Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions

Boundary value problems for nonlinear fractional differential equations have recently been investigated by several researchers. The study of fractional equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. A strong motivation for studying fractional differential equations comes from the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, chemistry, economics and biology, etc. [4,7,10,14]


Introduction
Boundary value problems for nonlinear fractional differential equations have recently been investigated by several researchers. The study of fractional equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. A strong motivation for studying fractional differential equations comes from the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, chemistry, economics and biology, etc. [4,7,10,14] Ahmad and Ntouyas [6] investigated the existence of solutions for a fractional boundary value problem with fractional separated boundary conditions given by α 1 x(0) + β 1 ( c D p x(0)) = γ 1 , α 2 x(1) + β 2 ( c D p x(1)) = γ 2 , 0 < p < 1.
Where c D q denotes the Caputo fractional derivative of order q, f is continuous function on [0, T ] × R and α i , β i , γ i , (i = 1, 2) are real constants, with α i 0.
Also Xiaoyou Liu and Zhenhai Liu [17] investigated the existence and uniqueness of solutions for the nonlinear fractional boundary value problem with fractional separated boundary conditions given by : Where c D α denotes the Caputo fractional derivative of order α, f is continuous function on [0, T ]×R×R and a i , b i , c i , (i = 1, 2) are real constants, with a 1 0 and T > 0. Bashir Ahmad, Juan J, Nieto and Ahmed Alsaedi [5] investigated the existence and uniqueness of the solutions for a new class of boundary value problems of nonlinear fractional differential equations with non-separated type integral boundary conditions. Precisely, they consider the following problem Where c D q denotes the Caputo fractional derivative of order q, and f, g, h : [0, T ] × R → R are given continuous functions and λ 1 , λ 2 , µ 1 , µ 2 ∈ R with λ 1 1, λ 2 1.
In this paper, we discuss the existence and uniqueness of solutions for a new class of boundary value problems of nonlinear fractional differential equations depending with non-separated type integral boundary conditions. Precisely, we consider the following problem Where c D q denotes the Caputo fractional derivative of order q, and f ∈ C([0, T ] × R × R, R), g, h : [0, T ] × R → R are given continuous functions and λ 1 , λ 2 , µ 1 , µ 2 ∈ R with λ 1 1, λ 2 1.
The rest of the paper is arranged as follows. In Section 2, we establish a basic result that lays the foundation for defining a fixed point problem equivalent to the given problem (1.1). The main results, based on Banach's contraction mapping principal, Schauder fixed point theorem and nonlinear alternative of Leray-Schauder type, are obtained in Section 3. Illustrating examples are discussed in Section 4.

Preliminaries
For convenience of the reader, we present here some necessary definitions about fractional calculus theory, which can be found in [1,9,12,13].
Definition 2.1. The Riemann-Liouville fractional integral of order q for a continuous function f is defined as Definition 2.2. For a at least n-times continuously differentiable function f : (0, ∞) −→ R, the Caputo derivative of order q > 0 is defined as where [q] denotes the integer part of the real number q. Lemma 2.3. Let α > 0, then the differential equation has solutions h(t) = c 0 + c 1 t + c 2 t 2 + . . . + c n−1 t n−1 and here c i ∈ R, i = 0, 1, 2, . . . , n − 1 and n = [α] + 1.
Theorem 2.4. (Schauder fixed Point theorem )(see [2]) Let U be a closed, convex and nonempty subset of a Banach space X, let P : U −→ U be a continuous mapping such that P(U) is a relatively compact subset of X. Then P has at least one fixed point in U.
Theorem 2.5. (Nonlineair alternative of Leray-Schauder type )(see [2]) Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and 0 ∈ U. Suppose that F :Ū −→ C is a continuous, compact (that is, F(Ū) is a relatively compact subset of C ) map. Then either (i) F has a fixed point inŪ, or (ii) there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λF(x).

Existence and uniqueness results
Let I = [0, T ] and C(I, R) be the space of all continuous real functions defined on I. Define the space X = {x(t) : x(t) ∈ C(I, R) and c D r x ∈ C(I, R)}, (0 < r ≤ 1) endowed with the norm ) is a Banach space. Now we present the Green's function for boundary value problem of fractional differential equation.
is given by : where G(t, s) is the Green's function given by

2)
Proof. We omit the proof as it employs the standard arguments for instance, see [3].
In this section, we given some existence results for the problem (1.1) In view of Lemma 3.1 we define an operator F : .
It is clear that the problem (1.1) has solutions if and only if the operator equation F x = x has fixed points. For any x ∈ X, let Since the function f is continuous and .
We know that the operator F maps X into X. Here k is constant given by Observe that problem (1.1) has solution if the operator Eq. (3.3) has fixed points, our first result is based on the Banach fixed point theorem (see [11]).
Theorem 3.2. We suppose that such that where : Then the boundary value problem (1.1) has a unique solution.
Proof. Let us set From the above inequalities, we obtain : Now, for any x, y ∈ X and for each t ∈ [0, T ], we obtain We obtain : Similary, we have : From the above inequalities, we obtain Which implies that F is a contraction mapping. By means of the Banach contraction mapping principle, F has a unique fixed point which is a unique solution of the boundary value problem (1.1). Now, we state a known result due to Schauder which is needed to prove the existence of at least one solution of (1.1).  Proof. Schauder's Fixed point theorem is used to prove that F defined by Eq. (3.3) has a fixed point. The proof will be given in several steps.
Step 1: F maps the bounded sets into the bounded sets in X.
x ≤ R} and R > 0 is a positive number. It is clear that B R is a closed, bounded and convex subset of the Banach space X. For any x ∈ B R , we have:

So, we have
Then from Eq. (3.4), we have So, we have: From above inequalities, we obtain Denote: .
Now let R be a positive number such that: Then it is obvious that for any x ∈ B R , This implies that F : B R → B R .
Step 3: F(B R ) is equicontinuous with B R defined as in Step 2.
Since f is continuous, we can assume, without any loss of generality, that | f (t, x(t), c D r x(t))| ≤ N 1 and |h(t, x(t))| ≤ N 2 for any x ∈ B R and t ∈ [0, T ]. Now let, 0 ≤ t 1 ≤ t 2 ≤ T. Then we have , So, we have: we find that .
Then the problem (1.1) has at least one solution.
Proof. It is trivially that F : X −→ X. We have shown in Theorem 3.3 that F is continuous.
Firstly, LetB be a uniformly bounded subset of X and let R > 0 be such that x ≤ R for all x ∈B. We prove that F :B →B. For any x ∈B, we have Finely, we have Where: Hence Fu is uniformly bounded.
Secondly, we prove the compactness of the operator F, we define | f (t, x(t), c D r x(t))| ≤ N 1 , |h(t, x(t))| ≤ N 2 . For any t 1 , t 2 ∈ [0, T ] are such that t 1 ≤ t 2 , we have the following facts:
Thirdly, the result will follow from the Leray-Schauder nonlinear alternative (Theorem 2.5) once we have proved the boundeness of the set of all solutions to equations x = λF x for λ ∈ (0, 1).
By the definition of the Caputo fractional derivative with 0 < r ≤ 1, | c D r (F x)(t)| ≤ Therefore, we can obtain that F x(t) ≤ c 1 (A 1 + A 2 |x(t)| + A 3 | c D r x(t)|) + c 2 (A 1 + A 2 |x(t)| + A 3 | c D r x(t)|) Suppose there exists a x ∈ ∂U and a λ ∈ (0, 1) such that x = λF x, then for this x and λ we have which is a contradiction. By Theorem 2.5, there exists a fixed point x ∈Ū of F. This fixed point is a solution of (1.1) and the proof is complete.

Examples
Example 4.1. Consider the following boundary value problem : Example 4.2. Consider the following boundary value problem :