Existence and uniqueness criteria for the higher-order Hilfer fractional boundary value problems at resonance

This investigation is devoted to the study of a certain class of coupled systems of higher-order Hilfer fractional boundary value problems at resonance. Combining the coincidence degree theory with the Lipschitz-type continuity conditions on nonlinearities, we present some existence and uniqueness criteria. Finally, to practically implement the obtained theoretical criteria, we give an illustrative application.


Introduction
In this paper, we study the coupled systems of the higher-order Hilfer fractional differential equations Since the second half of the seventeenth century by now, almost over 370 years, applicability potential of the differential/difference equations and their mutual inverses, integral/finite sum equations in description of the natural phenomena besides theoretical sciences tied up with mathematics have been proved. Meanwhile, the fractional differential calculus that includes arbitrary-order differential/difference and integral/finite sum operators, as a result of more rigorous frame to describe biological, chemical, and physical models of natural phenomena, possesses an elegant position in comparison with the integer-order differential calculus. Reasonability of our claim can be verified in the monographs [4,6,7,18,26,27,[33][34][35][36][37][38].
Restricting ourselves to the investigation of the solvability of fractional-order differential equations opens an independent wide research world in front of us. As some of the generally known techniques to solve a given fractional differential equation, we may mention the fixed point theory (via Green functions, controlling the growth of nonlinearities or measures of noncompactness), the operational calculus, the approximation theory, and the coincidence degree theory. Here we suggest (absolutely not comprehensive) a collection of the pioneering research papers [1-3, 5, 8-14, 17, 21, 23-25, 32, 40, 42-47], and [19,22,[28][29][30][31], respectively, and cited bibliography for more consultation of the interested followers.
Among this variety, we are interested in the investigation of solvability of higher-order generalized fractional differential systems using coincidence degree theory due to Mawhin [33]. More precisely, we are going to study fractional coupled systems at resonance. To specify the concept of resonance, let us consider the differential system Lu = F, BCu = 0, (1.3) in which L, F, and BC denote a differential operator, nonlinearity, and boundary conditions of the system, respectively. We say that the differential system (1.3) is of resonance category if in the corresponding homogenous differential system Lu = 0, BCu = 0, (1.4) the differential operator L is invertible with respect to the boundary conditions BC, that is, there is at least one nontrivial solution for the homogenous differential system (1.4). Otherwise, we are concerned with the nonresonance case. For some of the most motivating research works on the fractional-order resonance phenomenon, we refer to [22,[28][29][30][31]. The authors in [16] considered the coupled resonant system of the higher-order Caputotype fractional -difference boundary value problems where i = 2, 3, . . . , N , N -1 < α ≤ N , N ∈ N 2 , and a ∈ Z 1 , b ∈ Z 2 with a < b, α * denotes the Caputo-type fractional -difference of order α > 0, and f , g : N b+N-α-1 a+N-α-2 × R N → R are continuous functions. Using the coincidence degree theory, the authors obtained some existence and uniqueness criteria for the discrete fractional coupled system (1.5)-(1.6).
The authors in [19] studied the following higher-order Caputo fractional resonant system: subject to the boundary conditions (1.8) Thanks to the coincidence degree theory, they obtained existence criteria for at least one solution of this fractional-order coupled system.

Technical requirements
This section begins with a quick overview on those parts of the fractional calculus that will be needed in this paper. So, we start with the definitions of the Riemann-Liouville fractional integrals and derivatives.
The left-and right-sided Riemann-Liouville fractional integrals of order α ≥ 0 for a function f ∈ L 1 [a, b] are given by (2.1) The left-and right-sided Riemann-Liouville fractional derivatives of order α ≥ 0 for a function f ∈ AC n (a, b) are defined by where n = [α] + 1.
Interchanging the affection position of the nth-order derivative d n dt n as follows gives us the left-and right-sided Caputo fractional derivatives The power rules of the Riemann-Liouville fractional operators are given in the following lemma.

Lemma 2.3 ([26])
Let n -1 < α ≤ n, n ∈ N. Then for each β > -1, we have Now we are ready to define the Hilfer fractional derivatives and their basic properties.
Next, we present the inversion rules of the Hilfer fractional derivatives.
Here we begin the second part of this section, which includes a quick overview on the coincidence degree theory. For detailed discussions, we refer to Chapters IV and V of [15,22], and [29]. Definition 2.8 Let X and Y be real normed spaces. A linear mapping L : dom L ⊂ X → Y is called a Fredholm mapping if the following conditions hold: (i) ker L has a finite dimension, (ii) Im L is closed and has a finite codimension.
Let L be a Fredholm mapping. Then its index is given by Let L is a Fredholm mapping with index zero and suppose that there exist continuous projectors P : X → X and Q : Y → Y such that It follows that the mapping If L is a Fredholm mapping of index zero, then for every isomorphism J : Im Q → ker L, the mapping JQ + K P,Q : Z → dom L is an isomorphism, and for every u ∈ dom L, In addition, we say that N is L-completely continuous if it is L-compact on every bounded subset E ⊂ X. Theorem 2.10 Let ⊂ X be open and bounded, let L be a Fredholm mapping of index zero, and let N be L-compact on . Assume that the following conditions are satisfied: At the end of this part, we reveal why the Hilfer fractional coupled system (1.1)-(1.2) is of resonance category. Let us consider the homogeneous Hilfer fractional system (2.15) subject to the boundary conditions where i = 2, 3, . . . , n. Having identity (2.10) in hand and imposing the boundary conditions it follows that c k = 0, k = 0, 1, . . . , n -3, whereas the coefficients c n-2 and c n-1 are arbitrary. Now imposing the boundary condition yields c n-2 = 0. So, we can conclude that the homogeneous Hilfer fractional system (2.15)-(2.16) has a nontrivial solution of the form Therefore our situation lies in the resonance category. We finalize this section by presenting appropriate function spaces and their relevant norms. We first define the following Banach spaces: Accordingly, the appropriate Banach spaces in this paper are given as follows: (2.17)

Main results
In this section, we obtain some existence and uniqueness criteria for the Hilfer fractional resonant system (1.1)-(1.2). To this aim, as explained in the previous section, we are going to apply the coincidence degree theory. So, based on Theorem 2.10, we first transform the Here we construct required elements of this transformation: Here we present the structure of the abstract nonlinearity N : Considering the setting (3.1)-(3.3), we come to the conclusion that the Hilfer resonant We further have to implement the basis of the coincidence degree theory, step by step as follows.
In the first step, we prove that the operator L(u, v) defined by (3.1) is a Fredholm operator of index 0.
Proof Thanks to identity (2.10) in Lemma 2.6, focussing on the operator L defined by (3.1) with dom L defined by (3.3), we get that According to the dom L, we conclude that c i = d i = 0, i = 0, 1, . . . , n -2. Thus we arrive at Next, implementing the boundary conditions into equalities (3.5) gives us the structure of the Im L as follows: In other words, To get the desired outcome, we define the operator Q : It is easy to check that Im L = ker Q and . This is the expected opportunity to complete the proof.
The definitions of the operators P i , i = 1, 2, immediately give us P 2 i = P i , i = 1, 2, that is, Definitions (3.9) and (3.10) imply that , we can derive that X := ker P + ker L. On the other hand, since ker P ∩ ker L = {(0, 0)}, we have X := ker P ⊕ ker L. So, according to Definition 2.8, and the fact that ker Q = Im L, we conclude that Now the proof is complete.
In this position, we begin the second step proving the L-compactness of the operator N defined by (3.4). Let us define the operator K P : Im L → dom L ∩ ker P by So we directly get that for each (x, y) ∈ Im L, y). (3.13) Next, let (u, v) ∈ dom L ∩ ker P. Therefore, in the identities all the coefficients c i and d i vanish for i = 0, 1, . . . , n -1, which yields (3.14) Finally, relying on (3.13) and (3.14), we conclude that K P = (L dom L∩ker P ) -1 .
Here we prove the L-compactness of the operator N in the following lemma.
and, similarly, In addition, and, similarly, At the and, because of the uniform continuity of the functions ρ(t) := (ta) α and ] are bounded and equicontinuous. This guarantees that the operator K P,Q := K P (I -Q)N is compact on , which yields the L-compactness of the operator N .
Turning to Theorem 2.10, we have just been proved that the operators L defined by (3.1) and N defined by (3.4) are a Fredholm operator of index 0 and an L-compact operator, respectively. In what follows, we are going to identify conditions (i)-(iii) in this theorem. To this aim, we first consider the following hypotheses: (H 1 ) There exist positive real constants d i , i = 1, 2, b k , c k , θ k , and λ k with θ k , λ k ∈ [0, 1] for k = 1, 2, . . . , n such that for each (x 1 , x 2 , . . . , x n ) ∈ R n ,  where is given by (3.28). Here we define Indeed, if we prove that the subsets i ⊂ X, i = 1, 2, 3, and 3 ⊂ X are bounded, then relying on Theorem 2.10, we can directly conclude the existence of at least one solution for the Hilfer fractional resonant system (1.1)-(1.2). So, we start to prove the boundedness of these sets.

Now taking D n-(n-α)(1-β)-1,β a +
on both sides of these equalities, we get Setting t = t 0 in the first equality and t = t 1 in the second equality gives us (nα)) .
Applying hypothesis (H 1 ) to these inequalities, we get Let (u, v) ∈ 1 . Then (u, v) ∈ dom L\ ker L. Since P 2 = P, (I -P)(u, v) ∈ dom L ∩ ker P and LP(u, v) = (0, 0). So, by inequality (3.16) it follows that Accordingly, based on Remark 3.3 and (3.29), we have To complete the proof, since the right-hand side of inequality (3.30) depends on four cases, we will divide our boundedness estimation into four cases as follows: The resources of case i imply that We have Case iv.
The next case deals with the boundedness of the subset 3 .

Lemma 3.6 3 defined by (3.24c) is a bounded subset of X.
Proof Our proof basically depends on hypothesis (H 2 ). Let (u, v) ∈ 3 . Thus (u, v) ∈ ker L, that is, On the other hand, since ker L = Im Q, from λ(u, v) -(1λ)QN (u, v) = (0, 0) we deduce that The assumption |c n-1 |, |d n-1 | > B, together with the hypothesis (H 2 ), leads us to Concentrating on the second parts of hypothesis (H 2 ), that is, the right-hand side inequalities in (3.19) and (3.20), helps us to prove boundedness of 3 ⊂ X. So, we present this result without proof in the next lemma.
as a bounded open subset of X. According to the Lemma 3.2, we know that N is an L-compact operator on . Besides, via Lemmas 3.4-3.7, we get the following: To prove this, we define the homotopy According to the degree property of invariance under a homotopy, if u ∈ ker L ∩ ∂ , then Therefore, since assumption (iii) in Theorem 2.10 is satisfied, we conclude that the Hilfer fractional resonant system (1.1)-(1.2) has at least one solution in X.
As explained before, our investigation is divided into the couple of stages including the existence and uniqueness of solutions for the Hilfer fractional resonant system (1.1)-(1.2). So far, the existence of at least one solution is proved. So, in what follows, we state and prove a uniqueness criterion for the solutions of the coupled system (1.1)-(1.2). To do this, we first need the following hypotheses.
Since hypotheses (H 2 )-(H 3 ) are satisfied, we just need to check hypothesis (H 1 ). To do this, for each i = 1, 2, . . . , n, assume that y i = 0. Also suppose that In the traditional way, we consider the solutions (u k , v k ) ∈ X, k = 1, 2, of the coupled system (1.1)-(1.2) and will prove that u 1 = u 2 and v 1 = v 2 . As supposed, for k = 1, 2, Let us consider u := u 1u 2 and v := v 1v 2 . It follows that and So, in view of hypothesis (H 2 ), we get the following inequality: Therefore this inequality gives us From this inequality we immediately derive that If we take t = t 3 in (3.64) and t = t 2 in (3.65), then (3.28), together with inequalities (3.62) and (3.63), yields Here we recall once again inequality (3.30): Similarly to case i, (3.32), case ii, (3.34), case iii, (3.36), and case iv, (3.38), we conclude that and (3.71) Finally, if we impose conditions (3.51)-(3.54) into inequalities (3.68)-(3.71), respectively, then it follows that u = v = 0. More precisely, we have proven that u 1 = u 2 and v 1 = v 2 . This completes the proof of uniqueness of the solutions of the Hilfer fractional resonant system (1.1)-(1.2).

An application
In this section, we present an application to illustrate the obtained theoretical existence and uniqueness criteria in the frame of Theorems 3.8 and 3.9.

Discussion and concluding remarks
In this paper, we studied the higher-order Hilfer fractional resonant system (1.1)-(1.2). Our aim in this investigation was to apply the coincidence degree theory to obtain at least one solution for the resonant system (1.1)-(1.2). Besides, having certain conditions on the nonlinearities, we presented a uniqueness criterion. One of the advantages of this investigation is that, to the best of our knowledge, this is the first time in the literature that the Hilfer fractional differential equations have been considered to establish the fractional resonant problems.
At this position, we discuss other advantages of the Hilfer fractional derivatives. As mentioned in Definition 2.4, taking β = 0 gives us the Riemann-Liouville fractional resonant system which generalizes all the Riemann-Liouville-based fractional resonant problems with boundary conditions of the form (5.2). Also, taking β := 1, we get the following Caputo fractional resonant system: . . , v (n-1) ), a < t < b, c D α a + v(t) = (t, u, u , u , . . . , u (n-1) ), a < t < b, which coincides with the main problem in [19], that is, (1.7)-(1.8), and it generalizes [20] and [39]. We believe that due to the unifying characteristics of the Hilfer fractional derivatives, differential equations equipped with this generalized derivatives have much more potential to reach new results both in the theory and in the applications of the fractional-order problems. For instance, as a future research works, one may consider the half-linear Hilfer fractional differential equations and try to extract the corresponding Lyapunov-type inequalities to describe the qualitative dynamics of these problems involving stability, disconjugacy, nonexistence, algebraic properties of nontrivial solutions, and so on. Furthermore, applicability of the half-linear dynamical systems in the porous medium opens a new research line to find more applications of the Hilfer fractional dynamical systems.