Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance

This paper aims to investigate a class of fractional multi-point boundary value problems at resonance on an infinite interval. New existence results are obtained for the given problem using Mawhin’s coincidence degree theory. Moreover, two examples are given to illustrate the main results.


Introduction
Fractional calculus is a generalization of classical integer-order calculus and has been studied for more than 300 years. Unlike integer-order derivatives, the fractional derivative is a non-local operator, which implies that the future states depend on the current state as well as the history of all previous states. From this point of view, fractional differential equations provide a powerful tool for mathematical modeling of complex phenomena in science and engineering practice (see [1][2][3][4][5][6][7]). For example, an epidemic model of non-fatal disease in a population over a lengthy time interval can be described by fractional differential equations: where 0 < α ≤ 1, D α 0 is the Caputo fractional derivative of order α, x(t) represents the number of susceptible individuals, y(t) expresses the number of infected individuals that can spread the disease to susceptible individuals through contact, and z(t) is the number of isolated individuals who cannot contract or transmit the disease for various reasons (see [1]). In [2], Ateş and Zegeling investigated the following fractional-order advection-diffusion-reaction boundary value problem (BVP): where 1 < α ≤ 2, 0 < ε ≤ 1, γ ∈ R, C D α is the Caputo fractional derivative of order α and S(t) is a spatially dependent source term.
Throughout this paper, we assume the following conditions hold: where we let Σ : (H 3 ) ∆ := a 11 a 22 − a 12 a 21 = 0, where A BVP is called a resonance problem if the corresponding homogeneous BVP has nontrivial solution. According to (H 1 ), we will consider the following homogeneous BVP of fractional BVP (1): By Lemma 2 (see Section 2), BVP (2) has nontrivial solution u(t) = at α−1 + bt α−2 , a, b ∈ R, which implies that BVP (1) is a resonance problem and the kernel space of linear operator Lu = D α 0+ u is two-dimensional, i.e., dimKerL = 2 (see Section 3, Lemma 7).
In this paper we aim to show the existence of solutions for BVP (1). To the authors' knowledge, the existence of solutions for fractional BVPs at resonance with dimKerL = 2 on an infinite interval has not been reported. Thus, this article provides new insights. Firstly, our paper extends results from dimKerL = 1 to dimKerL = 2 [27,29] and from finite interval to infinite interval [26]. Secondly, we generalize the results of [37,38] to fractional-order cases. Meanwhile, in the previously literature [37,38] authors established the existence results are based on similar conditions to (H 4 ) and (H 5 ) (see Section 3, Theorem 1). In the present paper we also show that existence results can be obtained by imposing sign conditions (see Section 3, Theorem 2).
The main difficulties in solving the present BVP are: Constructing suitable Banach spaces for BVP (1); Since [0, +∞) is noncompact, it is difficult to prove that operator N is L-compact; The theory of Mawhin's continuation theorem is characterized by higher dimensions of the kernel space on resonance BVPs, therefore, constructing projections P and Q is difficult; Estimating a priori bounds of the resonance problem on an infinite interval with dim KerL = 2 (see Section 3, . The rest of this paper is organized as follows. Section 2, we recall some preliminary definitions and lemmas; Section 3, existence results are established for BVP (1) using Mawhin's continuation theorem; Section 4 provides two examples to illustrate our main results; Finally, conclusions of this work are outlined in Section 5.

Preliminaries
In this section, we recall some definitions and lemmas which are used throughout this paper.
Let (X, · X ) and (Y, · Y ) be two real Banach spaces. Suppose L : domL ⊂ X → Y is a Fredholm operator with index zero then there exist two continuous projectors P : X → X and Q : Y → Y such that Im P = KerL, Im L = KerQ, X = KerL ⊕ KerP, Y = Im L ⊕ Im Q, and the mapping L | domL∩KerP : domL → Im L is invertible. We denote K p = (L | domL∩KerP ) −1 . Let Ω be an open bounded subset of X and domL ∩Ω = ∅. The map N : X → Y is called L-compact onΩ, if QN (Ω) is bounded and K P,Q N(Ω) = K p (I − Q)N :Ω → X is compact (see [39,40]).

Main Result
Let It is easy to check that (X, · X ) and (Y, · Y ) are two Banach spaces.
Define the linear operator L : domL ⊂ X → Y and the nonlinear operator N : X → Y as follows: Then BVP (1) is equivalent to Lu = Nu.

Lemma 6.
(see [34]). Let M ⊂ X be a bounded set. Then M is relatively compact if the following conditions hold: (i) the functions from M are equicontinuous on any compact interval of [0, +∞) ; (ii) the functions from M are equiconvergent at infinity.

Lemma 7.
Assume that (H 1 ) and (H 3 ) hold. Then we have

Proof. By Lemmas 2 and 3 and boundary conditions, we obtain
KerL= Now, we prove that Im L = {y ∈ Y : Q 1 y = Q 2 y = 0} . In fact, if y ∈ Im L, then there exists a function u ∈ domL, such that y(t) = D α 0+ u(t). By Lemma 2, we have Using Lemmas 3 and 4 and boundary condition u(0) = 0, we have c 3 = 0, Thus, On the other hand, for any y ∈ Y satisfying (3), take Define the linear operators T 1 , T 2 : Y → Y by where ∆, a ij (i, j = 1, 2) are the constants which have been given in (H 3 ).
Lemma 8. Define the operators P : where X 1 := KerL, Y 1 := Im Q. Then L is a Fredholm operator with index zero.
Proof. Obviously, P is a projection operator and Im Noting that the definitions of the operators T 1 and T 2 , we see Q is a linear operator. On the other hand, for y ∈ Y, a routine computation gives It follows that Q 2 y = Q(Qy) = Qy. Thus, Q is a projection operator. Let y = (y − Qy) + Qy, then Qy ∈ Im Q and Q(y − Qy) = 0, which together with (H 3 ), yields that Hence, Y = Im L + Im Q. If y ∈ Im L ∩ Im Q, then y = Qy = 0. Therefore, Y = Im L ⊕ Im Q and dim KerL=codim Im L=2. Consequently, we infer that L is a Fredholm operator with index zero. Lemma 9. Define operator K p : Im L → domL ∩ KerP by Then K p is the inverse operator of L | domL∩KerP and K p y X ≤ y L 1 .
Proof. For any y ∈ Im L ⊂ Y, then Q 1 y = Q 2 y = 0 and K p y = I α 0+ y. By Lemma 4 and condition (H 1 ), it is not difficult to verify that K p y ∈ domL ∩ KerP. Hence, K p is well defined. We now prove that K p = (L | domL∩KerP ) −1 . In fact, for u ∈ domL ∩ KerP, by Lemma 3, we have Since K p Lu ∈ domL ∩ KerP, then K p Lu(0) = 0 and P(K p Lu) = 0, which yields that c 1 = c 2 = c 3 = 0. Therefore, K p Lu = u, for any u ∈ domL ∩ KerP. In view of Lemma 4, it is straightforward to show that LK p y = y for any y ∈ Im L. Then It remains to show that K p y X ≤ y L 1 . Indeed, Thus we arrive at the conclusion that K p y X ≤ y L 1 for any y ∈ Im L.

Lemma 10.
Suppose that (H 2 ) holds and Ω is an open bounded subset of X such that domL ∩Ω = ∅, then N is L-compact onΩ.

Proof.
Since Ω is bounded in X, there exists a constant r > 0 such that u X ≤ r for any u ∈Ω. Then, by (H 2 ), we have This means that QN (Ω) is bounded. Next, we show that K P,Q N (Ω) on [0, +∞) is compact. To this end, we divide our proof in three steps. First, we need to prove that K P,Q N :Ω → Y is bounded. In fact, for any u ∈Ω, we have Then Thus we conclude that K P,Q N (Ω) is bounded. The next thing to do in the proof is that K P,Q N (Ω) is equicontinuous on any subcompact interval of [0, +∞) . Indeed, for u ∈Ω, by (H 2 ), we have Let κ be any finite positive constant on [0, +∞), then for any t 1 , t 2 ∈ [0, κ] (without loss of generality we assume that t 1 < t 2 ), we obtain K P,Q Nu(t 1 ) Proceeding as in the proof of above, we can obtain Consequently, we infer that K P,Q N (Ω) is equicontinuous on [0, κ] . Finally, we have to show that K P,Q N (Ω) is equiconvergent at infinity. As a matter of fact, for any u ∈Ω, we have Hence, for given ε > 0, there exists a positive constant L such that On the other hand, since lim t→+∞ (t − L) α−1 1 + t α−1 = 1 and lim t→+∞ t − L 1 + t = 1, then for above ε > 0 there exists a constant T > L > 0 such that for any t 1 , t 2 ≥ T and 0 ≤ s ≤ L, we have and Thus, for any t 1 , t 2 ≥ T > L > 0, by (4)-(6), we get K P,Q Nu(t 1 ) Using the similar argument as in the proof of above, we can show that Thus we arrive at the conclusion that K P,Q N (Ω) is equiconvergent at infinity. According to Lemma 6, it follows that K P,Q N (Ω) is relatively compact. Therefore, N is L-compact onΩ. (H 4 ) There exist positive constants A and B such that, for all u(t) ∈ domL\KerL, if one of the following conditions is satisfied: then either Q 1 Nu = 0 or Q 2 Nu = 0. (H 5 ) There exists a positive constant C such that, for every a, b ∈ R satisfying |a| > C or |b| > C, then either or Then boundary value problem (1) has at least one solution in X provided that To prove the Theorem 1, we need several lemmas.

Lemma 12.
Assume that (H 5 ) holds, then the set Proof. Let u ∈ Ω 2 , then u can be written as u = at α−1 + bt α−2 , a, b ∈ R and Q 1 Nu = Q 2 Nu = 0. According to the assumption (H 5 ), it follows that |a| ≤ C and |b| ≤ C. Hence, we have and sup t≥0 |u| Thus we conclude that Ω 2 is bounded.
Since ∆ = 0, we obtain a = b = 0. For λ ∈ (0, 1), by λJu = (1 − λ)QNu, we have from which we deduce that In view of ∆ = 0, we get λa = (1 − λ)Q 1 Nu, We are now in a position to claim that |a| ≤ C and |b| ≤ C. If the assertion would not hold, then by (7), we obtain This leads to a contradiction. Consequently, we infer that Ω 3 is bounded.
We now turn to the proof of Theorem 1.
We finally remark that deg{QN| KerL , Ω ∩ KerL, 0} = 0. To show this, we define From Lemma 13 we conclude that H(u, λ) = 0 for any u ∈ KerL ∩ ∂Ω, λ ∈ [0, 1]. Hence, by the homotopy of degree, we have According to Lemma 1, it follows that Lu = Nu has at least one solution in domL ∩Ω, that is, (1) has at least one solution in X.
Proof. For u ∈ Ω 1 , we get Nu ∈ Im L = KerQ. By (H 6 ) and (H 7 ), there exist constants t 1 ∈ [0, +∞), This together with the Lemma 5 implies that Then, we obtain On the other hand, by Lemma 2, for u ∈ Ω 1 ⊂ domL, we have it follows that By solving the above equations, we obtain These together with the inequalities (15) and (16), we find Substituting (18) into (17), one has From this it follows that Combining formulas (15), (16) and (19) gives Noting that Lu = λNu, by (H 2 ), we have It follows from (20) and (21) that Thus we arrive at the conclusion that Ω_1 is bounded.
Proof. For any u ∈ Ω 2 , then u can be expressed as u(t) = at α−1 + bt α−2 , a, b ∈ R, t ∈ [0, +∞) and Q 1 Nu = Q 2 Nu = 0. Using the same argument as in the proof of Lemma 12, to get the desired result, we just need to show that |a| and |b| are bounded. By (H 6 ) and (H 7 ), there exist constants t 3 ∈[0, +∞) and Then, we obtain The proof is completed.
With the help of the preceding three lemmas we can now prove the Theorem 2.

Proof.
Set Ω ⊂ X be a bounded open set such that ∪ 2 i=1Ω i ∪Ω 4 ⊂ Ω . Using Lemma 10, N is L-compact onΩ . It follows from Lemma 14 and Lemma 15 that conditions (i) and (ii) of Lemma 1 hold. In what follows, we prove that condition (iii) is satisfied. To this end, we set By Lemma 16, we obtain H(u, µ) = 0 for any u ∈ KerL ∩ ∂Ω , µ ∈ [0, 1]. Based on the homotopy of degree, we have According to Lemma 1, the equation Lu = Nu has at least one solution in domL ∩Ω , which means (1) has at least one solution in X.
By Theorem 1, BVP (24) has at least one solution.

Conclusions
In the present work, we considered a class of fractional differential equations with multi-point boundary conditions at resonance on an infinite interval. With the aid of Mawhin's continuation theorem, we obtained existence results for solutions of BVP (1). Two practical examples were presented to illustrate the main results. BVPs of fractional differential equations on an infinite interval have been widely discussed in recent years. However, there is still more work to be done in the future on this interesting problem. For example, establishing the existence of solutions for fractional differential equations with infinite-point boundary conditions, as well as the existence of non-negative solutions for fractional BVPs, at resonance on an infinite interval in the case of dimKerL = 2.