Existence Results for the Distributed Order Fractional Hybrid Differential Equations

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 ∫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 0 |x s |ds. 3.3 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 a A is Lipschitz constant α, b B is completely continuous, c x AxBy ⇒ x ∈ S for all y ∈ S, d αM < 1, whereM ‖B S ‖ sup{‖B x ‖ : x ∈ S}. Then the operator equation AxBx x has a solution in S. At this point, we consider some hypotheses as follows. A0 The function x → x/f t, x is increasing in R almost everywhere for t ∈ J . 4 Abstract and Applied Analysis A1 There exists a constant L > 0 such that ∣∣f t, x − f(t, y)∣∣ ≤ L∣∣x − y∣∣, 3.4 for all t ∈ J and x, y ∈ R. A2 There exists a function h ∈ L1 J,R and a real nonnegative upper bound h∗ such that ∣∣g t, x ∣∣ ≤ h t ≤ h∗, 3.5 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: F s O 1 , |s| −→ ∞, F s O ( 1 |s| ) , |s| −→ 0, 3.6 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: f t L−1{F s ; t} 1 π ∫∞ 0 e−rt ( F ( re−iπ )) dr. 3.7 Definition 3.3. Suppose that X, d be a metric space and let B ⊆ C X,R . Then, B is equicontinuous if for all > 0 there exists δ > 0 such that for all f ∈ B and a, x ∈ X d x, a < δ ⇒ ∣∣f x − f a ∣∣ < . 3.8 Theorem 3.4 Arzela-Ascoli theorem 17 . Let X, d be a compact metric space and let B ⊂ C X,R . Then, B is compact if and only if B is closed, bounded, and equicontinuous. Theorem 3.5 Lebesgue dominated convergence theorem 18 . Let {fn} be a sequence of realvalued measurable functions on a measure space S,Σ, μ . Also, suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that ∣∣fn x ∣∣ ≤ g x , 3.9 for all numbers n in the index set of the sequence and all points x in S. Then, f is integrable and

Cite this paper

@inproceedings{Noroozi2014ExistenceRF, title={Existence Results for the Distributed Order Fractional Hybrid Differential Equations}, author={Hossein Noroozi and Alireza Ansari and Mohammad Shafi Dahaghin and Yongfu Su}, year={2014} }