constant c^>0 if ℙℍdWO~ϖ1∗,O~ϖ2∗≤c^dWϖ1∗,ϖ2∗,∀ϖ1∗,ϖ2∗∈W. Also, O~ is a contraction if c^∈0,1 [27].We represent the collection of all existing selections ofO~ at point ϖ∗∈Cℝ0,1 by
(14)SELO~,ϖ∗≔κ^∈Lℝ10,1:κ^z∈O~z,ϖ∗z,for almost all z∈0,1 [27, 28]. We note that ϖ∗∈W is an endpoint for given set-valued operator O~:W⟶PW whenever we have O~ϖ∗=ϖ∗ [29]. Also, the mapping O~ possesses an approximate endpoint property (APXEndP-property) whenever
(15)infϖ1∗∈Wsupϖ2∗∈O~ϖ1∗dWϖ1∗,ϖ2∗=0,[29]. We need next results.Theorem 3 (Closed graph theorem [30]).
LetW be a separable Banach space, O~:0,1×W⟶Pcmp,cvxW be L1-Carathéodory and ∐:LW10,1⟶CW0,1 be a linear continuous map. Then, ∐∘SELO~:CW0,1⟶Pcmp,cvxCW0,1 is another operator in CW0,1×CW0,1 with action ϖ∗↦∐∘SELO~ϖ∗=∐SELO~,ϖ∗ having closed graph property.Theorem 4 (Dhage’s theorem [31]).
Consider the Banach algebraW, and the operators A1∗:W⟶W and A2∗:W⟶Pcmp,cvxW satisfying the following:
(i)
A1∗ is Lipschitzian (with l∗>0)(ii)
A2∗ is compact upper semicontinuous(iii)
2l∗O^<1 with O^=A2∗W
Then, either1iO∗=ϖ∗∈W∣α0ϖ∗∈A1∗ϖ∗A2∗ϖ∗,α0>1 is unbounded, or 2i a solution, belonging to W, exists for which ϖ∗∈A1∗ϖ∗A2∗ϖ∗.Theorem 5 (Endpoint theorem [29]).
Suppose thatW,dW be complete and ψ:ℝ≥0⟶ℝ≥0 admits the upper semicontinuity via ψz<z and liminfz⟶∞z−ψz>0,∀z>0. Besides, we assume that O~:W⟶Pcls,bndW is such that ℙℍdWO~ϖ1∗,O~ϖ2∗≤ψdWϖ1∗,ϖ2∗ for each ϖ1∗,ϖ2∗∈W. Then, an endpoint (uniquely) exists for O~iffO~ involves the APXEndP-property.


## 3. New Existence Criteria

In two previous sections, we assembled some auxiliary and useful notions to achieve our main goals. Now in the following, we first establish a required lemma to derive the main existence results. To do this, we need to consider a sup–norm given byϖ∗W=supz∈0,1ϖ∗z on the space W=ϖ∗z:ϖ∗z∈Cℝ0,1. In this case, the Banach space W,·W along with the multiplication action defined as ϖ1∗·ϖ2∗z=ϖ1∗zϖ2∗z is a Banach algebra for all ϖ1∗,ϖ2∗∈W.Lemma 6.
Leth∗∈W, a≥0, σ∗∈1,2, μ~∈0,1, μ∗>0, m~1,m~2∈ℝ≠0, and s1∗,s2∗∈ℝ+. An element ϖ0∗∈W is a solution for given φ-hybrid fractional equation
(16)DCaσ∗;φϖ∗zN∗z,ϖ∗z=h∗z,z∈a,T,σ∗∈1,2,supplemented with separated mixed φ-integro-derivative boundary conditions
(17)m~1ϖ∗zN∗z,ϖ∗zz=a=s1∗+m~2ϖ∗zN∗z,ϖ∗zz=a,m~1RLIaμ∗;φDCaμ~;φϖ∗zN∗z,ϖ∗zz=T=s2∗+m~2RLIaμ∗;φDCa1;φϖ∗zN∗z,ϖ∗zz=T,which is given by the following:
(18)ϖ∗z=N∗z,ϖ∗zs1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1h∗qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1h∗qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2h∗qdq,
so thatm~∗ is a positive real constant given as
(19)m~∗≔m~1φT−φa1+μ∗−μ~Γ2+μ∗−μ~−m~2φT−φaμ∗Γ1+μ∗≠0.Proof.
At first, the elementϖ0∗ is assumed to be a solution for the hybrid φ-Caputo differential Equation (16). Then, there exist k~0∗,k~1∗∈ℝ such that
(20)ϖ0∗zN∗z,ϖ0∗z=IRLaσ∗;φh∗z+k~0∗+k~1∗φz−φa,or more precisely, we have
(21)ϖ0∗z=N∗z,ϖ0∗z∫azφ′qφz−φqσ∗−1Γσ∗h∗qdq+k~0∗+k~1∗φz−φa.
In view of the notion of fractional derivative in theφ-Caputo framework, we get the following relations for μ~∈0,1:
(22)DCa1;φϖ0∗zN∗z,ϖ0∗z=∫azφ′qφz−φqσ∗−2Γσ∗−1h∗qdq+k~1∗,(23)DCaμ~;φϖ0∗zN∗z,ϖ0∗z=∫azφ′qφz−φqσ∗−μ~−1Γσ∗−μ~h∗qdq+k~1∗φz−φa1−μ~Γ2−μ~.
In the following, by taking integral of orderμ∗>0 in the φ-Riemann-Liouville settings on both sides of (22) and (23), we obtain
(24)IRLaμ∗;φDCa1;φϖ0∗zN∗z,ϖ0∗z=∫azφ′qφz−φqσ∗+μ∗−2Γσ∗+μ∗−1h∗qdq+k~1∗φz−φaμ∗Γμ∗+1,IRLaμ∗;φDCaμ~;φϖ0∗zN∗z,ϖ0∗z=∫azφ′qφz−φqσ∗+μ∗−μ~−1Γσ∗+μ∗−μ~h∗qdq+k~1∗φz−φ−a1+μ∗−μ~Γμ∗−μ~+2.
In this step, by considering the first boundary condition in (17), we find that m~1−m~2k~0∗=s1∗ and so
(25)k~0∗=s1∗m~1−m~2.
In addition, the second integro-derivative boundary condition given in (17) yields
(26)k~1∗=s2∗m~∗−m~1m~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1h∗qdq+m~2m~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2h∗qdq.
In the last step, if we insert the valuesk~0∗ and k~1∗ obtained in (25) and (26) into (21), then we get
(27)ϖ0∗z=N∗z,ϖ0∗zs1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1h∗qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1h∗qdq+m~2φz−öam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2h∗qdq.
The resultant integral equation confirms thatϖ0∗ satisfies the mentioned φ-integral Equation (18), and the proof is completed.Now, by considering Lemma6, we can present the following definition.Definition 7.
An absolutely continuous functionϖ∗:a,T⟶ℝ is called a solution function for the φ-hybrid inclusion BVP in the sense of φ-Caputo (2) and (3) if there is κ^∈L1a,T,ℝ with κ^z∈O~z,ϖ∗z for almost all z∈a,T which satisfies separated mixed φ-integro-derivative boundary conditions
(28)m~1ϖ∗zN∗z,ϖ∗zz=a=s1∗+m~2ϖ∗zN∗z,ϖ∗zz=a,m~1IRLaμ∗;φDCaμ~;φϖ∗zN∗z,ϖ∗zz=T=s2∗+m~2IRLaμ∗;φDCa1;φϖ∗zN∗z,ϖ∗zz=T,and also
(29)ϖ∗z=N∗z,ϖ∗zs1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,for each z∈a,T.Now, we are in a position that we can prove the first existence result about the hybridφ-Caputo inclusion BVP (2) and (3).Theorem 8.
Assume thatO~:a,T×W⟶Pcmp,cvxW is a set-valued operator and a function N∗:a,T×W⟶W\0 is continuous. In addition, let
(C1) a bounded function χ:a,T⟶ℝ+ exists such that for each ϖ1∗,ϖ2∗∈W and z∈a,T(30)N∗z,ϖ1∗z−N∗z,ϖ2∗z≤χzϖ1∗z−ϖ2∗z
(C2) O~ is L1-Caratheodory
(C3) a function Yz∈L1a,T,ℝ+ exists such that
(31)O~z,ϖ∗=supκ^:κ^∈O~z,ϖ∗z≤Yz,for all ϖ∗∈W and for almost all z∈a,T
(C4) a real number ε^∈ℝ exists so that
(32)ε^>F∗Π∗YL11−χ∗Π∗YL1,where F∗=supz∈a,TN~∗z,0, YL1=∫aTYqdq, χ∗=supz∈a,Tχz, and
(33)Π∗=s1∗m~1−m~2+φT−φaσ∗Γσ∗+1+s2∗m~∗φT−φa+m~1φT−φaσ∗+μ∗−μ~+1m~∗Γσ∗+μ∗−μ~+1+m~2φT−φaσ∗+μ∗m~∗Γσ∗+μ∗
Ifχ∗Π∗YL1<1/2, then the φ-hybrid inclusion BVP (2) and (3) has at least a solution.Proof.
For eachϖ∗∈W, the collection of all existing selections of O~ is defined as
(34)SELO~,ϖ∗≔κ^∈Lℝ1a,T:κ^z∈O~z,ϖ∗z,for every ϖ∗∈W and for almost all z∈a,T. Define a set-valued map K:W⟶PW by
(35)Kϖ∗=ζ∗∈W:ζ∗z=ψ∗z,where
(36)ψ∗z=N∗z,ϖ∗zs1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,for some κ^∈SELO~,ϖ∗ and for almost all z∈a,T. It is obvious that the function ψ0∗ is a solution to the φ-hybrid BVP (2) and (3) if ψ0∗ is a fixed point of K. Now, define A1∗:W⟶W by A1∗ϖ∗z=N∗z,ϖ∗z and A2∗:W⟶PW by
(37)A2∗ϖ∗z=h∗∈W:h∗z=ξ∗z,where
(38)ξ∗z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,
for someκ^∈SELO~,ϖ∗ and for almost all z∈a,T. This implies Kϖ∗=A1∗ϖ∗A2∗ϖ∗. We show that both operators A1∗ and A2∗ satisfy Theorem 4. We at first prove that A1∗ is Lipschitzian. Let ϖ1∗,ϖ2∗∈W. We have
(39)A1∗ϖ1∗z−A1∗ϖ2∗z=N∗z,ϖ1∗z−N∗z,ϖ2∗z≤χzϖ1∗z−ϖ2∗z,for all z∈a,T. Therefore, for all ϖ1∗,ϖ2∗∈W, we get
(40)A1∗ϖ1∗−A1∗ϖ2∗W≤χ∗ϖ1∗−ϖ2∗W.
Hence,A1∗ is Lipschitz with constant l∗=χ∗>0. In the current moment, we check the convexity of A2∗. For this, let ϖ1∗,ϖ2∗∈A2∗ϖ∗. Choose κ^1,κ^2∈SELO~,ϖ∗ such that
(41)ϖi∗z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^iqdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^iqdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^iqdq,i=1,2,for almost all z∈a,T. Let 0<η<1. Then,
(42)ηϖ1∗z+1−ηϖ2∗z=s1∗m~1−m~2+s2∗m~∗φz−φa+1Γσ∗∫azφ′qφz−φqσ∗−1ηκ^1q+1−ηκ^2qdq−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1ηκ^1q+1−ηκ^2qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2ηκ^1q+1−ηκ^2qdq,for almost all z∈a,T. With due attention to the convexity of O~, SELO~,ϖ∗ has convex values. This implies that
(43)ηκ^1z+1−ηκ^2z∈SELO~,ϖ∗,for almost all z∈a,T. Therefore, A2∗ϖ∗ is convex for each ϖ∗∈W. Next, we claim that A2∗ is completely continuous. To confirm this claim, we verify that the set A2∗W is equicontinuous and uniformly bounded. Firstly, we prove that A2∗ corresponds bounded sets to bounded sets contained in W. For α∗∈ℝ+, define the bounded ball Bα∗=ϖ∗∈W:∥ϖ∗∥W≤α∗. For every ϖ∗∈Bα∗ and ξ∗∈A2∗ϖ∗, there exists a function κ^∈SELO~,ϖ∗ such that
(44)ξ∗z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,
for almost allz∈a,T. Then,
(45)ξ∗z≤1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa+m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq+s1∗m~1−m~2≤1Γσ∗∫azφ′qφz−φqσ∗−1Yqdq+s2∗∣m~∗∣φz−φa+m~1φz−φa∣m~∗∣Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1Yqdq+∣m~2∣φz−φa∣m~∗∣Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2Yqdq+s1∗∣m~1−m~2∣≤φT−φaσ∗Γσ∗+1+s2∗m~∗φT−φa+m~1φT−φaσ∗+μ∗−μ~+1m~∗Γσ∗+μ∗−μ~+1+m~2φT−φaσ∗+μ∗m~∗Γσ∗+μ∗+s1∗m~1−m~2YL1=Π∗∥Y∥L1,where Π∗ is given in (33). Thus, ξ∗≤Π∗YL1, and so the set A2∗W is uniformly bounded. Now, we want to prove that A2∗ corresponds bounded sets to equicontinuous sets. Take ϖ∗∈Bα∗, ξ∗∈A2∗ϖ∗ and choose κ^∈SELO~,ϖ∗ so that
(46)ξ∗z=1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq+s1∗m~1−m~2,for almost all z∈a,T. Let z1,z2∈a,T with z1<z2, Then,
(47)ξ∗z2−ξ∗z1≤1Γσ∗∫az1φ′qφz2−φqσ∗−1−φz1−φqσ∗−1κ^qdq+1Γσ∗∫z1z2φ′qφz2−φqσ∗−1κ^qdq+s2∗m~∗φz2−φz1+m~1φz2−φz1m~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz2−φz1m~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq≤1Γσ∗∫az1φ′qφz2−φqσ∗−1−φz1−φqσ∗−1Yqdq+1Γσ∗∫z1z2φ′qφz2−φqσ∗−1Yqdq+s2∗m~∗φz2−φz1+m~1φz2−φz1m~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1Yqdq+m~2φz2−φz1m~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2Yqdq.
The right-hand side of the latter inequalities tends to zero (independent ofϖ∗∈Bα∗) as z1 tends to z2. Application of Arzela–Ascoli theorem gives the complete continuity of A2∗. We here discuss that A2∗ has a closed graph, and this finding implies that A2∗ is upper semicontinuous. To achieve this aim, let ϖn∗∈Bα∗ and ξn∗∈A2∗ϖn∗ with ϖn∗⟶ϖ∗∗ and ξn∗⟶ξ~∗. We claim that ξ~∗∈A2∗ϖ∗∗. For every n≥1 and ξn∗∈A2∗ϖn∗, choose κ^n∈SELO~,ϖn∗ such that
(48)ξn∗z=1Γσ∗∫azφ′qφz−φqσ∗−1κ^nqdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^nqdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^nqdq+s1∗m~1−m~2,for almost all z∈a,T. It is suffices to find that there is a member κ∧∗∈SELO~,ϖ∗∗ so that
(49)ξ~∗z=1Γσ∗∫azφ′qφz−φqσ∗−1κ∧∗qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ∧∗qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ∧∗qdq+s1∗m~1−m~2,for almost all z∈a,T. Define a linear continuous operator ∐:L1a,T,ℝ⟶W=Ca,T,ℝ as
(50)∐κ^z=ϖ∗z=1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq+s1∗m~1−m~2,for almost all z∈a,T. Hence,
(51)ξn∗z−ξ~∗z=1Γσ∗∫azφ′qφz−φqσ∗−1κ^nq−κ∧∗qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^nq−κ∧∗qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^nq−κ∧∗qdq⟶0.
Application of Theorem3 shows that ∐∘SELO~ has a closed graph. Besides, since ξn∗∈∐SELO~,ϖn and ϖn⟶ϖ∗∗, so there exists κ∧∗∈SELO~,ϖ∗∗ such that
(52)ξ~∗z=1Γσ∗∫azφ′qφz−φqσ∗−1κ∧∗qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ∧∗qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ∧∗qdq+s1∗m~1−m~2,for almost all z∈a,T. Hence, ξ~∗∈A2∗ϖ∗∗, and so A2∗ possesses closed graph which implies that A2∗ is upper semicontinuous. On the other hand, because of the compactness of values of A2∗, it is immediately deduced that A2∗ is compact and upper semicontinuous. Utilizing (C3), we get
(53)O^=A2∗W=supA2∗ϖ∗:ϖ∗∈W=s1∗m~1−m~2+φT−φaσ∗Γσ∗+1+s2∗m~∗φT−φa+m~1φT−φaσ∗+μ∗−μ~+1m~∗Γσ∗+μ∗−μ~+1+m~2φT−φaσ∗+μ∗m~∗Γσ∗+μ∗YL1=Π∗∥Y∥L1.
Putl∗=χ∗. We have O^l∗<1/2. Utilizing Theorem 4, we prove that one of the items (i) or (ii) is possible. First, we check that the item (i) is not the case. From Theorem 4 and the assumption (C4), consider an arbitrary member ϖ0∗ of Σ∗ with ∥ϖ0∗∥=ε^. Then, α0ϖ0∗z∈A1∗ϖ0∗A2∗ϖ0∗z for all α0>1. Choosing a function κ^∈SELO~,ϖ0∗, for each α0>1, we have
(54)ϖ0∗z=1α0N∗z,ϖ0∗zs1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,for almost all z∈a,T. Thus, one can write
(55)ϖ0∗z=1α0N∗z,ϖ0∗zs1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa+m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq≤1α0N∗z,ϖ0∗z−N∗z,0+N∗z,0×s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa+m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq≤χ∗ε^+F∗Π∗YL1.
Hence, we get(56)ε^≤YL1F∗Π∗1−Π∗χ∗YL1,which is a contradiction. Hence, the item (ii) indicated in Theorem 4 is valid. Thus, ϖ∗∈W exists so that ϖ∗∈A1∗ϖ∗A2∗ϖ∗. In consequence, the operator K has a fixed point. So the φ-hybrid inclusion BVP (2) and (3) has a solution, and this completes the proof.Definition 9.
An absolutely continuous functionϖ∗:a,T⟶ℝ is called a solution for the non-φ-hybrid inclusion BVP (4) in the sense of Caputo if there is κ^∈L1a,T,ℝ with κ^z∈O~z,ϖ∗z for almost all z∈a,T which satisfies separated mixed φ-integro-derivative boundary conditions
(57)m~1ϖ∗a=s1∗+m~2ϖ∗a,m~1IRLaμ∗;φDCaμ~;φϖ∗T=s2∗+m~2IRLaμ∗;φDCa1;φϖ∗T,(58)ϖ∗z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,
for almost allz∈a,T.For eachϖ∗∈W, the collection of all existing selections of O is defined as
(59)SELO,ϖ∗=κ^∈L1a,T:κ^z∈Oz,ϖz,for almost all z∈a,T. Define F:W⟶PW as
(60)Fϖ∗=ϑ∈W:ϑz=ρz,where
(61)ϱz=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq,κ^∈SELO,ϖ∗.By making use of endpoints for the multifunctionF, we prove the following theorem.Theorem 10.
ConsiderO:a,T×W⟶PcpW as a set-valued operator. Let
(C5) ψ:0,∞⟶0,∞ be increasing and upper semicontinuous with liminfz⟶∞z−ψz>0 and z>ψz,∀z>0
(C6) the multifunction O:a,T×W⟶PcpW be integrable and bounded so that O·,ϖ∗:a,T⟶PcpW be measurable for each ϖ∗∈W
(C7) a function ϱ∈Ca,T,0,∞ exists such that
(62)ℙℍdWOz,ϖ1∗,Oz,ϖ′1∗≤ϱzψϖ1∗z−ϖ′1∗z1Ω∗∗,for almost all z∈a,T and all ϖ1∗,ϖ′1∗∈W, where supz∈a,Tϱz=ϱ and
(63)Ω∗∗=1Γσ∗+1−∣m~1∣φT−φaσ∗+μ∗−μ~+1m~∗Γσ∗+μ∗−μ~+1+∣m~2∣φT−φaσ∗+μ∗m~∗Γσ∗+μ∗ϱ
(C8) the operator F given by (60) possesses APXEndP-property
Then, a solution exists to the non-φ-hybrid inclusion FBVP (4).Proof.
In such an argument, we try to prove the existence of endpoint to the set-valued operatorF:W⟶PW defined by (60). To proceed this, we first investigate that Fϖ∗ is closed for each ϖ∗∈W. By taking into account the hypothesis (C6), z↦Oz,ϖ∗z is a closed-valued measurable multifunction for each ϖ∗∈W. In consequence, O has a measurable selection SELO,ϖ∗≠∅. Now, we show that Fϖ∗⊆W is closed for all ϖ∗∈W. Consider the sequence ϖn∗n≥1 contained in Fϖ∗ with ϖn∗⟶μ. For each n, there exists κ^n∈SELO,ϖ∗ such that
(64)ϖn∗z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^nqdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^nqdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^nqdq,for almost all z∈a,T. Since O is compact multifunction, we acquire a subsequence κ^nn≥1 tending to κ^∈L1a,T. Hence, we have κ^∈SELO,ϖ∗ and
(65)limn⟶∞ϖn∗z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^qdq=ϖ∗z,
for almost allz∈a,T. Hence, ϖ∗∈F which indicates that F is closed-valued. In addition, Fϖ∗ is bounded for each ϖ∗∈W since O is compact. Finally, we investigate if ℙℍdWFϖ∗,Fϖ~∗≤ψϖ∗−ϖ~∗ holds. Let ϖ∗,ϖ~∗∈W, and x1∈Fϖ~∗. Choose κ^1∈SELO,ϖ~∗ such that
(66)x1z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^1qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^1qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^1qdq,for almost all z∈a,T. Since
(67)ℙℍdWOz,ϖ∗,Oz,ϖ~∗≤ϱzψϖ∗z−ϖ~∗z1Ω∗∗,for almost all z∈a,T, so there exists k∗∈Oz,ϖ∗ such that
(68)κ^1z−k∗≤ψϖ∗z−ϖ~∗zϱzΩ∗∗,for almost all z∈a,T. Define the multifunction A:a,T⟶PW given by
(69)Az=k∗∈W:κ^1z−k∗≤ϱzψϖ∗z−ϖ~∗z1Ω∗∗.
Sinceκ^ and σ=ϱψϖ∗−ϖ~∗1/Ω∗∗ are measurable, thus we choose κ^2z∈Oz,ϖ∗z such that
(70)κ^1z−κ^2z≤ψϖ∗z−ϖ~∗zϱzΩ∗∗,for almost all z∈a,T. Select x2∈Fϖ∗ such that
(71)x2z=s1∗m~1−m~2+1Γσ∗∫azφ′qφz−φqσ∗−1κ^2qdq+s2∗m~∗φz−φa−m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^2qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^2qdq,
for almost allz∈a,T. Hence, we get
(72)x1z−x2z≤1Γσ∗∫azφ′qφz−φqσ∗−1κ^1q−κ^2qdq+m~1φz−φam~∗Γσ∗+μ∗−μ~∫aTφ′qφT−φqσ∗+μ∗−μ~−1κ^1q−κ^2qdq+m~2φz−φam~∗Γσ∗+μ∗−1∫aTφ′qφT−φqσ∗+μ∗−2κ^1q−κ^2qdq≤1Γσ∗+1ϱψϖ∗−ϖ~∗1Ω∗∗−m~1φT−φaσ∗+μ∗−μ~+1m~∗Γσ∗+μ∗−μ~+1ϱψϖ∗−ϖ~∗1Ω∗∗+m~2φT−φaσ∗+μ∗m~∗Γσ∗+μ∗ϱψϖ∗−ϖ~∗1Ω∗∗=1Γσ∗+1−m~1φT−φaσ∗+μ∗−μ~+1m~∗Γσ∗+μ∗−μ~+1+m~2φT−φaσ∗+μ∗m~∗Γσ∗+μ∗ϱψϖ∗−ϖ~∗1Ω∗∗=Ω∗∗ψ∥ϖ∗−ϖ~∗∥1Ω∗∗=ψ∥ϖ∗−ϖ~∗∥.
This givesx1−x2≤ψϖ∗−ϖ~∗ and shows that ℙℍdWFϖ∗,Fϖ~∗≤ψϖ∗−ϖ~∗ for all ϖ∗,ϖ~∗∈W. Also from (C8), we realize that F has approximate endpoint property. Application of Theorem 5 gives that K has a unique endpoint, i.e., there exists ϖ∗∗∈W such that Fϖ∗∗=ϖ∗∗. In conclusion, ϖ∗∗ is a solution to the non-φ-hybrid inclusion BVP (4).


## 4. Some Examples

This section involves two different numerical simulation examples corresponding to the relevantφ-hybrid and non-φ-hybrid fractional inclusion boundary problems to guarantee the applicability of proved theorems.Example 1.
With due attention to (2) and (3), we design the Caputo φ-hybrid differential inclusion BVP as
(73)DC01.62;z+2ϖ∗zzsinϖ∗z/42+1/4∈sinϖ∗zz2+sinz,cosϖ∗z41+cosϖ∗z+35,
supplemented with separated mixedφ-integro-derivative boundary conditions
(74)0.8ϖ∗zzsinϖ∗z/42+1/4z=0=1.2+0.4ϖ∗zzsinϖ∗z/42+1/4z=0,0.8IRL01.4;z+2DC00.4;z+2ϖ∗zzsinϖ∗z/42+1/4z=1=0.9+0.4IRL01.4;z+2DC01;z+2ϖ∗zzsinϖ∗z/42+1/4z=1,where z∈0,1, σ∗=1.62, μ~=0.4, μ∗=1.4, m~1=0.8, m~2=0.4, s1∗=1.2, s2∗=0.9, and φz=z+2. We define N∗:0,1×ℝ⟶ℝ0 by N∗z,ϖ∗z=zsinϖ∗z/42+1/4 which is nonzero and continuous. Notice that F∗=supz∈0,1N~∗z,0=1/4. Moreover, the function N∗ is Lipschitz, that is, for each φ1∗,φ2∗∈ℝ, we have
(75)N∗z,ϖ1∗z−N∗z,ϖ2∗z=zsinϖ1∗z42−zsinϖ2∗z42≤z42ϖ1∗z−ϖ2∗z.
Ifχz=z/42, then χ∗=supz∈0,1χz=1/42. Now, define O~:0,1×ℝ⟶Pℝ by
(76)O~z,ϖ∗z=sinϖ∗zzsinz+2z,cosϖ∗z4cosϖ∗z+1+35.
Since(77)ζ≤maxsinϖ∗zzsinz+2z,cosϖ∗z4cosϖ∗z+1+35≤1,therefore, we have
(78)O~z,ϖ∗z=supκ^:κ^∈O~z,ϖ∗z≤Yz=1.
Then,YL1=∫01Yqdq=∫011·dq=1. By using above values, we have Π∗=20.1356266. Also, we can find ε^ with ε^>9.67254. Finally, we have χ∗Π∗YL1=0.47949<1/2. Thus, all assertions of Theorem 8 are verified. Hence, the φ-hybrid Caputo differential inclusion BVP (73) supplemented with separated mixed φ-integro-derivative boundary conditions (74) has a solution.Example 2.
With due attention to (4), we design the Caputo non-φ-hybrid differential inclusion BVP as
(79)DC01.4;2z+3/2ϖ∗z∈0,2ϖ∗z31/2+z2+ϖ∗z,
supplemented with separated mixedφ-integro-derivative boundary
(80)0.73ϖ∗0=0.7+0.3ϖ∗0,0.73RLI01.6;2z+3/2DC00.73;2z+3/2ϖ∗1=0.6+0.3IRL01.6;2z+3/2DC01;2z+3/2ϖ∗1.where DC01.4;2z+3/2 denotes the φ-CF derivative of order σ∗=1.4, z∈0,1, μ~=0.6, μ∗=1.6, m~1=0.73, m~2=0.3, s1∗=0.7, s2∗=0.6, and φz=2z+3/2. Using these values, we have m~∗=0.15516. We consider the Banach space W=ϖ∗z:ϖ∗z∈C0,1,ℝ equipped with ϖ∗W=supz∈0,1ϖ∗z. Now, we define a multivalued map O~:0,1×W⟶PW by
(81)O~z,ϖ∗z=0,2ϖ∗z361/2+z2+ϖ∗z,for almost all z∈0,1. We define ψ:;0,∞⟶0,∞ by ψz=z/3,∀z>0. Obviously, liminfz⟶∞z−ψz>0 and ψz<z for all z>0. Now, for each ϖ1∗,ϖ2∗∈W, we have
(82)ℙℍdWO~z,ϖ1∗z,O~z,ϖ2∗z≤2361/2+zϖ1∗z−ϖ2∗z≤ψϖ1∗z−ϖ2∗zρzΩ∗∗,
whereΩ∗∗=0.21377 and ϱ∈C0,1,0,∞ is defined as ϱz=2/121/2+z for all z. Then, ϱ=supz∈0,1=1/3. Lastly, we introduce F:W⟶PW by
(83)Fϖ∗=ϑ∈W:thereexistsκ~∈SELO~,ϖ∗s.t.ϑz=ρz,∀z∈0,1,in which
(84)ρz=0.70.43+1Γ1.4∫0zφ′qφz−φq0.4κ^qdq+0.60.15516φz−φ0−0.73φz−φ00.15516Γ2.4∫0Tφ′qφ1−φq1.4κ^qdq+0.3φz−φ00.15516Γ2∫0Tφ′qφ1−φqκ^qdq.
Thus, all assertions of Theorem10 are verified. Hence, the non-φ-hybrid Caputo differential inclusion BVP (79) with separated mixed φ-integro-derivative boundary (80) has a solution.


## 5. Conclusion

In the current research study, we derived some theoretical criteria to prove the existence results to a newφ-hybrid fractional differential inclusion in the Caputo settings depending on the increasing function φ with separated mixed φ-hybrid-integro-derivative boundary conditions. The applied method to achieve desired purposes is based on Dhage’s fixed point result. In addition, we discussed a special case of the proposed φ-inclusion problem in the non-φ-hybrid structure with the help of the endpoint notion. To confirm the applicability of our theoretical findings, two specific numerical examples are provided which simulate both φ-hybrid and non-φ-hybrid cases. Hence, this research work can motivate other researchers in this field to concentrate on various investigations of different φ-hybrid structures formulated by other fractional operators.



---

*Source: 4512223-2021-09-02.xml*