Solvability of nonlinear fractional integro-differential equation with nonlocal condition

Purpose – This study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractionaldifferentialequationforwhichtheweaksolution ’ sexistenceanduniquenessareproved.Theauthors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results. Design/methodology/approach – Thefunctionalanalysismethodisthe apriori estimatemethodorenergy inequality method. Findings – The results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions. Research limitations/implications – The authors ’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order. Originality/value – Theauthorsconfirmthatthisworkisoriginalandhasnotbeenpublishedelsewhere,nor is it currently under consideration for publication elsewhere.


Introduction
Fractional order partial differential equations have become one of the most popular areas of research in mathematical analysis. Their application has been utilized in various scientific fields, such as optimal control theory, chemistry, physics, mathematics, biology, finance and engineering [1][2][3][4][5].
It is important to establish effective methods to solve fractional differential equations (FDEs). Recently, a great deal of attention was dedicated to FDE solutions utilizing different methods, including the Adomian decomposition method [16,17], the Laplace transform method [18], exponential differential operators [19], the F-expansion method [20], non-Nehari manifold method [21] and the reproducing kernel space method [22,23], in the search for exact or analytical solutions. The applicability of most techniques becomes difficult with the presence of the integral condition. The energy inequality method is a useful tool for studying nonlocal fractional and classical problems. Compared with other techniques, it has an essential role in establishing the solution's existence and uniqueness proof and depends on density arguments and certain a priori bounds.
There have been few articles related to nonlinear fractional partial equations that employ the energy inequality method [24]. Furthermore, for partial differential equations with classical order, many results have utilized this method [25][26][27][28]. Motivated by the previous results, the authors studied a nonlocal nonlinear time-fractional order problem. Moreover, we demonstrate the solution's uniqueness, existence and dependence on the given data.
This article is outlined in the following way: in Section 2, we present the main problem. The next section is focused on posing the associated linear problem and introducing some required preliminaries and functional spaces. Then, in Section 4, we develop the energy inequality method to demonstrate the linear problem's strong solution's uniqueness. In addition, we prove the strong solution's existence in Section 5. Moreover, we derive a priori bound and demonstrate the generated operator range density in a Hilbert space. We solve the nonlinear problem in Section 6 by utilizing the results achieved in Sections 4 and 5, and an iteration process.

Statement of problem
with 0 < β < 1. Associated with initial condition and the boundary condition Such that the known functions γ, η and ξ verify Assumption 1, and data functions f, w and ψ belong to suitable function spaces as mentioned in Section 3.
In the Caputo definition for a function v, the fractional derivatives of order β þ 1 with 0 < β < 1 is defined as where Γ : ð Þ is the gamma function AJMS and the Riemann-Liouville integral of order 0 < β < 1, which is given by 3. Technical tools and associated linear problem We define some function spaces and tools required to investigate the following linear problem associated with problems (1)- (3).
First, we convert problems (1)-(3) into an equivalent operator form where the unbounded operator L ¼ L; and v also verify the initial condition. Here E is Banach space containing elements having the finite norm and F is Hilbert space composed of functions normed with Lemma 1. [29] Let S(t) a nonnegative absolute continuous function verifying the inequality C v α t SðtÞ ≤ c 1 SðtÞ þ c 2 ðtÞ; 0 < α < 1; for almost all t ∈ [0, T], where c 1 is a positive constant and c 2 (t) is an integrable nonnegative function on [0, T]. Then, where are Mittag-Leffler functions.
[30] For any n ∈ N, we have where I 2n which holds for arbitrary a and b, and all « > 0.

A priori
where C > 0 constant independent of v.
Proof. We take the scalar product L 2 D τ ð Þ of equality (6) and the integro-differential operator Mv ¼ −2I 2 x vv vt , such that τ ∈ [0, T], we have The integration of the first three terms on the left-hand side (LHS) of Equation (20), taking into consideration initial and boundary conditions (2)À(3), gives Substituting (21)À(23) into (20) yields By applying inequality (16), we estimate the first and the last two terms on the right-hand side (RHS) of (24); as such it follows that Solvability of nonlinear fractional equation By Lemma 2, the first term on the LHS of (20) becomes Hence, by Formulas (25)- (29) and Assumption (1), we obtain where then We need to drop the last term on the RHS of (31). Therefore, we use Gronwall's lemma, which yields where Now, by discarding the last two terms on the LHS of (32) then posing S τ where Combining (32)À(33) yields where From given inequality we reduce inequality (34) as follows Solvability of nonlinear fractional equation Since the RHS of estimate (36) is independent of τ, we can take the supremum on the LHS with respect to τ over [0, T]. Thus, we get the desired inequality (19). Theorem (4) proof is complete. -

Existence of the linear problem solution
The current section's aim is to prove the existence of the strong solution of problems (6)- (8). It remains to demonstrate the density of the range R(L). Defining the operator equation solution as a strong solution of problems (6)- (8). The inequality (19) can be extended into the inequality demonstrated above assures the strong solution uniqueness. Proof. Introducing a new function σ(x, t) verifies conditions (2) and (3), and σ, σ x ,I t σ x , I t σ ¼0: AJMS Now, we consider the function g x;t ð Þ¼−I t I 2 x σ: Obviously, the function g included in L 2 (D). Equations (39)-(40) lead to Note that the function σ verifies conditions (2)À(3), then we have Insertion of Equations (42)-(44) into (41), yields According to Lemma 1, we bound the first term on the LHS of (45); we have Also, we bound the last three terms on the RHS of (45) utilizing inequality 17, and we then get The insertion of estimates (46)-(48) in Equation (45) gives Eliminating the first term on the LHS of (50), using Lemma 2, with Similarly, by discarding the second integral on the LHS of (50) and applying (53), we obtain it follows that for any τ ∈ [0, T]. Hence inequality (55) shows that g 5 0 ae in D. Continuing Theorem 7 proof, we assume that for a certain function G ¼ g;g 0 ;g 1 ð Þ∈R L ð Þ ⊥ , we have then we should show that g 0 5 0, g 1 5 0. Putting v ∈ D(L), verifying homogenous initial conditions into (56), yields By applying Proposition (8) to (57), we see that g 5 0. Consequently, (56) becomes Since l 1 v and l 2 v are independent and their ranges l 1 and l 2 are everywhere dense in L 2 Ω ð Þ, we conclude that g 0 5 g 1 5 0, this complete the proof of Theorem 7. -

Solvability of nonlinear fractional equation
Integrations by parts all terms of (81), by using conditions (78)À(79), proceeding as in the establishment of Theorem 4, yields On the other hand, applying to Equation (77) the operator I x , and taking into consideration condition (79), multiplying the resulting equation with vT ðnÞ vx and integrating over D τ , we get After integration by parts of all the terms of (83) and taking into consideration conditions (78), (79) and using inequality (17) Combination of inequalities (83)À(84) gives Eliminating the last term on the RHS of (85), by using Gronwall's lemma, it comes where To discard the last integral on the RHS of inequality 86 ð Þ, we drop the three first elements then use the Gronwall's lemma, it follows On the other hand, via the condition (65), we get Combining (86)-(88) and by using (35), we get , then the problems (62)-(64) admit a weak solution in L 2 (0, T, H 1 (Ω)). Now, we prove the uniqueness of problems (62)-(64). Proof. Suppose that the problems (62)-(64) admit v 1 and v 2 as solutions in L 2 (0, T, H 1 (Ω)), then H 5 v 1 À v 2 belongs to L 2 (0, T, H 1 (Ω)) and verifies