On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

Abstract: This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional (GPF ) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-GPF operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations (FVFIEs) via generalized fuzzy Hilfer-GPF Hukuhara differentiability (HD) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy FVFIEs which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.


Introduction
Recently, fractional calculus has attained assimilated bounteous flow and significant importance due to its rife utility in the areas of technology and applied analysis. Fractional derivative operators have given a new rise to mathematical models such as thermodynamics, fluid flow, mathematical biology, and virology, see [1][2][3]. Previously, several researchers have explored different concepts related to fractional derivatives, such as Riemann-Liouville, Caputo, Riesz, Antagana-Baleanu, Caputo-Fabrizio, etc. As a result, this investigation has been directed at various assemblies of arbitrary order differential equations framed by numerous analysts, (see [4][5][6][7][8][9][10]). It has been perceived that the supreme proficient technique for deliberating such an assortment of diverse operators that attracted incredible presentation in research-oriented fields, for example, quantum mechanics, chaos, thermal conductivity, and image processing, is to manage widespread configurations of fractional operators that include many other operators, see the monograph and research papers [11][12][13][14][15][16][17][18][19][20][21][22].
In [23], the author proposed a novel idea of fractional operators, which is called GPF operator, that recaptures the Riemann-Liouville fractional operators into a solitary structure. In [24], the authors analyzed the existence of the FDEs as well as demonstrated the uniqueness of the GPF derivative by utilizing Kransnoselskii's fixed point hypothesis and also dealt with the equivalency of the mixed type Volterra integral equation.
Fractional calculus can be applied to a wide range of engineering and applied science problems. Physical models of true marvels frequently have some vulnerabilities which can be reflected as originating from various sources. Additionally, fuzzy sets, fuzzy real-valued functions, and fuzzy differential equations seem like a suitable mechanism to display the vulnerabilities marked out by elusiveness and dubiousness in numerous scientific or computer graphics of some deterministic certifiable marvels. Here we broaden it to several research areas where the vulnerability lies in information, for example, ecological, clinical, practical, social, and physical sciences [25][26][27].
In 1965, Zadeh [28] proposed fuzziness in set theory to examine these issues. The fuzzy structure has been used in different pure and applied mathematical analyses, such as fixed-point theory, control theory, topology, and is also helpful for fuzzy automata and so forth. In [29], authors also broadened the idea of a fuzzy set and presented fuzzy functions. This concept has been additionally evolved and the bulk of the utilization of this hypothesis has been deliberated in [30][31][32][33][34][35] and the references therein. The concept of HD has been correlated with fuzzy Riemann-Liouville differentiability by employing the Hausdorff measure of non-compactness in [36,37].
Numerous researchers paid attention to illustrating the actual verification of certain fuzzy integral equations by employing the appropriate compactness type assumptions. Different methodologies and strategies, in light of HD or generalized HD (see [38]) have been deliberated in several credentials in the literature (see for instance [39][40][41][42][43][44][45][46][47][48][49]) and we presently sum up quickly a portion of these outcomes. In [50], the authors proved the existence of solutions to fuzzy FDEs considering Hukuhara fractional Riemann-Liouville differentiability as well as the uniqueness of the aforesaid problem. In [51,52], the authors investigated the generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions. Bede and Stefanini [39] investigated and discovered novel ideas for fuzzy-valued mappings that correlate with generalized differentiability. In [43], Hoa introduced the subsequent fuzzy FDE with order ϑ ∈ (0, 1) : A suitable assumption was provided so that this correspondence would be effective. Hoa et al. [53] proposed the Caputo-Katugampola FDEs fuzzy set having the initial condition: denotes the fuzzy Caputo-Katugampola fractional generalized Hukuhara derivative and a fuzzy function is F : [σ 1 , σ 2 ] × E → E. An approach of continual estimates depending on generalized Lipschitz conditions was employed to discuss the actual as well as the uniqueness of the solution. Owing to the aforementioned phenomena, in this article, we consider a novel fractional derivative ( merely identified as Hilfer GPF -derivative). Consequently, in the framework of the proposed derivative, we establish the basic mathematical tools for the investigation of GPF -F F HD which associates with a fractional order fuzzy derivative. We investigated the actuality and uniqueness consequences of the clarification to a fuzzy fractional IVP by employing GPF generalized HD by considering an approach of continual estimates via generalized Lipschitz condition. Moreover, we derived the F VF IE using a generalized fuzzy GPF derivative is presented. Finally, we demonstrate the problems of actual and uniqueness of the clarification of this group of equations. The Hilfer-GPF differential equation is presented as follows: where D ϑ,q,β σ + 1 (.) is the Hilfer GPF -derivative of order ϑ ∈ (0, 1), I 1−γ,β σ 1 (.) is the GPF integral of order 1 − γ > 0, R j ∈ R, and a continuous function F : [σ 1 , T ] × R → R with ν j ∈ [σ 1 , T ] fulfilling σ < ν 1 < ... < ν m < T for j = 1, ..., m. To the furthest extent that we might actually know, nobody has examined the existence and uniqueness of solution (1.3) regarding F VF IEs under generalized fuzzy Hilfer-GPF -HD with fuzzy initial conditions. An illustrative example of fractional-order in the complex domain is proposed and provides the exact solution in terms of the Fox-Wright function.
The following is the paper's summary. Notations, hypotheses, auxiliary functions, and lemmas are presented in Section 2. In Section 3, we establish the main findings of our research concerning the existence and uniqueness of solutions to Problem 1.3 by means of the successive approximation approach. We developed the fuzzy GPF Volterra-Fredholm integrodifferential equation in Section 4. Section 5 consists of concluding remarks.
For any δ ∈ R and Φ 1 , Φ 2 ∈ E, then the sum Φ 1 + Φ 2 and the product δΦ 1 are demarcated as: ˇq is the usual sum of two intervals of R and δ[Φ 1 ]ˇq is the scalar multiplication between δ and the real interval.
For any Φ ∈ E, the diameter of theq-level set of Φ is stated as diam[µ]ˇq =μ(q) − µ(q). Now we demonstrate the notion of Hukuhara difference of two fuzzy numbers which is mainly due to [54].
, then the GPF integral of order ϑ of the fuzzy function Φ is stated as: and 0 < ϑ < 1, we can write the fuzzy GPF -integral of the fuzzy mapping Φ depend on lower and upper mappingss, that is, where and (2.9) Definition 2.10. For n ∈ N, order ϑ and type q hold n − 1 < ϑ ≤ n with 0 ≤ q ≤ 1. The left-sided fuzzy Hilfer-proportional gH-fractional derivative, with respect to ζ having β ∈ (0, 1] of a function , E) and the fractional generalized Hukuhara GPF -derivative of fuzzy-valued function Φ is stated as: Definition 2.12. We say that a point ζ 0 ∈ (σ 1 , σ 2 ), is a switching point for the differentiability of F , if in any neighborhood U of ζ 0 there exist points ζ 1 < ζ 0 < ζ 2 such that Type I. at ζ 1 (2.11) holds while (2.12) does not hold and at ζ 2 (2.12) holds and (2.11) does not hold, or Type II. at ζ 1 (2.12) holds while (2.11) does not hold and at ζ 2 (2.11) holds and (2.12) does not hold.
Then for any β ∈ (0, 1], we have . Taking into account Definition 2.10, we have This completes the proof.
Utilizing the Definitions 2.6, 2.10 and Lemma 2.13 with the initial condition ( Now considering Proposition 1, Lemma 2.13 and Lemma 2.14, we obtain where we have used the fact e Proof. The proof is simple and can be derived as parallel to Theorem 2.2 in [53].
Proof. The proof is simple and can be derived as parallel to Theorem 2.3 in [53].

Main results and discussion
In this investigation, we find the existence and uniqueness of solution to problem 1.3 by utilizing the successive approximation technique by considering the generalized Lipschitz condition of the righthand side.
In our next result, we use the following assumption. For a given constant > 0 , and let B(
In order to find the analytical view of (3.12), we utilized the technique of successive approximation. Putting Φ 0 (ζ) = Φ 0 and Letting n = 1, δ > 0, assuming there is a d-increasing mapping Φ, then we have In contrast, if we consider δ < 0 and Φ is d-decreasing, then we have For n = 2, we have if δ > 0 and there is d-increasing mapping Φ, we have and there is δ < 0, and d-increasing mapping Φ. So, continuing inductively and in the limiting case, when n → ∞, we attain the solution for every δ > 0 and Φ is d-increasing, or δ < 0 and Φ is d-decreasing, accordingly. Therefore, by means of Mittag-Leffler function E ϑ,q (Φ) = ∞ l=1 Φ κ Γ(lϑ+q) , ϑ, q > 0, the solution of problem (3.12) is expressed by for every of δ > 0 and Φ is d-increasing. Alternately, if δ < 0 and Φ is d-decreasing, then we get the solution of problem (3.12)

Consider IVP
where β ∈ (0, 1] and ϑ ∈ (0, 1) is a real number and the operation gH D ϑ σ + 1 denote the GPF derivative of order ϑ, F : [ζ 0 , T ] × E × E × E → E is continuous in ζ which fulfills certain supposition that will be determined later, and Now, we investigate the existence and uniqueness of the solution of problem (4.1). To establish the main consequences, we require the following necessary results.
is described for any n ∈ N. Then the sequence {Φ n } converges to fixed point of problem (4.1) which is Proof. We now prove that the sequence {Φ n }, given in (4.16), is a Cauchy sequence in C([ζ 0 , T ], E). To do just that, we'll requirē Since δ < 1 promises that the sequence {Φ n } is a Cauchy sequence in C([ζ 0 , T ], E). Consequently, there exist Φ ∈ C([ζ 0 , T ], E) such that {Φ n } converges to Φ. Thus, we need to illustrate that Φ is a solution of the problem (4.1).

Conclusions
The present investigation deal with an IVP for GPF fuzzy FDEs and we employ a new scheme of successive approximations under generalized Lipschitz condition to obtain the existence and uniqueness consequences of the solution to the specified problem. Furthermore, another method to discover exact solutions of GPF fuzzy FDEs by utilizing the solutions of integer order differential equations is considered. Additionally, the existence consequences for F VF IDEs under GPF -HD with fuzzy initial conditions are proposed. Also, the uniqueness of the so-called integrodifferential equations is verified. Meanwhile, we derived the equivalent integral forms of the original fuzzy F VF IDEs whichis utilized to examine the convergence of these arrangements of conditions. Two examples enlightened the efficacy and preciseness of the fractional-order HD and the other one presents the exact solution by means of the Fox-Wright function. For forthcoming mechanisms, we will relate the numerical strategies for the estimated solution of nonlinear fuzzy FDEs.