Existence Results for the Distributed Order Fractional Hybrid Differential Equations

and Applied Analysis 3 Definition 2.3. The distributed order fractional hybrid differential equation DOFHDEs , involving the Riemann-Liouville differential operator of order 0 < q < 1 with respect to the nonnegative density function b q > 0, is defined as ∫1 0 b ( q ) D [ x t f t, x t ] dq g t, x t , t ∈ J, ∫1 0 b ( q ) dq 1, x 0 0. 2.3 Moreover, the function t → x/f t, x is continuous for each x ∈ R, where J 0, T is bounded in R for some T ∈ R. Also, f ∈ C J × R,R \ {0} and g ∈ C J × R . 3. The Main Theorems In this section, we state the existence theorem for the DOFHDE 2.3 on J 0, T . For this purpose, we define a supremum norm of ‖ · ‖ in C J,R as ‖x‖ sup t∈J |x t |, 3.1 and for x, y ∈ C J,R ( xy ) t x t y t , 3.2 is a multiplication in this space. We consider C J,R is a Banach algebra with respect to norm ‖ · ‖ and multiplication 3.2 . Moreover the norm ‖ · ‖L1 for x ∈ C J,R is defined by ‖x‖L1 ∫T


Introduction
The differential equations involving Riemann-Liouville differential operators of fractional order 0 < q < 1 are very important in the modeling of several physical phenomena 1, 2 . In recent years, quadratic perturbations of nonlinear differential equations and first-order ordinary functional differential equations in Banach algebras, have attracted much attention to researchers. These type of equations have been called the hybrid differential equations 3-8 . One of the important first-order hybrid differential equations HDE is defined as 4, 9 d dt x t f t, x t g t, x t , t ∈ J, where J t 0 , t 0 a is a bounded interval in R for some t 0 and a ∈ R with a > 0. Also, f ∈ C J × R, R \ {0} and g ∈ C J × R , such that C J × R, R is the class of continuous 2 Abstract and Applied Analysis functions and C J × R is called the Caratheodory class of functions g : J × R → R which are Lebesgue integrable bounded by a Lebesgue integrable function on J. Moreover i the map t → g t, x is measurable for each x ∈ R, ii the map x → g t, x is continuous for each t ∈ J.
For the above hybrid differential equation, Dhage and Lakshmikantham 9 established existence, uniqueness, and some fundamental differential inequalities. Also, they stated some theoretical approximation results for the extremal solutions between the given lower and upper solutions 10 . Later, Zhao. et al. 11 developed the following fractional hybrid differential equations involving the Riemann-Liouville differential operators of order 0 < q < 1, They established the existence, uniqueness, and some fundamental fractional differential inequalities to prove existence of the extremal solutions of 1.2 . Also, they considered necessary tools under the mixed Lipschitz and Caratheodory conditions to prove the comparison principle. Now, in this article in view of the distributed order fractional derivative 12-14 , we develop the distributed order fractional hybrid differential equations DOFHDEs with respect to a nonnegative density function.
In this regard, in Section 2 we introduce the distributed order fractional hybrid differential equation. Section 3 is about some main theorems which are used in this paper. In Section 4, we prove the existence theorem for this class of equations, and we express some special cases for the density function used in the distributed order fractional hybrid differential equation. Finally, the main conclusions are set.

The Fractional Hybrid Differential Equation of Distributed Order
In this section, we recall some definitions which are used throughout this paper. Definition 2.1 see 1, 2 . The fractional integral of order q with the lower limit t 0 for the function f is defined as Definition 2.2 see 1, 2 . The Riemann-Liouville derivative of order q with the lower limit t 0 for the function f : t 0 , ∞ → R can be written as Abstract and Applied Analysis 3 Definition 2.3. The distributed order fractional hybrid differential equation DOFHDEs , involving the Riemann-Liouville differential operator of order 0 < q < 1 with respect to the nonnegative density function b q > 0, is defined as x 0 0.

The Main Theorems
In this section, we state the existence theorem for the DOFHDE 2.3 on J 0, T . For this purpose, we define a supremum norm of · in C J, R as and for x, y ∈ C J, R xy t x t y t , 3.2 is a multiplication in this space. We consider C J, R is a Banach algebra with respect to norm · and multiplication 3.2 . Moreover the norm · L 1 for x ∈ C J, R is defined by Now, for expressing the existence theorem for the DOFHDE 2.3 , we state a fixed point theorem in the Banach algebra.

Theorem 3.1 see 15 .
Let S be a nonempty, closed convex, and bounded subset of the Banach algebra X and let A : X → X and B : S → X be two operators such that

Then the operator equation AxBx x has a solution in S.
At this point, we consider some hypotheses as follows.
A 1 There exists a constant L > 0 such that for all t ∈ J and x, y ∈ R.
A 2 There exists a function h ∈ L 1 J, R and a real nonnegative upper bound h * such that for all t ∈ J and x ∈ R.
Theorem 3.2 Titchmarsh theorem 16 . Let F s be an analytic function which has a branch cut on the real negative semiaxis. Furthermore, F s has the following properties: for any sector | arg s | < π − η, where 0 < η < π. Then, the Laplace transform inversion f t can be written as the Laplace transform of the imaginary part of the function F re −iπ as follows:

Existence Theorem for the DOFHDEs
We apply the following lemma to prove the main existence theorem of this section.
Lemma 4.1. Assume that hypothesis (A 0 ) holds in pervious section, then for any h ∈ L 1 J, R and 0 < q < 1, the function x ∈ C J, R is a solution of the DOFHDE 2.3 if and only if x satisfies the following equation Proof. Applying the Laplace transform on both sides of 2.3 and letting Now, using the inverse Laplace transform on both sides of 4.6 and applying the convolution product, we get or equivalently Since B s is an analytic function which has a branch cut on the real negative semiaxis, according to the Titchmarsh Theorem 3.2 we get According to hypothesis A 0 , the map x → x/f 0, x is injective in R and hence x 0 0. Next, with dividing 4.9 by f t, x t and using the Laplace transform operator on both sides of this equation, 4.6 also holds. Since Y 0 0, we obtain 4.4 and by applying the inverse Laplace transform, 2.3 also holds. Proof. We set X C J, R as a Banach algebra and define a subset S of X by such that It is obvious that S is closed and if x 1 , x 2 ∈ R, then x 1 ≤ N and x 2 ≤ N, also by properties of the norm, we get Therefore, S is a convex and bounded and by applying Lemma 4.1, DOFHDE 2.3 is equivalent to 4.1 . Define operators A : X → X and B : S → X by

thus, from 4.1 , we obtain the operator equation as follows:
Ax t Bx t x t , t ∈ J.

4.17
If operators A and B satisfy all the conditions of Theorem 3.1, then the operator equation 4.17 has a solution in S. For this paper, let x, y ∈ X which by hypothesis A 1 we have and if for all x, y ∈ X take a supremum over t, then we have Therefore, A is a Lipschitz operator on X with the Lipschitz constant L > 0, and the condition a from Theorem 3.1 is held. Now, for checking the condition b from this theorem, first, we shall show that B is continuous on S. Let {x n } be a sequence in S such that lim n → ∞ x n x 4.20 8 Abstract and Applied Analysis with x ∈ S. By applying the Lebesgue-dominated convergence Theorem 3.5 for all t ∈ J, we get

4.21
Thus, B is a continuous operator on S. In next stage, we shall show that B is a compact operator on S. For this paper, we shall show that B s is a uniformly bounded and eqicontinuous set in X. Let x ∈ S, then by hypothesis A 2 for all t ∈ J we have

4.22
Let s t − τ such that 0 ≤ τ ≤ t ≤ T . Then by the existence Laplace transform theorem 19 , there exists a constant M > 0 such that for a constant c that s > c,

4.23
Hence, we find an upper bound for the integral of 4.22 as follows: Thus, B is uniformly bounded on S.
In this stage, now we show that B S is an equicontinuous set in X. Let t 1 , t 2 ∈ J, with t 1 < t 2 . In this respect, we have for all

4.28
If we set s 1 t 1 − τ and s 2 t 2 − τ, then by Laplace transform definition and 4.23 , for s 1 > c and s 2 > c we can write

4.29
10 Abstract and Applied Analysis Therefore, we have

4.30
Also, by 4.24 we have

4.31
Finally, with respect to 4.28 , 4.30 , and 4.31 we obtain

4.32
Hence, for > 0, there exists δ > 0 such that if |t 1 − t 2 | < δ, then for all t 1 , t 2 ∈ J and all x ∈ S we have |Bx t 1 − Bx t 2 | < , 4.33 which implies that B S is an equicontinuous set in X and according to the Arzela-Ascoli Theorem 3.4, B is compact. Therefore B is continuous and compact operator on S into X and B is a completely continuous operator on S and the condition b from the Theorem 3.1 is held.
For checking the condition c of Theorem 3.1, let x ∈ X and y ∈ S be arbitrary such that x AxBy. Then, by hypothesis A 1 we get

y τ dτ
Abstract and Applied Analysis 11

4.34
Therefore, which by taking a supremum over t, we obtain Thus, the condition c of Theorem 3.1 is satisfied. If we consider the hypothesis d of Theorem 3.1 is satisfied. Hence, all the conditions of Theorem 3.1 are satisfied and therefore the operator equation AxBx x has a solution in S. As a result, the DOFHDE 2.3 has a solution defined on J and proof is completed.

Some Special Cases
In this section, we discuss some special cases of the density function b q for the DOFHDE 2.3 and we find the operators A and B which introduce in Theorem 4.2. In proof of Lemma 4.1, the following equation is equivalent to the DOFHDE 2.3 , Thus, where Ei t is the exponential integral defined by Therefore, for this case, the DOFHDE 2.3 is x 0 0,

5.6
and it is equivalent to the following equation: 2 Two-term equation: Let b q a 1 δ q − q 1 a 2 δ q − 0 , which 0 < q 1 < 1 also, a 1 and a 2 are nonnegative constant coefficients and δ is the Dirac delta function. Then by the following inverse Laplace transform 2 : where E λ,μ z is the Mittag-Leffler function in two parameters Abstract and Applied Analysis 13 we get the DOFHDE 2.3 as x 0 0.

5.11
It is equivalent to the following equation such that the operators A and B in Theorem 4.2 are

5.17
We get the DOFHDE 2.3 as x t f t, x t g t, x t , x 0 0.

and the operators A and B in Theorem 4.2 are given by
Ax t f t, x t , Bx t t 0 G n t − τ g τ, x τ dτ. 5.26

Conclusions
In this paper, we introduced a new class; the fractional hybrid differential equations of distributed order and stated an existence theorem for it. We pointed out a fixed point theorem in the Banach algebra for the existence of solution. Basis of this theorem is on finding two operator equations which in special cases for multiterms fractional hybrid equations are given with respect to the derivatives of Mittag-Leffler function.