Positive solutions for Hadamard fractional differential equations on infinite domain

This paper investigates the existence of nonnegative multiple solutions for nonlinear fractional differential equations of Hadamard type, with nonlocal fractional integral boundary conditions on an unbounded domain by means of Leggett-Williams and Guo-Krasnoselskii’s fixed point theorems. Two examples are discussed for illustration of the main work.


Introduction
Fractional calculus has gained considerable attention from both theoretical and the applied points of view in recent years. There are numerous applications in a variety of fields such as electrical networks, chemical physics, fluid flow, economics, signal and image processing, viscoelasticity, porous media, aerodynamics, modeling for physical phenomena exhibiting anomalous diffusion, and so on. In contrast to integer-order differential and integral operators, fractional-order differential operators are nonlocal in nature and provide the means to look into hereditary properties of several materials and processes. The monographs [-] are commonly cited for the theory of fractional derivatives and integrals and applications to differential equations of fractional order. For more details and examples, see [-] and the references therein.
However, it has been noticed that most of the work on the topic is concerned with Riemann-Liouville or Caputo type fractional differential equations. Besides these fractional derivatives, another kind of fractional derivatives found in the literature is the fractional derivative due to Hadamard introduced in  [], which differs from the aforementioned derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed description of Hadamard fractional derivative and integral can be found in [, -].
Boundary value problems on infinite intervals appear often in applied mathematics and physics, such as in unsteady flow of gas through a semi-infinite porous medium, the theory of drain flows, etc. More examples and a collection of works on the existence of solutions of boundary value problems on infinite intervals for differential, difference and integral equations may be found in the monographs [, ]. For boundary value problems of fractional order on infinite intervals we refer to [-].
Zhao and Ge [] studied the existence of unbounded solutions for the following boundary value problem on the infinite interval: where D α  denotes Riemann-Liouville fractional derivative of order α, and  < β, ξ < ∞. Zhang et al. [] studied the existence of nonnegative solutions for the following boundary value problem for fractional differential equations with nonlocal boundary conditions on unbounded domains: where D α  denotes Riemann-Liouville fractional derivative of order α, f ∈ C([, ∞) × R, R + ) and  < β, η < ∞. The Leray-Schauder nonlinear alternative is used.
Liang and Zhang [] used a fixed-point theorem for operators on a cone, proved the existence of positive solutions to the following fractional boundary value problem: we studied a new class of boundary value problems on fractional differential equations with m-point Erdélyi-Kober fractional integral boundary conditions on an infinite interval of the form where D α  denotes the Riemann-Liouville fractional derivative of order α, I is the Erdélyi-Kober fractional integral of order δ i >  with η i > , γ i ∈ R, i = , , . . . , m -, β i ∈ R, and ξ i ∈ (, ∞), i = , , . . . , m - are given constants. We proved the existence and uniqueness of unbounded solution of the boundary value problem (.)-(.) by using the Leray-Schauder nonlinear alternative and Banach contraction principle.
In this paper, we aim to investigate the existence criteria of positive solutions for fractional differential equations of Hadamard type, with integral boundary condition on infinite intervals. Precisely, we consider the following boundary value problem for Hadamard fractional differential equations: where D α denotes the Hadamard fractional derivative of order α, η ∈ (, ∞), and I β i is the Hadamard fractional integral of order β i > , i = , , . . . , m, and λ i ≥ , i = , , . . . , m are given constants.
To the best of our knowledge, there are no papers devoted to the study of positive solutions for Hadamard fractional differential equations on infinite intervals; we fill the gap in this paper. As we know, [, ∞) is noncompact. In the literature a special Banach space were introduced. Unfortunately this Banach space is not suitable for Hadamard fractional differential equations. In order to overcome this difficulty, a special Banach space is introduced so that we can establish some inequalities, which guarantee that the functionals defined on [, ∞) have better properties. Applying first the well-known Leggett-Williams fixed point theorem, we obtain a new result on the existence of at least three distinct nonnegative solutions under some conditions. Next we prove the existence of at least one positive solution by using Guo-Krasnoselskii's fixed point theorem.
The rest of the paper is organized as follows: In Section , we present some preliminaries and lemmas that will be used to prove our main results. We also obtain the corresponding Green's function and some of its properties. The main result is formulated and proved in Sections  and . Especially in Section  we prove the existence of at least three distinct nonnegative solutions while, in Section , we prove the existence of at least one positive solution. Examples illustrating our results are presented in Section .

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [] and present preliminary results needed in our proofs later.

Definition . [] The Hadamard fractional integral of order q for a function g is defined as
provided the integral exists.

Definition . []
The Hadamard derivative of fractional order q for a function g : where [q] denotes the integer part of the real number q and log(·) = log e (·).
. Then the Hadamard fractional differential equation D q x(t) =  has the solutions and the following formula holds: where c i ∈ R, i = , , . . . , n, and n - < q < n.
Then the unique solution of the following fractional differential equation: subject to the boundary conditions is given by the integral equation and Proof Applying the Hadamard fractional integral of order α to both sides of (.), we have The first condition of (.) implies c  = . Therefore, In accordance with Lemma ., we have The second condition of (.) leads to where is defined by (.). Therefore, the unique solution of fractional boundary value problem (.)-(.) is The proof is completed.
Lemma . The Green's function G(t, s) defined by (.) satisfies the following conditions: Proof It is easy to check that (C  ) and (C  ) hold.
In this paper, we will use the Banach space E, which is defined by and is equipped with the norm n= be any Cauchy sequence in the space (E, · E ). Then ∀ε > , ∃N >  such that n= is a Cauchy sequence in R. In this way, we can associate to each t ∈ [, ∞) for a unique u(t). Letting n → ∞, we obtain |u(t)u m (t)| ≤ ε for all m > N and t ∈ [, ∞). It is easy to show that u m → u in E as m → ∞. Therefore, we see that (E, · E ) is Banach space.
for any t  , t  ≥ T and u ∈ U.
Proof Evidently, it is sufficient to prove that U is totally bounded. In the following we divide the proof into two steps.
Step . Let us consider the case t ∈ [, T]. Define Then clearly, U [,T] , with the norm u ∞ = sup t∈[,T] |u(t)| +(log t) α- is a Banach space. The condition (i) combined with the Arzelá-Ascoli theorem indicates that U [,T] is relatively compact, hence U [,T] is totally bounded, namely, for any ε > , there exist finitely many balls B ε (u i ) such that Step . Define T] . Now let us take u i ∈ U i ; then U can be covered by the balls B ε (u i ), i = , , . . . , n where In fact, for u(t) ∈ U, the arguments in Step  imply that there exist i such that u [,T] ∈ B ε (u i ). Hence, for t ∈ [, T], we have For t ∈ [T, +∞), (.), and (.) yield Equations (.) and (.) show that u(t)u i (t) E < ε. Therefore, U is totally bounded and Lemma . is proved.
We define the cone P ⊂ E by and the operator T : P → E by where G(t, s) defined by (.). Throughout this paper, we assume that the following conditions hold: Proof We divide the proof into four steps.
Step : We show that T is uniformly bounded on P.
From the definition of E, we can choose r  such that sup n∈N u n E < r  . Let B r  = sup{f (( + (log t) α- )u), u ∈ [, r  ]} and be any bounded subset of P, then there exists r >  such that u E ≤ r for all u ∈ . From (C  ), it follows that for u ∈ . Therefore T is uniformly bounded.
Step : We show that T is equicontinuous on any compact interval of [, ∞). For any S > , t  , t  ∈ [, S], and u ∈ , without loss of generality, we assume that t  < t  . In fact, uniformly as t  → t  , and uniformly as t  → t  . Hence, from (.), (.), and (.), we get Thus T is equicontinuous on [, ∞).
Step : We show that T is equiconvergent at ∞.
Hence, T is equiconvergent at infinity.
Step : We show that T is continuous.
Hence the Lebesgue dominated convergence and the continuity of f guarantee that Therefore, we get So, T is continuous. Using Lemma ., we see that T : P → P is completely continuous. The proof is completed.

Existence of at least three positive solutions
In this section we use the Leggett-Williams fixed point theorem to prove the existence of at least three positive solutions.
For convenience, we denote Definition . A continuous mapping θ : P → [, ∞) is said to be a nonnegative continuous concave functional on the cone P of a real Banach space E provided that for all u, v ∈ P and λ ∈ [, ].

Theorem . []
Let P be a cone in the real Banach space E and c >  be a constant. Assume that there exists a concave nonnegative continuous functional θ on P with θ (u) ≤ u for all u ∈ P c . Let T : P c → P c be a completely continuous operator. Suppose that there exist constants  < a < b < d ≤ c such that the following conditions hold: Theorem . Suppose conditions (A  ) and (A  ) hold. Let  < a < b < d ≤ c and suppose that f satisfies the following conditions: Then fractional boundary value problem (.)-(.) has at least three positive solutions u  , u  , and u  satisfying Proof We will show that the conditions of the Leggett-Williams fixed point theorem are satisfied for the operator T defined by (.). We defined a nonnegative functional on E by For u ∈ P c , we have u E ≤ c, that is, Then assumption (A  ) implies Therefore, Hence, T : P c → P c and by Lemma ., T is completely continuous. It follows from condition (A  ), using the above argument, that if u ∈ P a then Tu E < a. Therefore, condition (ii) of Theorem . holds. Let It is obvious that u * (t) ∈ P and u * = b+c  < c. From the definition of θ (u), then Hence, we get Then assumption (A  ) implies So, we have Thus θ (Tu) > b for all u ∈ P (θ , b, d). This shows that condition (i) of Theorem . holds. Finally, we assume that u ∈ P (θ , b, c) with Tu E > d; then u E ≤ c and b ≤ u(t) +(log t) α- ≤ c and from assumption (A  ) we can show θ (Tu) > b. So, condition (iii) of Theorem . is satisfied. As a consequence of Theorem . implies that boundary value problem (.)-(.) has at least three positive solutions u  , u  , and u  such that u  E < a, b < θ (u  ) and a < u  E with θ (u  ) < b.
The proof is completed.

Existence of at least one positive solution
In this section we use the Guo-Krasnoselskii fixed point theorem to prove the existence of at least one positive solution.

Theorem . []
Let E be a Banach space, and let P ⊂ E be a cone. Assume that  ,  are open subsets of E with  ∈  ,  ⊂  , and let T : P ∩ (  \  ) → P be a completely continuous operator such that: (i) Tu ≥ u , u ∈ P ∩ ∂  , and Tu ≤ u , u ∈ P ∩ ∂  ; or (ii) Tu ≤ u , u ∈ P ∩ ∂  , and Tu ≥ u , u ∈ P ∩ ∂  . Then T has a fixed point in P ∩ (  \  ). Then fractional boundary value problem (.)-(.) has at least one positive solution u such that r  < u E < r  .
Proof We will show that the condition (i) of Theorem . is satisfied. By Lemma ., the operator T : P → P is completely continuous.

Examples
Example . We consider the following Hadamard fractional differential equation with nonlocal boundary conditions on an unbounded domain: