# An Analytical Survey on the Solutions of the Generalized Double-Orderφ-Integrodifferential Equation

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

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## Abstract

We study the existence of solutions for a newly configured model of a double-order integrodifferential equation includingφ-Caputo double-order φ-integral boundary conditions. In this way, we use the Krasnoselskii and Leray-Schauder fixed point results. Also, we invoke the Banach contraction principle to confirm the uniqueness of the existing solutions. Finally, we provide three examples to illustrate our analytical findings.


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## Body

## 1. Introduction

An arbitrary order calculus, specifically the fractional order calculus, has been one of the most important subbranches of mathematics in other existing computational and applied sciences. This amount of applicability is for the sake of the high ability of the relations and operators formulated in the mentioned theory. For this reason, most researchers have utilized numerous applied fractional operators in the past years to model various forms of natural processes that occurred in the world. The applicability and diversity of mentioned fractional operators in modeling can be observed in many literatures including [1–16]. In addition, since by making the model on the basis of fractional operators, we obtain more accurate computational results than existing usual models on the basis of integer order operators; thus, this subject motivates all researchers to construct new generalizations of these fractional operators. In other words, some generalized formulations of these operators have been defined to combine the previous operators effectively to avoid confusion when using existing fractional operators (see [17–20]).In 2017, an extension of the usual Caputo operator entitledφ-Caputo derivative (φ-CapFr) is introduced by Almeida ([21]) in which the kernel of mentioned operator depends on an increasing function φ. The most applied advantage of the φ-CapFr derivative is its flexibility to combine all fractional derivatives introduced before. This extended operator possesses the semigroup property which is vital to obtain the structure of solutions. Hence, φ-CapFr derivative is regarded as a generalized construction of arbitrary order derivatives. By invoking the newly defined φ-CapFr operator and its other generalizations, several limited studies have been done which we refer the reader to those including [22–27].To review several previous research notes implemented in terms ofφ-CapFr operators, we point out a study discussed by Belmor et al. ([28]). In the mentioned article, the authors configured a φ-fractional differential inclusion endowed with φ-integral conditions as
(1)CD0+σ∗;φϖ∗z∈P~z,ϖ∗z,ϖ∗0−ρφ∗ϖ∗0=m1∗RLI0+ζ1;φℏ~∗1c1,ϖ∗c1,ϖ∗k−ρφ∗ϖ∗k=m2∗RLI0+ζ2;φℏ~∗2c2,ϖ∗c2,where z∈0,k, σ∗∈1,2, c1,c2∈0,k, and m1∗,m2∗ are chosen as arbitrary constants and P~:0,k×ℝ→Pℝ introduces a multifunction and ℏ~∗1,ℏ~∗2:0,k×ℝ→ℝ are continuous along with ρφ∗=1/φ′zd/dz. The authors deduced the required criteria for the existence by the aid of an endpoint concept on the category of the ψ-weak contractions ([28]).In the same year, Wahash et al. ([29]) turned to a structure involving the generalized φ-Caputo equation subject to integral conditions as
(2)CD0+σ∗;φϖ∗z=ℏ~∗z,ϖ∗z,ϖ∗0=m∗∫01ξquqdq+ν,respectively.where z∈0,1, σ∗∈0,1, m∗≥0, and ν∈ℝ+, and further, ℏ∗:0,1×ℝ+→ℝ+ illustrates a continuous function along with ξ∈Lℝ+10,1. The upper-lower solution is the method utilized in this work by authors in which the authors invoke a fixed point result on cones. In addition to this, an upper-lower control maps are built with respect to the nonlinear term without a specific monotone condition ([29]).We can find many works in the literature for describing different phenomena or physical meanings of fractional integrodifferential equations. But, some generalizations of fractional notions have no physical meaning at this time and maybe it will find some new physical interpretations for those in the future. Our results satisfy this second version. There are many works which apply different fractional derivatives such as Caputo-Fabrizio (see for example, [4, 30–33]) and discuss different views on applied mathematical modelings (see for example, [34–39]).Inspired by the above-implemented works, we intend to formulate and generalize a new category of the nonlinear fractional double-order integrodifferential equation in the generalized fractionalφ-operators settings given by
(3)m1∗CDs0σ∗;φ+1−m1∗CDs0ρ∗;φϖ∗z=p1∫s0zφ′qφz−φqδ1∗−1Γδ1∗h^∗q,ϖ∗qdq+p2∫s0zφ′qφz−φqδ2∗−1Γδ2∗f^∗q,ϖ∗qdq,furnished with double-order integral boundary conditions in the φ-Riemann-Liouville frame
(4)ϖ∗s0=0,m2∗∫s0Tφ′qφT−φqθ1∗−1Γθ1∗ϖ∗qdq+1−m2∗∫s0Tφ′qφT−φqθ2∗−1Γθ2∗ϖ∗qdq=0,so that z∈s0,T with s0≥0 and T∈ℝ+. Moreover, let σ∗,ρ∗∈1,2 so that σ∗−ρ∗>1 and δ1∗,δ2∗∈0,1, θ1∗,θ2∗>0, m1∗,m2∗∈0,1, and p1,p2∈ℝ+. Both real-valued functions h^∗,f^∗:s0,T×ℝ→ℝ are supposed to be continuous. The notation CDs0η;φ represents the φ-Caputo derivative (φ-CapFr derivative) of arbitrary order η∈σ∗,ρ∗, and RLIs0ρ;φ illustrates the φ-Riemann-Liouville integral (φ-RLFr integral) of arbitrary order ρ∈δ1∗,δ2∗,θ1∗,θ2∗.It is necessary that all researchers pay attention to this subject that the proposed double-order double-φ-CapFr-integrodifferential equation has a novel and unique structure. In other words, the formulated structure for the given fractional double-φ-CapFr-integrodifferential problem (3)-(4) includes two φ-CapFr-derivatives and also four φ-RLFr-integrals with different orders. This combined boundary value problem (BoVaPr) covers many different special cases of various nonlinear integrodifferential equations. Therefore, we emphasize that this kind of the nonlinear double-φ-CapFr-integro-differential BoVaPr has not been investigated in any literature so far. In this direction, we apply well-known analytical techniques to derive desired criteria which guarantee the existence aspects of desired solutions for the proposed double-order φ-CapFr-integrodifferential BoVaPr (3)-(4).The organization of the contents of the current manuscript is as follows. In the next section, some required notions in the context of the generalizedφ-calculus are assembled. Section 3 is devoted to establish the main theorems in which the existence criteria can be obtained under some required conditions. In Section 4, we present three simulative examples to confirm the validity of our analytical findings.


## 2. Fundamental Preliminaries

In the current section, we collect and review some fundamental and auxiliary notions in the framework of our analytical methods applied in this paper. As you know, the concept of the Riemann-Liouville integral of orderσ∗>0 for a function ϖ∗:0,+∞→ℝ is defined as
(5)RLI0σ∗ϖ∗z=∫0zz−qσ∗−1Γσ∗ϖ∗qdq,provided that the value of the integral is finite [40, 41]. In this position, let us assume that σ∗∈n−1,n so that n=1+σ∗. For a continuous function ϖ∗:0,+∞→ℝ, the Riemann-Liouville derivative of order ρ∗ is given by
(6)RLD0σ∗ϖ∗z=ddzn∫0zz−qn−σ∗−1Γn−σ∗ϖ∗qdq,provided that the value of the integral is finite [40, 41]. In the next step, for an absolutely continuous function ϖ∗∈ACℝn0,+∞, the fractional derivative of Caputo type is given by
(7)CD0σ∗ϖ∗z=∫0zz−qn−σ∗−1Γn−σ∗ϖ∗nqdq,provided that the integral is finite-valued [40, 41].Definition 1 ([41–43]).
Letφ∈Cns0,b,ℝ be an increasing function so that φ′z>0 for any z∈s0,b. Then, the φ-Riemann-Liouville integral of order σ∗ for an integrable function ϖ∗:s0,b→ℝ with respect to another increasing function φ is defined as
(8)RLIs0σ∗;φϖ∗z=1Γσ∗∫s0zφ′qφz−φqσ∗−1ϖ∗qdq,provided that the R.H.S. of above equality is finite-valued.It is to be noted that ifφz=z and s0=0, then clearly the φ-RLFr integral (8) reduces to the standard Riemann-Liouville integral (5).Definition 2 ([41–43]).
Forφ∈Cns0,b,ℝ as above and for a continuous function ϖ∗:s0,+∞→ℝ, the φ-Riemann-Liouville derivative of order σ∗ is given by
(9)RLDs0σ∗;φϖ∗z=1Γn−σ∗1φ′zddzn∫s0zφ′qφz−φqn−σ∗−1ϖ∗qdq,provided that the R.H.S. of the above equality is finite-valued and n=1+σ∗.In a manner, ifφz=z and s0=0, then it is obvious that the φ-RLFr derivative (9) reduces to the standard Riemann-Liouville derivative (6). Inspired by these operators, Almeida presented a new φ-version of the Caputo derivative as the following formulation.Definition 3 ([21]).
Forϖ∗∈Cns0,b, φ-version of the Caputo derivative is illustrated as
(10)CDs0σ∗;φϖ∗z=1Γn−σ∗∫s0zφ′qφz−φqn−σ∗−11φ′qddqnϖ∗qdq,provided that the R.H.S. of the above equality is finite-valued and n=1+σ∗.It is notable that ifφz=z, then it is obvious that the φ-CapFr derivative (10) reduces to the standard Caputo derivative (7). In the following, some important properties of the φ-Caputo and φ-Riemann-Liouville fractional operators can be seen in the following lemmas.Lemma 4 ([21, 41–43]).
Assume thatσ∗>0, ρ∗>0, and β∗>0, and φ∈Cns0,b,ℝ is an increasing function so that φ′z>0 for any z∈s0,b. Then, the following statements hold:
(i1)
RLIs0σ∗;φRLIs0ρ∗;φϖ∗z=RLIs0σ∗+ρ∗;φϖ∗z(i2)
RLIs0σ∗;φφz−φs0β∗y=Γβ∗+1/Γσ∗+β∗+1φy−φs0σ∗+β∗(i3)
CDs0σ∗;φφz−φs0β∗y=Γβ∗+1/Γβ∗−σ∗+1φy−φs0β∗−σ∗,β∗>−1(i4)
RLDs0σ∗;φRLIs0ρ∗;φϖ∗z=RLIs0ρ∗−σ∗;φϖ∗z,σ∗<ρ∗For instance, in Figures1 and 2, we plot the graphs of the φ-CapFr derivative of two arbitrary functions ϖz=z6 and ϖz=z5/32 for φz=z2+1 and φz=z/2, respectively.Lemma 5 ([21]).
Letn−1<σ∗<n and φ∈Cns0,b,ℝ be an increasing function so that φ′z>0 for any z∈a,b. Then, for every ϖ∗∈Cn−1s0,b,ℝ, we have
(11)RLIs0σ∗;φCDs0σ∗;φϖ∗z=ϖ∗z−∑j=0n−1ρφjϖ∗s0j!φz−φs0j,ρφ=1φ′zddz.Figure 1
The CapFr-derivative ofϖz=z6 for φz=z2+1.Figure 2
The CapFr-derivative ofϖz=z5/32 for φz=z/2.In view of the above lemma, it is verified that the general solution of the homogeneous equationCDs0σ∗;φϖ∗z=0 is given by
(12)ϖ∗z=∑j=0n−1cj∗φz−φs0j=c0∗+c1∗φz−φs0+c2∗φz−φs02+⋯+cn−1∗φz−φs0n−1,where n−1<σ∗<n and c0∗,c1∗,⋯,cn−1∗∈ℝ ([21]). Both next theorems required analytical tools to derive the desired criteria in the direction of our goals. The first theorem is attributed to Krasnosel’skii˘ and the second one is due to Leray-Schauder.Theorem 6 [44].
The setB∗ is assumed to be a nonempty, convex, bounded, and closed subset of a given Banach space E. Let h^∗,f^∗:B∗→E be so that h^∗ω+f^∗ν∈B∗ for any ω,ν∈B∗, where h^∗ is regarded as a compact and continuous mapping and f^∗ a contraction. Then, an element υ∈B∗ exists so that υ=h^∗υ+f^∗υ.Theorem 7 [45].
Assume thatB∗ is a subset of a Banach space E with convexity and closeness property and G is an open subset of B∗ such that 0∈G. Then, a compact continuous function A^∗:G¯→B∗ possesses a fixed point in G¯ or there is a g∈∂G and 0<σ<1 such that g=σA^∗g, where ∂G stands for the boundary of GinB∗.


## 3. Main Analytical Results

Suppose thatE=Cs0,T,ℝ stands for the collection of all functions given on s0,T with continuity property. In this case, one can simply verify that E is a Banach space together with
(13)ϖ∗=supz∈s0,Tϖ∗z,ϖ∗∈E.We start with the next lemma which will be required in the subsequent sections.Lemma 8.
Takeh∗∈Cs0,T,ℝ and
(14)Y≔m2∗φT−φs01+θ1∗Γ2+θ1∗+1−m2∗φT−φs01+θ2∗Γ2+θ2∗≠0.Then,ϖ∗ satisfies the nonlinear double-order differential boundary problem of the fractional type
(15)m1∗CDs0σ∗;φ+1−m1∗CDs0ρ∗;φϖ∗z=h∗z,ϖ∗s0=0,m2∗RLIs0θ1∗;φϖ∗T+1−m2∗RLIs0θ2∗;φϖ∗T=0,iff ϖ∗ satisfies the fractional nonlinear double-order integral equation
(16)ϖ∗z=m1∗−1m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1ϖ∗qdq+1m1∗Γσ∗∫s0zφ′qφz−φqσ∗−1h∗qdq+φz−φs0Y−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ1∗−1ϖ∗qdq−m2∗m1∗Γσ∗+θ1∗∫s0Tφ′qφT−φqσ∗+θ1∗−1h∗qdq−1−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ2∗−1ϖ∗qdq−1−m2∗m1∗Γσ∗+θ2∗∫s0Tφ′qφT−φqσ∗+θ2∗−1h∗qdq.Proof.
To start the proof, assume thatϖ∗ satisfies the nonlinear double-order differential equation (15). We have
(17)CDs0σ∗;φϖ∗z=m1∗−1m1∗CDs0ρ∗;φϖ∗z+1m1∗h∗z.
By fractional integrating in the Riemann-Liouville setting of orderσ∗, we get
(18)ϖ∗z=m1∗−1m1∗RLIs0σ∗;φCDs0ρ∗;φϖ∗z+1m1∗RLIs0σ∗;φh∗z+c0∗+c1∗φz−φs0=m1∗−1m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1ϖ∗qdq+1m1∗Γσ∗∫s0zφ′qφz−φqσ∗−1h∗qdq+c0∗+c1∗φz−φs0,where c0∗,c1∗∈ℝ are arbitrary constants. By using the first boundary condition, we get c0∗=0 and so
(19)ϖ∗z=m1∗−1m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1ϖ∗qdq+1m1∗Γσ∗∫s0zφ′qφz−φqσ∗−1h∗qdq+c1φz−φs0.
On the other hand, if we takeη∗∈θ1∗,θ2∗, then we have
(20)RLIs0η∗;φϖ∗z=m1∗−1m1∗Γσ∗−ρ∗+η∗∫s0zφ′qφz−φqσ∗−ρ∗+η∗−1ϖ∗qdq+1m1∗Γσ∗+η∗∫s0zφ′qφz−φqσ∗+η∗−1h∗qdq+c11Γ2+η∗φz−φs0η∗+1.
Now, by utilizing the second condition, we have(21)0=m2∗m1∗−1m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φs0σ∗−ρ∗+θ1∗−1ϖ∗qdq+m2∗m1∗Γσ∗+θ1∗∫s0Tφ′qφT−φqσ∗+θ1∗−1h∗qdq+c1m2∗Γ2+θ1∗φT−φs01+θ1∗+1−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φs0σ∗−ρ∗+θ2∗−1ϖ∗qdq+1−m2∗m1∗Γσ∗+θ2∗∫s0Tφ′qφT−φs0σ∗+θ2∗−1h∗qdq+c11−m2∗Γ2+θ2∗φT−φs01+θ2∗.
By solving the above equation with respect toc1, we get
(22)c1∗=1Y−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ1∗−1ϖ∗qdq−m2∗m1∗Γσ∗+θ1∗∫s0Tφ′qφT−φqσ∗+θ1∗−1h∗qdq−1−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ2∗−1ϖ∗qdq−1−m2∗m1∗Γσ∗+θ2∗∫s0Tφ′qφT−φqσ∗+θ2∗−1h∗qdq,where Y is illustrated in (14). Inserting the obtained value for c1∗ into (19), we obtain the double-order integral equation illustrated by (16). On the other hand, clearly, ϖ∗ satisfies the nonlinear fractional double-order differential equation (15) whenever it satisfies the double-order integral equation (16) and this ends the proof.Keeping in view of the nonlinear double-integrodifferential problem (3) and (4) and by Lemma 8, we introduce a single-valued operator Q:E→E as
(23)Qϖ∗z=m1∗−1m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1ϖ∗qdq+p1m1∗Γσ∗+δ1∗∫s0zφ′qφz−φqσ∗+δ1∗−1h^∗q,ϖ∗qdq+p2m1∗Γσ∗+δ2∗∫s0zφ′qφz−φqσ∗+δ2∗−1f^∗q,ϖ∗qdq+φz−φs0Y−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ1∗−1ϖ∗qdq−p1m2∗m1∗Γσ∗+θ1∗+δ1∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ1∗−1h^∗q,ϖ∗qdq−p2m2∗m1∗Γσ∗+θ1∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ2∗−1f^∗q,ϖ∗qdq−1−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ2∗−1ϖ∗qdq−p11−m2∗m1∗Γσ∗+θ2∗+δ1∗∫s0Tφ′qφT−φqσ∗+δ1∗+θ2∗−1h^∗q,ϖ∗qdqp21−m2∗m1∗Γσ∗+θ2∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ2∗+δ2∗−1f^∗q,ϖ∗qdq,where ϖ∗∈E and z∈s0,T. From now on, we set
(24)Λ0=φT−φs0σ∗−ρ∗m1∗−1m1∗Γσ∗−ρ∗+1+φT−φs0σ∗−ρ∗+θ1∗+1m2∗m1∗−1Ym1∗Γσ∗−ρ∗+θ1∗+1+φT−φs0σ∗−ρ∗+θ2∗+11−m2∗m1∗−1Ym1∗Γσ∗−ρ∗+θ2∗+1,Λ1=φT−φs0σ∗+δ1∗p1m1∗Γσ∗+δ1∗+1+φT−φs0σ∗+θ1∗+δ1∗+1p1m2∗Ym1∗Γσ∗+θ1∗+δ1∗+1+φT−φs0σ∗+θ2∗+δ1∗+1p11−m2∗Ym1∗Γσ∗+θ2∗+δ1∗+1,Λ2=φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗∣Y∣m1∗Γσ∗+θ1∗+δ2∗+1+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗Ym1∗Γσ∗+θ2∗+δ2∗+1.We now derive the first criterion to confirm the existence of solutions for the proposed double-order integrodifferential problem (3)–(4). This purpose is achieved by utilizing Theorem 6 attributed to Krasnosel’skii˘.Theorem 9.
Both functionsh^∗,f^∗:s0,T×ℝ→ℝ are supposed to be continuous having the following assertions:
(H1) There is a constant N∗>0 so that we have the following inequality for any ϖ1∗,ϖ2∗∈E,
(25)h^∗z,ϖ1∗z−h^∗z,ϖ2∗z<N∗ϖ1∗z−ϖ2∗z,z∈s0,T
(H2) For any ϖ∗∈E, a continuous function μ∗ exists on s0,T such that
(26)f^∗z,ϖ∗z<μ∗z,z∈s0,T.Then, by assumingΛ0+N∗Λ1<1, at least one solution exists on s0,T for the nonlinear double-integrodifferential equation (3) furnished with double-order integral conditions (4), where Λ0 and Λ1 are illustrated by (24).Proof.
To start the proof, letμ∗=supz∈s0,Tμ∗z. Define Bε∗≔ϖ∗∈E:ϖ∗≤ε with
(27)ε≥μ∗Λ2+K∗Λ11−Λ0+N∗Λ1,so that K∗≔supz∈s0,Th^∗z,0 and Λ0 and Λ1 are given by (24). Obviously, nonempty ball Bε∗ is a convex, bounded, and closed set contained in the Banach space E. Besides, let us assume the Q:E→E is as in (23). It is an evident fact that all fixed points of Q will be all solutions of the proposed double-order double-integrodifferential BoVaPr (3)–(4) according to Lemma 8. Now, for each z∈s0,T, we split Q into Q1,Q2:Bε∗→E by
(28)Q1ϖ∗z=m1∗−1m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1ϖ∗qdq+p1m1∗Γσ∗+δ1∗∫s0zφ′qφz−φqσ∗+δ1∗−1h^∗q,ϖ∗qdq+φz−φs0Y−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ1∗−1ϖ∗qdq−p1m2∗m1∗Γσ∗+θ1∗+δ1∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ1∗−1h^∗q,ϖ∗qdq−1−m2∗m1∗−1m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ2∗−1ϖ∗qdq−p11−m2∗m1∗Γσ∗+θ2∗+δ1∗∫s0Tφ′qφT−φqσ∗+δ1∗+θ2∗−1h^∗q,ϖ∗qdq,Q2ϖ∗z=p2m1∗Γσ∗+δ2∗∫s0zφ′qφz−φqσ∗+δ2∗−1f^∗q,ϖ∗qdq+φz−φs0Y×−p2m2∗m1∗Γσ∗+θ1∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ2∗−1f^∗q,ϖ∗qdq−p21−m2∗m1∗Γσ∗+θ2∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ2∗+δ2∗−1f^∗q,ϖ∗qdq.
By the conditionH1, we have
(29)h^∗z,ϖ∗z≤h^∗z,ϖ∗z−h^∗z,0+h^∗z,0≤N∗ϖ∗+K∗≤N∗ε+K∗for each ϖ∗∈E and z∈s0,T. In consequence, for both ϖ1∗,ϖ2∗∈Bε∗ and z∈s0,T and in view of (24) and (27), we get
(30)∣Q1ϖ1∗z+Q2ϖ2∗z∣≤φT−φs0σ∗−ρ∗∣m1∗−1∣m1∗Γσ∗−ρ∗+1∥ϖ∗∥+φT−φs0σ∗+δ1∗p1m1∗Γσ∗+δ1∗+1N∗∥ϖ∗∥+K∗+φT−φs0σ∗−ρ∗+θ1∗+1m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ1∗+1∥ϖ∗∥+φT−φs0σ∗+θ1∗+δ1∗+1p1m2∗∣Y∣m1∗Γσ∗+θ1∗+δ1∗+1N∗∥ϖ∗∥+K∗+φT−φs0σ∗−ρ∗+θ2∗+11−m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ2∗+1∥ϖ∗∥+φT−φs0σ∗+θ2∗+δ1∗+1p11−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ1∗+1N∗∥ϖ∗∥+K∗+φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1∥μ∗∥+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗∣Y∣m1∗Γσ∗+θ1∗+δ2∗+1∥μ∗∥+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ2∗+1∥μ∗∥=φT−φs0σ∗−ρ∗∣m1∗−1∣m1∗Γσ∗−ρ∗+1+φT−φs0σ∗−ρ∗+θ1∗+1m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ1∗+1+φT−φs0σ∗−ρ∗+θ2∗+11−m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ2∗+1∥ϖ∗∥+φT−φs0σ∗+δ1∗p1m1∗Γσ∗+δ1∗+1+φT−φs0σ∗+θ1∗+δ1∗+1p1m2∗∣Y∣m1∗Γσ∗+θ1∗+δ1∗+1+φT−φs0σ∗+θ2∗+δ1∗+1p11−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ1∗+1N∗∥ϖ∗∥+K∗+φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗∣Y∣m1∗Γσ∗+θ1∗+δ2∗+1+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ2∗+1∥μ∗∥=Λ0∥ϖ∗∥+Λ1N∗∥ϖ∗∥+K∗+Λ2∥μ∗∥≤Λ0+N∗Λ1ε+Λ1K∗+Λ2∥μ∗∥≤ε,which implies ∥Q1ϖ1∗+Q2ϖ2∗∥≤ε and so Q1ϖ1∗z+Q2ϖ2∗z∈Bε∗ for all ϖ1∗,ϖ2∗∈Bε. Now, we prove that Q2 is continuous. A sequence ϖn∗ in Bε∗ is assumed to be convergent provided that ϖn∗→ϖ∗. Then, for any element z∈s0,T, one may write
(31)Q2ϖn∗z−Q2ϖ∗z≤φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1f^∗q,ϖn∗q−f^∗q,ϖ∗q+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗Ym1∗Γσ∗+θ1∗+δ2∗+1f^∗q,ϖn∗q−f^∗q,ϖ∗q+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗Ym1∗Γσ∗+θ2∗+δ2∗+1f^∗q,ϖn∗q−f^∗q,ϖ∗q.
Sincef^∗ is continuous, we get Q2ϖn∗−Q2ϖ∗→0 as ϖn∗→ϖ∗. From this, we realize that Q2 is continuous on Bε∗. In the sequel, to verify that Q2 is compact, we have to prove that Q2 has the uniform boundedness property. For every ϖ∗∈Bε∗ and all z∈s0,T, the following estimate for Q2 holds
(32)∣Q2ϖ∗z∣=∣p2m1∗Γσ∗+δ2∗∫s0zφ′qφz−φqσ∗+δ2∗−1f^∗q,ϖ∗qdq+φz−φs0Y−p2m2∗m1∗Γσ∗+θ1∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ2∗−1f^∗q,ϖ∗qdq−p21−m2∗m1∗Γσ∗+θ2∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ2∗+δ2∗−1f^∗q,ϖ∗qdq∣≤φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1∣f^∗q,ϖ∗q∣+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗∣Y∣m1∗Γσ∗+θ1∗+δ2∗+1∣f^∗q,ϖ∗q∣+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ2∗+1∣f^∗q,ϖ∗q∣≤∥μ∗∥φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗∣Y∣m1∗Γσ∗+θ1∗+δ2∗+1+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ2∗+1=∥μ∗∥Λ2,which implies that ∥Q2ϖ∗∥≤∥μ∗∥Λ2. In order to confirm the equicontinuity of Q2, suppose that z1,z2∈s0,T such that z2>z1. We investigate that Q2 corresponds bounded sets to equicontinuous sets. To guarantee this claim, for any ϖ∗z∈Bε∗, we obtain
(33)Q2ϖ∗z2−Q2ϖ∗z1≤p2∥μ∗∥m1∗Γσ∗+δ2∗+1φz1−φs0σ∗+δ2∗−φz2−φs0σ∗+δ2∗+∥μ∗∥p2φz2−φz1∣Y∣m2∗φT−φs0σ∗+θ1∗+δ2∗m1∗Γσ∗+θ1∗+δ2∗+1+1−m2∗φT−φs0σ∗+θ2∗+δ2∗m1∗Γσ∗+θ2∗+δ2∗+1.
It is to be notice that the R.H.S of (33) is not dependent to ϖ∗∈Bε∗ and also goes to 0 by assuming z1→z2. This describes that Q2 is equicontinuous. Hence, Q2 has the relative compactness property on Bε∗, and thus, application of the Arzelá-Ascoli result gives the complete continuity of Q2, and eventually on Bε∗, we reach the compactness of Q2. At last, we intend to conclude that Q1 is a contraction. For any ϖ1∗,ϖ2∗∈Bε∗, and z∈s0,T, we get
(34)∣Q1ϖ1∗z−Q1ϖ2∗z∣≤φT−φs0σ∗−ρ∗∣m1∗−1∣m1∗Γσ∗−ρ∗+1∣ϖ1∗z−ϖ2∗z∣+φT−φs0σ∗+δ1∗+1p1m1∗Γσ∗+δ1∗+1N∗∣ϖ1∗z−ϖ2∗z∣+φT−φs0σ∗−ρ∗+θ1∗+1m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ1∗+1∣ϖ1∗z−ϖ2∗z∣+φT−φs0σ∗+θ1∗+δ1∗+1p1m2∗∣Y∣m1∗Γσ∗+θ1∗+δ1∗+1N∗∣ϖ1∗z−ϖ2∗z∣+φT−φs0σ∗−ρ∗+θ2∗+11−m2∗m1∗−1∣Y∣m1∗Γσ∗−ρ∗+θ2∗+1∣ϖ1∗z−ϖ2∗z∣+φT−φs0σ∗+θ2∗+δ1∗+1p11−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ1∗+1N∗∣ϖ1∗z−ϖ2∗z∣≤Λ0+N∗Λ1∥ϖ1∗−ϖ2∗∥.
Hence,(35)∥Q1ϖ1∗−Q1ϖ2∗∥≤Λ0+N∗Λ1∥ϖ1∗−ϖ2∗∥.
SinceΛ0+N∗Λ1<1, Q1 is a contraction. Thus, all the conditions of Theorem 6 are verified. Thus, the double-φ-integrodifferential-BoVaPr (3) with double-order φ-RLFr integral boundary conditions (4) possesses at least one solution in s0,T and the proof is completed.In the next result, we prove the existence of a solution for the double-order double-φ-integrodifferential-BoVaPr (3)–(4) with the help of the Leray-Schauder nonlinear alternative.Theorem 10.
Considerh^∗,f^∗:s0,T×ℝ→ℝ as continuous mappings having the following assertions:
(C3) There exist continuouctions η1,η2:s0,T→ℝ+ and functions γ1,γ2∈Cs0,T,ℝ+ such that ∣h^∗z,ϖ∗∣≤γ1zη1∣ϖ∗∣ and ∣f^∗z,ϖ∗∣≤γ2zη2∣ϖ∗∣ for each z,ϖ∗∈s0,T×ℝ
(C4) There exists a constant Ω∗>0 such that Λ0<1 and
(36)1−Λ0Ω∗Λ1∥γ1∥η1Ω∗+Λ2∥γ2∥η2Ω∗>1,where Λ0,Λ1,Λ2 are given by (24).Then, the nonlinear double-φ-CapFr-integrodifferential equation (3) along with double-order φ-RLFr integral boundary conditions (4) possesses at least one solution in s0,T.Proof.
We utilize the Leray-Schauder nonlinear alternative to prove the current result. First, we prove thatQ formulated by (23) corresponds all bounded sets to all bounded sets in E. Corresponding to ε>0, build a bounded ball Bε∗=ϖ∗∈E:∥ϖ∗∥<ε in E. Then, for z∈s0,T, we have
(37)∣Qϖ∗z∣≤∣m1∗−1∣m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1∥ϖ∗∥dq+p1m1∗Γσ∗+δ1∗∫s0zφ′qφz−φqσ∗+δ1∗−1∥γ1∥η1∥ϖ∗∥dq+p2m1∗Γσ∗+δ2∗∫s0zφ′qφz−φqσ∗+δ2∗−1∥γ2∥η2ϖ∗dq+φz−φs0∣Y∣m2∗∣m1∗−1∣m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ1∗−1∥ϖ∗∥dq+p1m2∗m1∗Γσ∗+θ1∗+δ1∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ1∗−1∥γ1∥η1∥ϖ∗∥dq+p2m2∗m1∗Γσ∗+θ1∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ2∗−1∥γ2∥η2∥ϖ∗∥dq+1−m2∗∣m1∗−1∣m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ2∗−1∥ϖ∗∥dq+p11−m2∗m1∗Γσ∗+θ2∗+δ1∗∫s0Tφ′qφT−φqσ∗+δ1∗+θ2∗−1∥γ1∥η1∥ϖ∗∥dq+p21−m2∗m1∗Γσ∗+θ2∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ2∗+δ2∗−1∥γ2∥η2∥ϖ∗∥dq≤Λ0∥ϖ∗∥+Λ1∥γ1∥η1∥ϖ∗∥+Λ2∥γ2∥η2∥ϖ∗∥.
Consequently, we have(38)∥Qϖ∗∥≤Λ0∥ϖ∗∥+Λ1∥γ1∥η1∥ϖ∗∥+Λ2∥γ2∥η2∥ϖ∗∥.
The obtained inequality indicates thatQ is uniformly bounded. Now, let z1,z2∈s0,T with z1<z2 and ϖ∗∈Bε∗. Then, we have
(39)∣Qϖ∗z2−Qϖ∗z1∣≤ε∣m1∗−1∣m1∗Γσ∗−ρ∗+1φz1−φs0σ∗−ρ∗−φz2−φs0σ∗−ρ∗+∥γ1∥η1εp1m1∗Γσ∗+δ1∗+1φz1−φs0σ∗+δ1∗−φz2−φs0σ∗+δ1∗+∥γ2∥η2εp2m1∗Γσ∗+δ2∗+1φz1−φs0σ∗+δ2∗−φz2−φs0σ∗+δ2∗+φz2−φz1∣Y∣φT−φqσ∗−ρ∗+θ1∗εm2∗∣m1∗−1∣m1∗Γσ∗−ρ∗+θ1∗+1+∥γ1∥η1εφT−φqσ∗+θ1∗+δ1∗p1m2∗m1∗Γσ∗+θ1∗+δ1∗+1+∥γ2∥η2εφT−φqσ∗+θ1∗+δ2∗p2m2∗m1∗Γσ∗+θ1∗+δ2∗+1+εφT−φqσ∗−ρ∗+θ2∗1−m2∗∣m1∗−1∣m1∗Γσ∗−ρ∗+θ2∗+1+∥γ1∥η1εφT−φqσ∗+θ2∗+δ1∗p11−m2∗m1∗Γσ∗+θ2∗+δ1∗+1+∥γ2∥η2εφT−φqσ∗+θ2∗+δ2∗p21−m2∗m1∗Γσ∗+θ2∗+δ2∗+1.
This illustrates that∣Qϖ∗z2−Qϖ∗z1∣→0 when z1→z2 independent of ϖ∗∈Bε∗. Thus, the well-known result attributed to Arzelá-Ascoli confirms that Q:E→E is completely continuous. In the final stage, we intend to check that the collection of all solutions for the equation ϖ∗z=ϑQϖ∗z is bounded for ϑ∈0,1. For that, let ϖ∗ be a solution of ϖ∗z=ϑQϖ∗z for ϑ∈0,1. Then, for z∈s0,T, using the arguments given in the first step, we get that
(40)∥ϖ∗∥≤Λ0∥ϖ∗∥+Λ1∥γ1∥η1∥ϖ∗∥+Λ2∥γ2∥η2∥ϖ∗∥.
Consequently, we have(41)1−Λ0∥ϖ∗∥Λ1∥γ1∥η1∥ϖ∗∥+Λ2∥γ2∥η2∥ϖ∗∥≤1.
From condition (C4), there is a number Ω∗>0 so that ∥ϖ∗∥≠Ω∗. We construct a set
(42)∐=ϖ∗∈E:∥ϖ∗∥<Ω∗,and notice that Q:∐→E is continuous and completely continuous. For such set ∐, there is not an element ϖ∗∈∂∐ for which ϖ∗=ϑQϖ∗ holds for some 0<ϑ<1. Hence, in virtue of Theorem 7, it is concluded that the operator Q possesses a fixed point in ∐ meaning that a solution exists on s0,T for the nonlinear double-φ-CapFr-integrodifferential equation (3) along with double-order φ-RLFr integral boundary conditions (4), and in this case, the proof is ended.In our conclusive outcome, the uniqueness aspect of the obtained solutions for the proposed double-φ-CapFr-integrodifferential BoVaPr (3)-(4) is checked by the aid of the standard contraction result attributed to Banach.Theorem 11.
Assume that a functionh^∗:s0,T×ℝ→ℝ fulfills the presumption H1. Additionally, presume that f^∗:s0,T×ℝ→ℝ is Lipschitz, i.e., M∗ exists so that for any ϖ1∗,ϖ2∗∈ℝ,
(43)∣f^∗z,ϖ1∗−f^∗z,ϖ2∗∣≤M∗ϖ1∗−ϖ2∗.
Then, the nonlinear double-φ-CapFr-integrodifferential BoVaPr (3) with double-order φ-RLFr-integral conditions (4) possesses a unique solution on s0,T, provided that Λ0+N∗Λ1+M∗Λ2<1, where Λ0,Λ1,Λ2 are illustrated by (24).Proof.
The argument is straightforward. In other words, by lettingsupz∈s0,Th^∗z,0<A∗<∞ and supz∈s0,Tf^∗z,0<S∗<∞, we select ε>0 with
(44)ε≥A∗Λ2+S∗Λ11−Λ0+N∗Λ1+M∗Λ2.
We claim that the inclusionQBε∗⊆Bε∗ is valid in which Bε∗=ϖ∗∈E:ϖ∗≤ε. Simply, for any ϖ∗∈Bε∗, from the previous arguments done in the proof of Theorem 9, we get
(45)Qϖ∗≤Λ0+N∗Λ1+M∗Λ2ε+A∗Λ2+S∗Λ1,which illustrates that QBε∗⊆Bε∗. For any z∈s0,T and any ϖ1∗,ϖ2∗∈E, we have
(46)∣Qϖ1∗z−Qϖ2∗z∣≤∣m1∗−1∣m1∗Γσ∗−ρ∗∫s0zφ′qφz−φqσ∗−ρ∗−1∣ϖ1∗q−ϖ2∗q∣dq+p1m1∗Γσ∗+δ1∗∫s0zφ′qφz−φqσ∗+δ1∗−1∣h^∗q,ϖ1∗q−h^∗q,ϖ2∗q∣dq+p2m1∗Γσ∗+δ2∗∫s0zφ′qφz−φqσ∗+δ2∗−1∣f^∗q,ϖ1∗q−f^∗q,ϖ2∗q∣dq+φT−φs0∣Y∣m2∗∣m1∗−1∣m1∗Γσ∗−ρ∗+θ1∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ1∗−1∣ϖ1∗q−ϖ2∗q∣dq+p1m2∗m1∗Γσ∗+θ1∗+δ1∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ1∗−1∣h^∗q,ϖ1∗q−h^∗q,ϖ2∗q∣dq+p2m2∗m1∗Γσ∗+θ1∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ1∗+δ2∗−1∣f^∗q,ϖ1∗q−f^∗q,ϖ2∗q∣dq+1−m2∗∣m1∗−1∣m1∗Γσ∗−ρ∗+θ2∗∫s0Tφ′qφT−φqσ∗−ρ∗+θ2∗−1∣ϖ1∗q−ϖ2∗q∣dq+p11−m2∗m1∗Γσ∗+θ2∗+δ1∗∫s0Tφ′qφT−φqσ∗+δ1∗+θ2∗−1∣h^∗q,ϖ1∗q−h^∗q,ϖ2∗q∣dq+p21−m2∗m1∗Γσ∗+θ2∗+δ2∗∫s0Tφ′qφT−φqσ∗+θ2∗+δ2∗−1∣f^∗q,ϖ1∗q−f^∗q,ϖ2∗q∣dq≤φT−φs0σ∗−ρ∗∣m1∗−1∣m1∗Γσ∗−ρ∗+1∥ϖ1∗−ϖ2∗∥+φT−φs0σ∗+δ1∗p1m1∗Γσ∗+δ1∗+1N∗∥ϖ1∗−ϖ2∗∥+φT−φs0σ∗+δ2∗p2m1∗Γσ∗+δ2∗+1M∗∥ϖ1∗−ϖ2∗∥+φT−φs0σ∗−ρ∗+θ1∗+1m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ1∗+1∥ϖ1∗−ϖ2∗∥+φT−φs0σ∗+θ1∗+δ2∗+1p2m2∗∣Y∣m1∗Γσ∗+θ1∗+δ2∗+1M∗∥ϖ1∗−ϖ2∗∥+φT−φs0σ∗+θ1∗+δ1∗+1p1m2∗∣Y∣m1∗Γσ∗+θ1∗+δ1∗+1N∗∥ϖ1∗−ϖ2∗∥+φT−φs0σ∗−ρ∗+θ2∗+11−m2∗∣m1∗−1∣∣Y∣m1∗Γσ∗−ρ∗+θ2∗+1ϖ1∗−ϖ2∗+φT−φs0σ∗+θ2∗+δ1∗+1p11−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ1∗+1N∗ϖ1∗−ϖ2∗+φT−φs0σ∗+θ2∗+δ2∗+1p21−m2∗∣Y∣m1∗Γσ∗+θ2∗+δ2∗+1M∗∥ϖ1∗−ϖ2∗∥≤Λ0+N∗Λ1+M∗Λ2∥ϖ1∗−ϖ2∗∥.
Therefore,(47)Qϖ1∗−Qϖ2∗≤Λ0+N∗Λ1+M∗Λ2ϖ1∗−ϖ2∗.
AsΛ0+N∗Λ1+M∗Λ2<1, thus Q is a contraction. Therefore, in virtue of the standard contraction result attributed to Banach, a fixed point exists uniquely for Q. In consequence, the nonlinear double-φ-CapFr-integrodifferential BoVaPr (3) along with double-order φ-RLFr-integral conditions (4) possesses a unique solution on s0,T.


## 4. Illustrative Examples

We here design three numerical examples which simulate the structure of the proposed nonlinear double-φ-CapFr-integrodifferential BoVaPr (3) along with double-order φ-RLFr-integral conditions (4).Example 12.
As regards the nonlinear double-φ-CapFr-integrodifferential BoVaPr
(48)0.8CD01.5;z+1+1−0.8CD01.1;z+1ϖ∗z=0.3RLI00.5;z+1sinz1+z2+7z2ϖ∗z100+0.4RLI00.6;z+11z+341+ϖ∗z,furnished with double-order φ-RLFr-integral conditions
(49)ϖ∗0=0,0.7RLI01;z+1ϖ∗1+1−0.7RLI02;z+1ϖ∗1=0.Here, we have taken valuesm1∗=0.8, m2∗=0.7, σ∗=1.5, φz=z+1, ρ∗=1.1, p1=0.3, δ1∗=0.5, p2=0.4, δ2∗=0.6, θ1∗=1, θ2∗=2, z∈0,1, and h^∗,f^∗:0,1×ℝ→ℝ by rules
(50)h^∗z,ϖ∗=sinz1+z2+7z2ϖ∗z100,f^∗z,ϖ∗=1z+341+ϖ∗z.Now for eachϖ1∗,ϖ2∗∈ℝ, clearly, h^∗z,ϖ1∗−h^∗z,ϖ2∗≤N∗ϖ1∗−ϖ2∗ with N∗=0.07>0. On the other side, we obtain a continuous function μ∗z=1/z+34 such that f^∗z,ϖ∗≤μ∗z for all ϖ∗∈ℝ. Also, we have μ∗=supz∈0,1μ∗z≃0.012. Moreover, Y≃0.40 and Λ0+N∗Λ1≃0.61527<1. Obviously, all the presumptions of Theorem 9 are verified. Therefore, the nonlinear double-φ-CapFr-integrodifferential BoVaPr (48) along with double-order φ-RLFr-integral conditions (49) possesses a solution in 0,1.The following example illustrates Theorem10.Example 13.
Regard the nonlinear double-φ-CapFr-integrodifferential BoVaPr
(51)0.8CD01.5;z+1+1−0.8CD01.1;z+1ϖ∗z=0.3RLI00.5;z+114913+z331z+12+ϖ∗z2+0.4RLI00.6;z+1110+z320.9347z3+12+ϖ∗z17,furnished with double-order φ-RLFr-integral conditions
(52)ϖ∗0=0,0.7RLI01;z+1ϖ∗1+1−0.7RLI02;z+1ϖ∗1=0.Here, we have taken valuesm1∗=0.8, m2∗=0.7, σ∗=1.5, φz=z+1, ρ∗=1.1, p1=0.3, δ1∗=0.5, p2=0.4, δ2∗=0.6, θ1∗=1, θ2∗=2, z∈0,1, and h^∗,f^∗:0,1×ℝ→ℝ by rules
(53)h^∗z,ϖ∗=14913+z331z+12+ϖ∗z2,f^∗z,ϖ∗=110+z320.9347z3+12+ϖ∗z17.Clearly,(54)h^∗z,ϖ∗≤14913+z331+∥ϖ∗∥,f^∗z,ϖ∗≤110+z321+∥ϖ∗∥,with γ1z=1/4913+z33,γ2z=1/10+z32 and η1∥ϖ∗∥=η2∥ϖ∗∥=1+∥ϖ∗∥. Now, ∥γ1∥≃0.0588, ∥γ2∥≃0.01, and η1Ω∗=η2Ω∗=1+Ω∗. From the above values, we have Y≃0.4, Λ0≃0.59367, Λ1≃0.30859, and Λ2≃0.36938. By condition (H4), we obtain Ω∗>0.02282. Clearly, all the presumptions of Theorem 10 are verified. Therefore, the nonlinear double-φ-CapFr-integro-differential BoVaPr (51) along with double-order φ-RLFr-integral conditions (52) possesses a solution in 0,1.The following example illustrates Theorem11.Example 14.
As regards the nonlinear double-φ-CapFr-integrodifferential BoVaPr
(55)0.8CD01.5;z+1+1−0.8CD01.1;z+1ϖ∗z=0.3RLI00.5;z+13z2tanπ/4zϖ∗z731+ϖ∗z2+1+0.4RLI00.6;z+1zcotπ/4zϖ∗z101z+ϖ∗z2+2,furnished with double-order φ-RLFr-integral conditions
(56)ϖ∗0=0,0.7RLI01;z+1ϖ∗1+1−0.7RLI02;z+1ϖ∗1=0.Here, we have taken valuesm1∗=0.8, m2∗=0.7, σ∗=1.5, φz=z+1, ρ∗=1.1, p1=0.3, δ1∗=0.5, p2=0.4, δ2∗=0.6, θ1∗=1, θ2∗=2, z∈0,1, and h^∗,f^∗:0,1×ℝ→ℝ by rules
(57)h^∗z,ϖ∗=3z2tanπ/4zϖ∗z731+ϖ∗z2+1,f^∗z,ϖ∗=zcotπ/4zϖ∗z101z+ϖ∗z2+2.Then,N∗≃0.041 and M∗≃0.0099, since
(58)h^∗z,ϖ1∗−h^∗z,ϖ2∗≤0.0411ϖ1∗−ϖ2∗,∣f^∗z,ϖ1∗−f^∗z,ϖ2∗∣≤0.0099ϖ1∗−ϖ2∗.From the above-given values, we haveY≃0.40 and Λ0+N∗Λ1+M∗Λ2≃0.61892<1. Clearly, all the assumptions of Theorem 11 are verified. Thus, the nonlinear double-φ-CapFr-integro-differential BoVaPr (55) along with double-order φ-RLFr-integral conditions (56) possesses a solution in 0,1.


## 5. Concluding Notes

Fractional differential (FD) equations are used to model a number of natural phenomena arising in science and technology. For this reason, most researchers have utilized numerous applied fractional operators in the past years to model various forms of natural processes that occurred in the world. Here, we concentrate on the existence specifications of solutions in relation to a newly configured model of a double-order integrodifferential equation in the framework of theφ-CapFr derivative subject to double-order φ-integral boundary conditions in the context of the φ-RLFr integral. Accordingly, to arrive at this issue, we first extract one of the existence features by making use of the fixed point method expressed by Krasnosel’skii˘, and then, by using a nonlinear alternative criterion attributed to Leray-Schauder, another existence result of the solution is derived. In addition to these, we invoke the Banach principle to confirm the uniqueness of the existing solutions. Three examples are simulated to guarantee the consistency of the analytical findings. From a mathematical view, it is important that we try to obtain generalized notions for the existence definitions. By using this view, the novelty of this work provides an instance of the generalized structure. Also, the proposed structure in this work can be extended by designing different and complicated boundary conditions in the future.



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