Existence Results for the Distributed Order Fractional Hybrid Differential Equations

Abstract

We introduce the distributed order fractional hybrid differential equations (DOFHDEs) involving the Riemann-Liouville differential operator of order 0<𝑞<1 with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved via a fixed point theorem in the Banach algebras under the mixed Lipschitz and Caratheodory conditions.

1. 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] 𝑑𝑥(𝑡)𝑑𝑡𝑓((𝑡((𝑡𝑥𝑡𝑡,𝑥))=𝑔𝑡,𝑥)),𝑡∈𝐽,₀=𝑥₀, (1.1) where 𝐽=[𝑡₀,𝑡₀+𝑎) is a bounded interval in ℝ for some 𝑡₀ 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 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, 𝐷^𝑞𝑥(𝑡)𝑓((𝑡((𝑡𝑥(0)𝑡,𝑥))=𝑔𝑡,𝑥)),𝑡∈𝐽,=0, (1.2) where 𝐽=[0,𝑇] is bounded in ℝ for some 𝑇∈ℝ and 𝑓∈𝐶(𝐽×ℝ,ℝ⧵{0}), 𝑔∈𝒞(𝐽×ℝ).

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.

2. 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 𝑡₀ for the function 𝑓 is defined as 𝐼^𝑞𝑓(𝑡)=1Γ𝑞^𝑡_𝑡0𝑓(𝑠)(𝑡−𝑠)^1−𝑞𝑑𝑠,𝑡>𝑡₀,𝑞>0. (2.1)

Definition 2.2 (see [1, 2]). The Riemann-Liouville derivative of order 𝑞 with the lower limit 𝑡₀ for the function 𝑓∶[𝑡₀,∞)→ℝ can be written as 𝐷^𝑞𝑓(𝑡)=1Γ𝑑𝑛−𝑞𝑑𝑡^𝑛𝑛𝑡_𝑡0𝑓(𝑠)(𝑡−𝑠)^{𝑞+1−𝑛}𝑑𝑠,𝑡>𝑡₀,𝑛−1<𝑞<𝑛. (2.2)

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 ¹₀𝑏𝑞𝐷^𝑞𝑥(𝑡)𝑓((𝑡((𝑡𝑡,𝑥))𝑑𝑞=𝑔𝑡,𝑥)),𝑡∈𝐽,¹₀𝑏𝑞𝑥(0)𝑑𝑞=1,=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=^𝑇₀|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, where 𝑀=‖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.  (𝐴₁) There exists a constant 𝐿>0 such that 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝐿𝑥−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)(1),|𝑠|𝐹(𝑠)1=𝑂→∞,=𝑂|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)}=1;𝑡𝜋^∞₀𝑒^−𝑟𝑡ℑ𝐹𝑟𝑒^−𝑖𝜋𝑑𝑟. (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 𝜖>0 there exists 𝛿>0 such that for all 𝑓∈𝐵 and 𝑎,𝑥∈𝑋𝑑(𝑥,𝑎)<𝛿⇒𝑓(𝑥)−𝑓(𝑎)<𝜖. (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 {𝑓_𝑛} be a sequence of real-valued 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 𝑓_𝑛(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 lim_𝑆𝑓_𝑛𝑑𝜇=_𝑆𝑓𝑑𝜇. (3.10)

4. 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 (𝐴₀) holds in pervious section, then for any ℎ∈𝐿¹(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 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡−𝜏𝑔(𝜏,𝑥(𝜏))𝑑𝜏 (4.1) such that 0≤𝜏≤𝑡≤𝑇 and 𝐵(𝑠)=¹₀𝑏𝑞𝑠^𝑞𝑑𝑞. (4.2)

Proof. Applying the Laplace transform on both sides of (2.3) and letting 𝑌(𝑡)=𝑥(𝑡)𝑓((𝑡,𝑡,𝑥)) (4.3) we have ℒ¹₀𝑏𝑞𝐷^𝑞𝑌(𝑡)𝑔((𝑡=𝑑𝑞;𝑠=ℒ𝑡,𝑥));𝑠₁₀𝑏𝑞𝑠^𝑞𝑌(𝑠)−𝐷_𝑡^𝑞−1𝑌(0)𝑑𝑞=𝐺(𝑠). (4.4) Since 𝑌(0)=0, we have 𝑌(𝑠)¹₀𝑏𝑞𝑠^𝑞𝑑𝑞=𝐺(𝑠), (4.5) and hence, 𝑌(𝑠)=1𝐵(𝑠)𝐺(𝑠), (4.6) such that 𝐵(𝑠)=¹₀𝑏𝑞𝑠^𝑞𝑑𝑞. (4.7) Now, using the inverse Laplace transform on both sides of (4.6) and applying the convolution product, we get ℒ⁻¹{𝑌(𝑠=𝑥(𝑡))}𝑓(𝑡,𝑥(𝑡))=ℒ⁻¹1𝐵(𝑠)𝐺(𝑠=_𝑡0)ℒ⁻¹1𝐵(𝑠);𝑡−𝜏𝑔(𝜏,𝑥(𝜏))𝑑𝜏, (4.8) or equivalently 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀ℒ⁻¹1𝐵(𝑠);𝑡−𝜏𝑔(𝜏,𝑥(𝜏))𝑑𝜏. (4.9) 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 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))𝜋^𝑡₀^∞₀𝑒^{−𝑟(𝑡−𝜏)}ℑ1𝐵𝑟𝑒^−𝑖𝜋𝑔(𝜏,𝑥(𝜏))𝑑𝑟𝑑𝜏, (4.10) which by the Laplace transform definition, (4.1) is held. Conversely, let 𝑥 satisfies (4.1), therefore, 𝑥 satisfies the equivalent equation (4.9). By 𝑡=0 in (4.1), we have 𝑥(0)𝑓((000,𝑥))=0=𝑓().0,0 (4.11) According to hypothesis (𝐴₀), 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.

Theorem 4.2. Suppose that hypothesis (𝐴₀)–(𝐴₂) hold. Further, if 𝐿𝑀‖ℎ‖_𝐿¹𝜋<1,𝑀>0, (4.12) then, the DOFHDE (2.3) has a solution defined on J.

Proof. We set 𝑋=𝐶(𝐽,ℝ) as a Banach algebra and define a subset 𝑆 of 𝑋 by 𝑆={𝑥∈𝑋∣‖𝑥‖≤𝑁}, (4.13) such that 𝐹𝑁=⁰𝑀‖ℎ‖_𝐿¹𝜋−𝐿𝑀‖ℎ‖_𝐿¹,𝐹₀=sup_𝑡∈𝐽𝑓(𝑡,0). (4.14) It is obvious that 𝑆 is closed and if 𝑥₁,𝑥₂∈ℝ, then ‖𝑥₁‖≤𝑁 and ‖𝑥₂‖≤𝑁, also by properties of the norm, we get 𝜆𝑥₁+(1−𝜆)𝑥₂≤𝜆𝑥₁+(1−𝜆)𝑥₂≤𝜆𝑁+(1−𝜆)𝑁=𝑁. (4.15) 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 (𝑡)((𝑡(𝑡)=1𝐴𝑥=𝑓𝑡,𝑥)),𝑡∈𝐽,𝐵𝑥𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡−𝜏𝑔(𝜏,𝑥(𝜏))𝑑𝜏,𝑡∈𝐽, (4.16) 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 (𝐴₁) we have 𝐴𝑥(𝑡)−𝐴𝑦(𝑡)=𝑓(𝑡,𝑥(𝑡))−𝑓𝑡,𝑦(𝑡)≤𝐿𝑥(𝑡)−𝑦(𝑡)≤𝐿𝑥−𝑦,𝑡∈𝐽, (4.18) and if for all 𝑥,𝑦∈𝑋 take a supremum over 𝑡, then we have 𝐴𝑥−𝐴𝑦≤𝐿𝑥−𝑦. (4.19) 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) with 𝑥∈𝑆. By applying the Lebesgue-dominated convergence Theorem 3.5 for all 𝑡∈𝐽, we get lim_𝑛→∞𝐵𝑥_𝑛(𝑡)=lim_𝑛→∞1𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡−𝜏𝑔𝜏,𝑥_𝑛(𝜏=1)𝑑𝜏𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡−𝜏lim_𝑛→∞𝑔𝜏,𝑥_𝑛(𝜏)=1𝑑𝜏𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋𝑔((𝜏(𝑡).;𝑡−𝜏𝜏,𝑥))𝑑𝜏=𝐵𝑥 (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 (𝐴₂) for all 𝑡∈𝐽 we have |(𝑡=1𝐵𝑥)|𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋𝑔((𝜏≤1;𝑡−𝜏𝜏,𝑥))𝑑𝜏𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡−𝜏|ℎ(𝜏)|𝑑𝜏. (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, ℑ1𝐵𝑟𝑒^−𝑖𝜋≤𝑀^′𝑒^𝑐𝑟. (4.23) Hence, we find an upper bound for the integral of (4.22) as follows: ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡−𝜏=^∞₀𝑒^−𝑠𝑟ℑ1𝐵𝑟𝑒^−𝑖𝜋≤𝑑𝑟_∞0𝑒^−𝑠𝑟ℑ1𝐵𝑟𝑒^−𝑖𝜋𝑑𝑟≤^∞₀𝑀^′𝑒^{(𝑐−𝑠)𝑟}𝑀𝑑𝑟≤^′|𝑠−𝑐|≤𝑀, (4.24) such that 𝑀=sup_{0≤𝜏≤𝑡≤𝑇}𝑀^′||.𝑡−𝜏−𝑐 (4.25) Finally, with respect to the inequality (4.22) we obtain |(𝑡≤𝑀‖ℎ‖𝐵𝑥)|^𝐿1𝜋, (4.26) which by applying supremum over 𝑡, we get for all 𝑥∈𝑆‖‖≤𝑀𝐵𝑥𝜋‖ℎ‖_𝐿¹. (4.27) Thus, 𝐵 is uniformly bounded on 𝑆.
In this stage, now we show that 𝐵(𝑆) is an equicontinuous set in 𝑋. Let 𝑡₁,𝑡₂∈𝐽, with 𝑡₁<𝑡₂. In this respect, we have for all 𝑥∈𝑆𝐵𝑥𝑡₁−𝐵𝑥𝑡₂=1𝜋^𝑡1₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₁𝑔((𝜏−1−𝜏𝜏,𝑥))𝑑𝜏𝜋^𝑡2₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₂𝑔((𝜏≤1−𝜏𝜏,𝑥))𝑑𝜏𝜋^𝑡1₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₁𝑔((𝜏−1−𝜏𝜏,𝑥))𝑑𝜏𝜋^𝑡1₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₂𝑔((𝜏+1−𝜏𝜏,𝑥))𝑑𝜏𝜋^𝑡1₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₂𝑔((𝜏−1−𝜏𝜏,𝑥))𝑑𝜏𝜋^𝑡2₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₂−𝜏𝑔(𝜏,𝑥(𝜏))𝑑𝜏. (4.28) If we set 𝑠₁=𝑡₁−𝜏 and 𝑠₂=𝑡₂−𝜏, then by Laplace transform definition and (4.23), for 𝑠₁>𝑐 and 𝑠₂>𝑐 we can writeℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑠₁ℑ1−ℒ𝐵𝑟𝑒^−𝑖𝜋;𝑠₂=^∞₀𝑒^−𝑠1𝑟ℑ1𝐵𝑟𝑒^−𝑖𝜋𝑑𝑟−^∞₀𝑒^−𝑠2𝑟ℑ1𝐵𝑟𝑒^−𝑖𝜋≤𝑑𝑟_∞0𝑒^−𝑠1𝑟−𝑒^−𝑠2𝑟ℑ1𝐵𝑟𝑒^−𝑖𝜋𝑑𝑟≤𝑀^′∞₀𝑒^{(𝑐−𝑠1)𝑟}−𝑒^{(𝑐−𝑠2)𝑟}𝑑𝑟=𝑀^′1𝑠₁−1−𝑐𝑠₂.−𝑐 (4.29) Therefore, we have 1𝜋^𝑡1₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑠₁ℑ1−ℒ𝐵𝑟𝑒^−𝑖𝜋;𝑠₂𝑔((𝜏≤ℎ𝜏,𝑥))𝑑𝜏_∗𝜋^𝑡1₀𝑀^′1𝑡₁−1−𝜏−𝑐𝑡₂=𝑀−𝜏−𝑐𝑑𝜏_′ℎ^∗𝜋ln𝑐+𝑡₁−𝑡₂𝑐−𝑡₁𝑐𝑐−𝑡₂. (4.30) Also, by (4.24) we have 1𝜋^𝑡1_𝑡2ℒℑ1𝐵𝑟𝑒^−𝑖𝜋;𝑡₂𝑔((𝜏≤ℎ−𝜏𝜏,𝑥))𝑑𝜏_∗𝜋^𝑡1_𝑡2𝑀^′𝑡₂𝑀−𝜏−𝑐𝑑𝜏=^′ℎ^∗𝜋𝑐ln𝑐+𝑡₁−𝑡₂. (4.31) Finally, with respect to (4.28), (4.30), and (4.31) we obtain 𝐵𝑥𝑡₁−𝐵𝑥𝑡₂≤𝑀^′ℎ^∗𝜋ln𝑐+𝑡₁−𝑡₂𝑐−𝑡₁𝑐𝑐−𝑡₂𝑐+ln𝑐+𝑡₁−𝑡₂=𝑀^′ℎ^∗𝜋ln𝑐−𝑡₁𝑐−𝑡₂. (4.32) Hence, for 𝜖>0, there exists 𝛿>0 such that if |𝑡₁−𝑡₂|<𝛿, then for all 𝑡₁,𝑡₂∈𝐽 and all 𝑥∈𝑆 we have 𝐵𝑥𝑡₁−𝐵𝑥𝑡₂<𝜖, (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 (𝐴₁) we get |𝑥(𝑡=|(𝑡(𝑡=𝑓((𝑡1)|𝐴𝑥)|𝐵𝑦)𝑡,𝑥))𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋𝑔(𝜏)≤𝑓((𝑡(+𝑓(1;𝑡−𝜏𝜏,𝑦𝑑𝜏𝑡,𝑥))−𝑓𝑡,0)𝑡,0)𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋𝑔(𝜏)≤𝐿|𝑥(𝑡;𝑡−𝜏𝜏,𝑦𝑑𝜏)|+𝐹₀1𝜋^𝑡₀ℒℑ1𝐵𝑟𝑒^−𝑖𝜋ℎ(𝜏)≤𝐿|𝑥(𝑡;𝑡−𝜏𝑑𝜏)|+𝐹₀𝑀‖ℎ‖_𝐿¹𝜋. (4.34) Therefore, |𝑥(𝑡≤𝐹)|⁰𝑀‖ℎ‖_𝐿¹𝜋−𝐿𝑀‖ℎ‖_𝐿¹, (4.35) which by taking a supremum over 𝑡, we obtain ‖𝑥(𝑡≤𝐹)‖⁰𝑀‖ℎ‖_𝐿¹𝜋−𝐿𝑀‖ℎ‖_𝐿¹=𝑁. (4.36) Thus, the condition (c) of Theorem 3.1 is satisfied. If we consider 𝑀₁=‖𝐵(𝑠‖}≤𝑀)‖=sup{‖𝐵𝑥∶𝑥∈𝑆𝜋‖ℎ‖_𝐿¹,𝛼𝑀₁𝑀≤𝐿𝜋‖ℎ‖_𝐿¹<1, (4.37) 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.

5. 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), 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀ℒ⁻¹1𝐵(𝑠);𝑡−𝜏𝑔(𝜏,𝑥(𝜏))𝑑𝜏, (5.1) such that, 𝐵(𝑠)=¹₀𝑏𝑞𝑠^𝑞𝑑𝑞. (5.2)

(1) Let 𝑏(𝑞)=1. Then we have 𝐵(𝑠)=¹₀𝑠^𝑞𝑑𝑞=𝑠−1(𝑠).ln (5.3) Thus, ℒ⁻¹1𝐵(𝑠);𝑡−𝜏=𝑒^𝑡−𝜏Ei(𝑡−𝜏), (5.4) where Ei(𝑡) is the exponential integral defined by Ei(𝑡)=^∞_𝑡𝑒^−𝑢𝑢𝑑𝑢. (5.5) Therefore, for this case, the DOFHDE (2.3) is ¹₀𝐷^𝑞𝑥(𝑡)𝑓((𝑡((𝑡𝑥(0)𝑡,𝑥))𝑑𝑞=𝑔𝑡,𝑥)),𝑡∈𝐽,=0, (5.6) and it is equivalent to the following equation: 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀𝑒^𝑡−𝜏Ei(𝑡−𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏, (5.7) such that the operators 𝐴 and 𝐵 in Theorem 4.2 are 𝐴𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝐵𝑥(𝑡)=^𝑡₀𝑒^𝑡−𝜏Ei(𝑡−𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏. (5.8)

(2) Two-term equation: Let 𝑏(𝑞)=𝑎₁𝛿(𝑞−𝑞₁)+𝑎₂𝛿(𝑞−0), which 0<𝑞₁<1 also, 𝑎₁ and 𝑎₂ are nonnegative constant coefficients and 𝛿 is the Dirac delta function. Then by the following inverse Laplace transform [2]: ℒ⁻¹1𝐵(𝑠);𝑡−𝜏=ℒ⁻¹1𝑎₁𝑠^𝑞1+𝑎₂=1;𝑡−𝜏𝑎₁(𝑡−𝜏)^𝑞1−1𝐸_𝑞1,𝑞1−𝑎₂𝑎₁(𝑡−𝜏)^𝑞1, (5.9) where 𝐸_𝜆,𝜇(𝑧) is the Mittag-Leffler function in two parameters 𝐸_𝜆,𝜇(𝑧)=^∞_𝑗=0𝑧^𝑗Γ𝜆𝑗+𝜇𝜆,𝜇>0,𝑧∈ℂ (5.10) we get the DOFHDE (2.3) as 𝑎₁𝐷^𝑞1𝑥(𝑡)𝑓(𝑡,𝑥(𝑡))+𝑎₂𝑥(𝑡)𝑓((𝑡((𝑡𝑥(0)𝑡,𝑥))=𝑔𝑡,𝑥)),𝑡∈𝐽,=0. (5.11) It is equivalent to the following equation 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀1𝑎₁(𝑡−𝜏)^𝑞1−1𝐸_𝑞1,𝑞1−𝑎₂𝑎₁(𝑡−𝜏)^𝑞1𝑔(𝜏,𝑥(𝜏))𝑑𝜏, (5.12) such that the operators 𝐴 and 𝐵 in Theorem 4.2 are (𝑡)((𝑡(𝑡)=1𝐴𝑥=𝑓𝑡,𝑥)),𝐵𝑥𝑎₁^𝑡₀(𝑡−𝜏)^𝑞1−1𝐸_𝑞1,𝑞1−𝑎₂𝑎₁(𝑡−𝜏)^𝑞1𝑔(𝜏,𝑥(𝜏))𝑑𝜏. (5.13)

(3) Three-term equation: Let, 𝑏𝑞=𝑎₁𝛿𝑞−𝑞₁+𝑎₂𝛿𝑞−𝑞₂+𝑎₃𝛿𝑞−0, (5.14) which 1>𝑞₁>𝑞₂>0 and 𝑎₁, 𝑎₂, and 𝑎₃ are nonnegative constant coefficients and 𝛿 is the Dirac delta function. Then, by virtue of [2] ℒ⁻¹1𝐵(𝑠);𝑡−𝜏=ℒ⁻¹1𝑎₁𝑠^𝑞1+𝑎₂𝑠^𝑞2+𝑎₃;𝑡−𝜏=𝐺₃(𝑡−𝜏), (5.15) where 𝐺₃()=1𝑡−𝜏𝑎₁^∞_𝑘=0(−1)^𝑘𝑎𝑘!𝑎³₁^𝑘(𝑡−𝜏)^{𝑞1(𝑘+1)−1}𝐸_𝑞^(𝑘)_{1−𝑞2,𝑞1+𝑘𝑞2}−𝑎₂𝑎₁(𝑡−𝜏)^{𝑞1−𝑞2}, (5.16) and 𝐸^(𝑘)_𝜆,𝜇(𝑧) is the 𝑘th derivative of the Mittag-Leffler function in two parameters 𝐸^(𝑘)_𝜆,𝜇(𝑧)≡𝑑^𝑘𝑑𝑧^𝑘𝐸_𝜆,𝜇(𝑧)=^∞_𝑗=0𝑗+𝑘!𝑧^𝑗𝑗!Γ𝜆𝑗+𝜆𝑘+𝜇,𝑘=0,1,2,…. (5.17) We get the DOFHDE (2.3) as 𝑎₁𝐷^𝑞1𝑥(𝑡)𝑓(𝑡,𝑥(𝑡))+𝑎₂𝐷^𝑞2𝑥(𝑡)𝑓(𝑡,𝑥(𝑡))+𝑎₃𝑥(𝑡)𝑓((𝑡((𝑡,𝑥(0)𝑡,𝑥))=𝑔𝑡,𝑥))=0. (5.18) It is equivalent to the following equation: 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀𝐺₃(𝑡−𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏, (5.19) such that the operators 𝐴 and 𝐵 in Theorem 4.2 are 𝐴𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝐵𝑥(𝑡)=^𝑡₀𝐺₃(𝑡−𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏. (5.20)

(4) General Case: 𝑛-term equation: suppose that 𝑏𝑞=𝑎₀𝛿𝑞−𝑞₀+𝑎₁𝛿𝑞−𝑞₁+𝑎₂𝛿𝑞−𝑞₂+⋯+𝑎_𝑛𝛿𝑞−𝑞_𝑛, (5.21) which 1>𝑞_𝑛>𝑞_𝑛−1>⋯>𝑞₀>0 and 𝑎_𝑖 for 𝑖=0,1,2,…,𝑛 are nonnegative constant coefficients. Therefore, by the following inverse Laplace transform [2], we have ℒ⁻¹1𝐵(𝑠);𝑡−𝜏=ℒ⁻¹1𝑎₀𝑠^𝑞0+𝑎₁𝑠^𝑞1+⋯+𝑎_𝑛𝑠^𝑞𝑛;𝑡−𝜏=𝐺_𝑛(𝑡−𝜏), (5.22) where𝐺_𝑛()=1𝑡−𝜏𝑎_𝑛^∞_𝑚=0(−1)^𝑚𝑚!_{𝑘0+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚;𝑘₀,𝑘₁,…,𝑘_𝑛−2×^𝑛−2_𝑖=0𝑎_𝑖𝑎_𝑛^𝑘𝑖(𝑡−𝜏)^{(𝑞𝑛−𝑞𝑛−1)𝑚+𝑞𝑛+∑𝑛−2𝑗=0(𝑞𝑛−1−𝑞𝑗)𝑘𝑗−1}×𝐸_𝑞^(𝑚)_{𝑛−𝑞𝑛−1,+𝑞𝑛+∑𝑛−2𝑗=0(𝑞𝑛−1−𝑞𝑗)𝑘𝑗}−𝑎_𝑛−1𝑎_𝑛(𝑡−𝜏)^{𝑞𝑛−𝑞𝑛−1}. (5.23) Thus, for this case, the DOFHDE (2.3) is 𝑎₀𝐷^𝑞0𝑥(𝑡)𝑓(𝑡,𝑥(𝑡))+𝑎₁𝐷^𝑞1𝑥(𝑡)𝑓(𝑡,𝑥(𝑡))+⋯+𝑎_𝑛𝐷^𝑞𝑛𝑥(𝑡)𝑓((𝑡((𝑡,𝑥(0)𝑡,𝑥))=𝑔𝑡,𝑥))=0. (5.24) It is equivalent to the following equation: 𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))^𝑡₀𝐺_𝑛(𝑡−𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏 (5.25) and the operators 𝐴 and 𝐵 in Theorem 4.2 are given by 𝐴𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝐵𝑥(𝑡)=^𝑡₀𝐺_𝑛(𝑡−𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏. (5.26)

6. 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.

Acknowledgment

The authors have been partially supported by the Center of Excellence for Mathematics, Shahrekord University.