Solvability of a Hadamard Fractional Boundary Value Problem at Resonance on Infinite Domain

This paper investigates the existence of solutions for Hadamard fractional di ﬀ erential equations with integral boundary conditions at resonance on in ﬁ nite domain. By constructing two suitable Banach spaces, establishing an appropriate compactness criterion, and de ﬁ ning appropriate projectors, we study an existence theorem upon the coincidence degree theory of Mawhin. An example is given to illustrate our main result.


Introduction
In this paper, we study the following Hadamard fractional boundary value problem (BVP for short) on an infinite domain: where 2 < α ≤ 3, λ i ≥ 0, η ∈ 1,∞ , H D α is the Hadamard fractional derivative of α, and H I β i denotes the Hadamard fractional integral of β i ≥ 0, i = 1, 2, ⋯, m.
In [15], the authors discussed a Hadamard fractional differential equation on infinite intervals: where D α is the Hadamard fractional derivative of α and I β i denotes the Hadamard fractional integral of β i , and In [14], by using monotone iterative technique, the authors obtained the existence of positive solutions for a Hadamard fractional differential equation on infinite intervals: where H D α denotes the Hadamard fractional derivative of α, η ∈ 1,+∞ , and H I • is the Hadamard fractional integral.r, In [11], Li and Zhai obtained the existence and uniqueness of positive solutions to Hadamard fractional differential equations on infinite intervals by making use of a fixed point theorem for generalized concave operators: where H D α is the Hadamard fractional derivative of α and H I β i denotes the Hadamard fractional integral of β i , and The above three articles studied the solutions of Hadamard fractional differential equation at nonresonance, i.e., To the best of our knowledge, there are few papers that have investigated the boundary value problems of Hadamard fractional differential equations at resonance on infinite domain.Inspired by the excellent results in [18][19][20][21], we will discuss this problem by constructing two suitable Banach spaces, establishing an appropriate compactness criterion, defining appropriate operators and the coincidence degree theory due to Mawhin.In this paper, we will assume that the following conditions hold. (

Preliminaries and Lemmas
For convenience, we introduce some definitions and fundamental results of fractional calculus theory [22].
Definition 1.The Hadamard fractional integral of order q > 0 for a function g 1,+∞ ⟶ R is defined as provided that the integral exists and log = log e .
Definition 2. The Hadamard fractional derivative of order q > 0 for a function g 1,+∞ ⟶ R is given by where n = q + 1 and q denotes the integral part of number q.
Let X and Z be real Banach spaces, L dom L ⊂ X ⟶ Z be a Fredholm operator with index zero, and P X ⟶ X, Q Z ⟶ Z be projectors such that Theorem 5 (see [23]).Let Ω ⊂ X be open and bounded, let L dom L ⊂ X ⟶ Z be a Fredholm operator of index zero, and let N X ⟶ Z be L-compact on Ω. Assume that the following conditions are satisfied: Then, the equation Lu = Nu has at least one solution in dom L ∩ Ω.
Define the spaces with the norm where Similar to the proof of Lemma 2.4 in [20], we can prove that X, • ∞ and X, • X are Banach spaces.
Let Z = y t : y t , y t /t ∈ L 1 1,+∞ with the norm Then, it is easy for us to prove that Z, • Z is a Banach space.
We define the operator and we define N X ⟶ Z by Then, BVP (1) can be written as Lu = Nu.
Next, similar to the compactness criterion in [15,20], we establish the following criterion, and it can be proved in a similar way.Lemma 6. Assume V is bounded in X, and then, V is relatively compact in X if the following conditions hold: 1 + log t , and H D α−1 u t are equicontinuous on any compact interval of 1, +∞ (ii) For any ε > 0, there exists a constant for any u t ∈ V and t 1 , t 2 ≥ T.

The Main Results
For convenience, for y ∈ Z, we define the operator A by Proof.We can easily obtain that have a solution u t that satisfies the conditions in BVP (1).From Lemma 4, we have

Journal of Function Spaces
where c 1 , c 2 , and c 3 ∈ R. By u 1 = 0, we get c 3 = 0.By Lemma 3, we get From H D α−2 u 1 = 0, we obtain c 2 = 0. Hence, where B β i , α is the beta-function.Thus, On the other hand, if (26) holds, setting where b is an arbitrary constant, then u t is solution of (21).
Lemma 8.If (H 1 ) and (H 2 ) hold, then L dom L ⊂ X ⟶ Z is a Fredholm operator of index zero, and the linear continuous projectors P X ⟶ X and Q Y ⟶ Y can be defined as respectively, and the linear operator K p Im L ⟶ dom L ∩ Ker P can be given by Proof.We can easily get P 2 u = Pu, u ∈ X, and Q 2 y = Qy, y ∈ Z, which imply that P, Q are projectors.Clearly, Im P = Ker L, and Im L = Ker Q.For y ∈ Z, we have y = y − Qy + Qy.
From (H 2 ), we obtain b = 0 and y t = 0.So Im L ∩ Im Q = 0 and Z = Im L ⊕ Im Q. Obviously, dimKerL = dimIm Q = 1; this means that L is a Fredholm operator of index zero.
Similarly, we can get X = Ker L ⊕ Ker P.
For y ∈ Im L, we have And for u ∈ dom L ∩ Ker P, by Lemma 4 and K p Lu ∈ dom L, we get It follows from u ∈ Ker P that H D α−1 u 1 = 0. This, together with K p Lu ∈ Ker P, means that c = 0. So, K p L u t = u t .This shows that K p = L dom L∩Ker P −1 .
Proof.Let Ω ⊆ X be bounded; i.e., there exists r > 0 such that u X ≤ r, u ∈ Ω.By (H 3 ), we get Hence, QN Ω is bounded.Next, we will prove that Firstly, for u ∈ Ω, Next, we prove that for any u t ∈ Ω, there exists a constant T > M > 1 so that for any t 1 , t 2 ≥ T,

and
To obtain the existence of the solution for BVP (1), we give the following conditions: (H 4 ) There exists a constant (H 6 ) There exists a constant k > 0 such that for any u t = c log t α−1 with c > k, we have either cAN c log t α−1 > 0 or cAN c log t α−1 < 0.

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Journal of Function Spaces Take h t = 1/t 2 , and then,   Using the same method as calculating in (H 4 ), we can get ANu > 0. So the condition H 6 holds.Hence, by Theorem 13, BVP (64) has at least one solution.
condition (H 2 ) is satisfied.Obviously, the condition (H 3 ) holds.The condition (H 4 ) can be estimated as follows: