Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.


Introduction
Fractional differential equation can be extensively applied to various disciplines such as physics, mechanics, chemistry, and engineering, see 1-3 .Hence, in recent years, fractional differential equations have been of great interest and there have been many results on existence and uniqueness of the solution of FDE, see 4-8 .Especially, Diethelm and Ford 9 have gained existence, uniqueness, and structural stability of solution of the type of fractional differential equation D q y − T m−1 y x f x, y x , y k 0 y k 0 , k 0, 1, . . ., m − 1, 1.1 where q > 0 is a real number, D q denotes the Riemann-Liouville differential operator of order q, and T m−1 y is the Taylor polynomial of order m − 1 for the function y x at x 0 0. Recently, Daftardar-Gejji and Jafari 10 have discussed the existence, uniqueness, and stability of solution of the system of nonlinear fractional differential equation where m < α ≤ m 1 and D α * denotes Caputo fractional derivative see Definition 2.3 .Delbosco and Rodino 11 have proved existence and uniqueness theorems for the nonlinear fractional equation where 0 < δ < 1, D δ is the Riemann-Liouville fractional derivative.Zhang 12 used the method of the upper and lower solution and cone fixed point theorem to obtain the existence and uniqueness of positive solution to 1.3 .Yao 13 considered the existence of positive solution to 1.3 controlled by the power function employing Krasnosel'skii fixed point theorem of cone expansion-compression type.The existence of the local and global solution for 1.3 was obtained by Lakshmikantham and Vatsala 14 utilizing classical differential equation theorem.
More recently, Zhang 15 shows the existence of positive solutions to the singular boundary value problem for fractional differential equation where D α 0 is the Riemann-Liouville fractional derivative of order n − 1 < α ≤ 2, n ≥ 2. However, in the previous works, the nonlinear term has to satisfy the monotone or others control conditions.In fact, the fractional differential equations with nonmonotone function can respond better to impersonal law, so it is very important to weaken monotone condition.Considering this, in this paper, we mainly investigate the fractional differential Equation 1.2 without any monotone requirement on nonlinear term by constructing upper and lower control function and exploiting the method of upper and lower solutions and the Schauder fixed point theorem.The existence and uniqueness of positive solution for 1.2 are obtained.Some properties concerning the maximal and minimal solutions are also given.This work is motivated by the above references and my previous work 16, 17 .Other related results on the fractional differential equations can be found in 18-24 .This paper is organized as follow.In Section 2, we recall briefly some notions of the fractional calculus and the theory of the operators for integration and differentiation of fractional order.Section 3 is devoted to the study of the existence and uniqueness of positive solution for 1.2 utilizing the upper and lower solution method and the Schauder fixed point theorem.The existence of maximal and minimal solutions for 1.2 is given in Section 4.

Preliminaries and Notations
First, we give some basic definitions and theorems which are basically used throughout this paper.C 0, 1 denotes the space of continuous functions defined on 0, 1 and C n 0, 1 denotes the class of all real valued functions defined on 0, 1 which have continuous nth order derivative.Definition 2.1.Let f x ∈ C 0, 1 and α > 0, then the expression is called the left-sided Riemann-Liouville integral of order α.
is called the left-sided Riemann-Liouville derivative of f x of order α whenever the expression on the right-hand side is defined.
is called the left-sided Caputo derivative of f x of order α.
Lemma 2.4 see 25, 26 .Let f x ∈ C n 0, 1 and n − 1 < α ≤ n, n ∈ N, then we one has Proof.Suppose y t satisfies the initial value problem 1.2 , then applying I α to both sides of 1.2 and using Lemma 2.4 2.7 follows.Conversely, suppose y t satisfies 2.7 .Then observe that D m 1 y t exists and is integrable, because which exists and is integrable as f t, y t is * on both sides of 2.7 , one has Let X C 0, 1 be the Banach space endowed with the infinity norm and K a nonempty closed subset of X defined as The positive solution which we consider in this paper is a function such that y t ∈ K.
According to Lemma 2.5, 1.2 is equivalent to the fractional integral Equation 2.7 .The integral equation 2.7 is also equivalent to fixed point equation Ty t y t , y t ∈ C 0, 1 , where operator T : K → K is defined as then we have the following lemma.
Proof.Let M ⊂ K be bounded, that is, there exists a positive constant l * such that y ∞ ≤ l * for any y t ∈ M. Since f t, y t is a given continuous function, we have where Let L max t,y ∈D f t, y , then for any y t ∈ M, we have

2.12
Thus, Hence T : K → K is uniformly bounded.Now, we prove that T : K → K is continuous.Since f t, y t is continuous function in a compact set 0, 1 × 0, l , then it is uniformly continuous there.Thus given ε > 0, we can find μ > 0 such that f t, y − f t, z < ε * whenever y − z < μ, where ε * εΓ α 1 .Then proving the continuity of the operators T : K → K. Now, we will prove that the operator T :

2.15
The Arzela-Ascoli Theorem implies that T is completely continuous.The proof is therefore completed.
Lemma 2.7.If the operator A : X → X is the contraction mapping, where X is the Banach space, then A has a unique fixed point in X.
then the functions y t , y t are called a pair of order upper and lower solutions for 1.2 .

Existence and Uniqueness of Positive Solution
Now, we give and prove the main results of this paper.endowed with the norm z max t∈ 0,1 z t , then we have z ≤ b.Hence S is a convex, bounded, and closed subset of the Banach space X.According to Lemma 2.6, the operator T : K → K is completely continuous.Then we need only to prove T : S → S.
For any z t ∈ S, we have y t ≥ z t ≥ y t , then

3.3
Hence y t ≥ Tz t ≥ y t , 1 > t > 0, that is, T : S → S. According to Schauder fixed point theorem, the operator T exists at least one fixed point y t ∈ S, 0 < t < 1. Therefore the boundary value problem 1.2 exists at least one solution y t ∈ C 0, 1 , and y t ≥ y t ≥ y t , t ∈ 0, 1 .Corollary 3.2.Assume f : 0, 1 × 0, l → 0, ∞ is continuous, and there exist p 2 > p 1 ≥ 0, such that then the boundary value problem 1.2 exists at least one positive solution y t ∈ C 0, 1 , moreover Proof.By assumption 3.4 and the definition of control function, we have Now, we consider the equation Obviously, 3.7 has a positive solution namely, w t is a upper solution of 1.2 .In the similar way, we obtain v t m k 0 t k /k! c k p 1 t α /Γ α 1 is the lower solution of 1.2 .An application of Theorem 3.1 now yields that the boundary value problem 1.2 exists at least one positive solution y t ∈ C 0, 1 , moreover then the boundary value problem 1.2 has at least one positive solution u t ∈ C 0, 1 .
Proof.By assumption 3.9 , there are positive constants N, R, such that f t, y ≤ N whenever u > R. Let M max 0≤t≤1,0≤y≤R f t, y , then f t, y ≤ N M, 0 ≤ y < ∞.By the definition of control function, one has H t, y ≤ N M, 0 ≤ t ≤ 1, 0 ≤ y < ∞.Now, we consider the equation Obviously, 3.11 has a positive solution namely, w t is the upper solution of 1.2 .In the similar way, we obtain v t m k 0 t k /k! c k ct α /Γ α 1 is the lower solution of 1.2 .Therefore, the boundary value problem of 1.2 has at least one positive solution y t ∈ C 0, 1 , what is more, we have then the boundary value problem 1.2 exists at least at one positive solution u t ∈ C 0, 1 .
Proof.According to c < lim y → ∞ max 0≤t≤1 f t, y /y M < ∞, there exists D > 0, such that for any y t ∈ X, we have f t, y t ≤ My t D.

3.15
By the definition of control function, we have H t, y t ≤ My t D.

3.16
We now consider the equation According to Lemma 2.5, 3.17 is equivalent to the integral equation

3.18
Let A : K → K be an operator as follows:

by Lemma 2.6, the operator
where R > 0 and satisfies that is convex, bounded, and closed subset of the Banach space C 0, 1 .For any y t ∈ B R , we have Hence, the Schauder fixed theorem assures that the operator A has at least one fixed point and then 3.17 has at least one positive solution y * t , therefore we have 3.24 Combining condition 3.16 , we have 3.25 Obviously, y * t is the upper solution of initial value problem 1.2 , and v t m k 0 c k /k! ct α /Γ α 1 is the lower solution.By Theorem 3.1, system 1.2 has at least one positive solution u t ∈ C 0, 1 .Corollary 3.5.Assume f : 0, 1 × 0, ∞ → c, ∞ is continuous and there exists d > 0, e > 0, such that
Proof.According to Theorem 3.1, if the conditions in Theorem 3.1 hold, then the boundary value problem 1.2 has at least one positive solution in S. Hence we need only to prove that the operator T defined in 2.10 is the contraction mapping in X.In fact, for any y 1 t , y 2 t ∈ X, by assumption 3.30 , we have

3.31
Thus, when l/Γ α 1 < 1, the operator T is the contraction mapping.Then by Lemma 2.7, the boundary value problem 1.2 has a unique positive solution y t ∈ S.

Maximal and Minimal Solutions Theorem
In this section, we consider the existence of maximal and minimal solutions for 1.2 .
Definition 4.1.Let m t be a solution of 1.2 in 0, 1 , then m t is said to be a maximal solution of 1.2 , if for every solution y t of 1.2 existing on 0, 1 the inequality y t ≤ m t , t ∈ 0, 1 holds.A minimal solution may be defined similarly by reversing the last inequality.

4.1
Then there exist maximal solution ϕ t and minimal solution η t of 1.2 on 0, 1 , moreover Proof.It is easy to know that m k 0 t k /k! c k μt α /Γ α 1 and m k 0 t k /k! c k λt α /Γ α 1 are the upper and lower solutions of 1.2 , respectively.Then by using

4.4
The

y 0 m k 0 t
t k /k! c k μt α /Γ α 1 , y 0 m k 0 t k /k! c kλt α /Γ α 1 as a pair of coupled initial iterations we construct two sequences {y m }, {y m }from the following linear iteration process: − s α−1 f s, y m−1 t ds.

4 . 3
It is easy to show from the monotone property of f t, y and condition 4.1 that the sequences {y m }, {y m } possess the following monotone property: y 0 ≤ y m ≤ y m 1 ≤ y m 1 ≤ y m ≤ y 0 6 Definition 2.8.Let y t , y t ∈ K, b ≥ y t ≥ y t ≥ a, and satisfy be a given function.Take a, b ∈ R , and a < b < l.For any y ∈ a, b one defines the upper-control function H t, y sup a≤η≤y f t, η , and lower-control function h t, y inf y≤η≤b f t, η , obviously H t, y , h t, y is monotonous nondecreasing on y and h t, y ≤ f t, y ≤ H t, y .
26then the boundary value problem 1.2 has at least one positive solution y t ∈ C 0, 1 , moreover Letting m → ∞ in 4.3 shows that ϕ t and η t satisfy the equations It is easy to verify that the limits ϕ t and η t are maximal and minimal solutions of 1.2 inS * {ψ t | ψ t ∈ K, m k 0 t k /k! c k λt α /Γ α 1 ≤ ψ t ≤ m k 0 t k /k! c k μt α /Γ α 1 , st ∈ 0, 1 , ψ t max 0≤t≤1 ψ t },respectively, furthermore, if ϕ t η t ≡ ζ t then ζ t is the unique solution in S * , and hence the proof is completed.