An Analytical Survey on the Solutions of the Generalized Double-Order 
 φ
 -Integrodifferential Equation

We study the existence of solutions for a newly configured model of a double-order integrodifferential equation including 
 
 φ
 
 -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.


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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][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 com-bine the previous operators effectively to avoid confusion when using existing fractional operators (see [17][18][19][20]).
In 2017, an extension of the usual Caputo operator entitled φ-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][23][24][25][26][27].
In the same year, Wahash et al. ( [29]) turned to a structure involving the generalized φ-Caputo equation subject to integral conditions as where z ∈ ½0, 1, σ * ∈ ð0, 1Þ, m * ≥ 0, and ν ∈ ℝ + , and further, ℏ * : ½0, 1 × ℝ + → ℝ + illustrates a continuous function along with ξ ∈ L 1 ℝ + ð½0, 1Þ. 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][31][32][33]) and discuss different views on applied mathematical modelings (see for example, [34][35][36][37][38][39]).
The organization of the contents of the current manuscript is as follows. In the next section, some required notions in the context of the generalized φ-calculus are assembled. Section 3 is devoted to establish the main theorems in which 2 Journal of Function Spaces 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.

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 order σ * > 0 for a function ϖ * : ½0,+∞Þ → ℝ is defined as provided that the value of the integral is finite [40,41]. In this position, let us assume that σ * ∈ ðn − 1, nÞ so that n = 1 + ½σ * . For a continuous function ϖ * : ½0,+∞Þ → ℝ, the Riemann-Liouville derivative of order ρ * is given by provided that the value of the integral is finite [40,41]. In the next step, for an absolutely continuous function ϖ * ∈ AC ðnÞ ℝ ð½0,+∞ÞÞ, the fractional derivative of Caputo type is given by provided that the integral is finite-valued [40,41].
In a manner, if φðzÞ = z and s 0 = 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.

Journal of Function Spaces
In view of the above lemma, it is verified that the general solution of the homogeneous equation ð C D σ * ;φ s 0 ϖ * ÞðzÞ = 0 is given by where n − 1 < σ * < n and c * 0 , c * 1 , ⋯, c * n−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 set B * is assumed to be a nonempty, convex, bounded, and closed subset of a given Banach space ;  Journal of Function Spaces whereĥ * is regarded as a compact and continuous mapping andf * a contraction. Then, an element υ ∈ B * exists so that υ =ĥ * υ +f * υ.
Theorem 7 [45]. Assume that B * 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 func-tionÂ * : G → B * possesses a fixed point in G or there is a g ∈ ∂G and 0 < σ < 1 such that g = σÂ * ðgÞ, where ∂G stands for the boundary of G in B * .

Main Analytical Results
Suppose that E = Cð½s 0 , T, ℝÞ stands for the collection of all functions given on ½s 0 , T with continuity property. In this case, one can simply verify that E is a Banach space together with We start with the next lemma which will be required in the subsequent sections.
Then, ϖ * satisfies the nonlinear double-order differential boundary problem of the fractional type iff ϖ * satisfies the fractional nonlinear double-order integral equation Proof. To start the proof, assume that ϖ * satisfies the nonlinear double-order differential equation (15). We have By fractional integrating in the Riemann-Liouville setting of order σ * , we get where c * 0 , c * 1 ∈ ℝ are arbitrary constants. By using the first boundary condition, we get c * 0 = 0 and so On the other hand, if we take η * ∈ fθ * 1 , θ * 2 g, then we have Now, by utilizing the second condition, we have By solving the above equation with respect to c 1 , we get where Y is illustrated in (14). Inserting the obtained value for c * 1 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 singlevalued operator Q : E → E as where ϖ * ∈ E and z ∈ ½s 0 , T. From now on, we set 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.
Proof. To start the proof, let kμ * k = sup so that K * ≔ sup z∈½s 0 ,T jĥ * ðz, 0Þj and Λ 0 and Λ 1 are given by (24). Obviously, nonempty ball B * ε 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 ∈ ½s 0 , T, we split Q into Q 1 , Q 2 : B * ε → E by By the condition ðH 1 Þ, we havê for each ϖ * ∈ E and z ∈ ½s 0 , T. In consequence, for both ϖ * 1 , ϖ * 2 ∈ B * ε and z ∈ ½s 0 , T and in view of (24) and (27), we get which implies ∥Q 1 ϖ * 1 + Q 2 ϖ * 2 ∥≤ε and so ðQ 1 ϖ * 1 ðzÞ + Q 2 ϖ * 2 ðzÞÞ ∈ B * ε for all ϖ * 1 , ϖ * 2 ∈ B ε . Now, we prove that Q 2 is continuous. A sequence fϖ * n g in B * ε is assumed to be convergent provided that ϖ * n → ϖ * . Then, for any element z ∈ ½s 0 , T, one may write Sincef * is continuous, we get kQ 2 ϖ * n − Q 2 ϖ * k → 0 as ϖ * n → ϖ * . From this, we realize that Q 2 is continuous on B * ε . In the sequel, to verify that Q 2 is compact, we have to prove that Q 2 has the uniform boundedness property. For every ϖ * ∈ B * ε and all z ∈ ½s 0 , T, the following estimate for Q 2 holds which implies that ∥Q 2 ϖ * ∥≤∥μ * ∥Λ 2 . In order to confirm the equicontinuity of Q 2 , suppose that z 1 , z 2 ∈ ½s 0 , T such that z 2 > z 1 . We investigate that Q 2 corresponds bounded sets to equicontinuous sets. To guarantee this claim, for any ϖ * ðzÞ ∈ B * ε , we obtain It is to be notice that the R.H.S of (33) is not dependent to ϖ * ∈ B * ε and also goes to 0 by assuming z 1 → z 2 . This describes that Q 2 is equicontinuous. Hence, Q 2 has the relative compactness property on B * ε , and thus, application of the Arzelá-Ascoli result gives the complete continuity of Q 2 , and eventually on B * ε , we reach the compactness of Q 2 . At last, we intend to conclude that Q 1 is a contraction. For any ϖ * 1 , ϖ * 2 ∈ B * ε , and z ∈ ½s 0 , T, we get