Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

Abstract

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations (𝒟^𝛼−𝜌𝑡𝒟^𝛽)𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)), 𝑡∈(0,1) with boundary conditions 𝑥(0)=𝑥₀,𝑥(1)=𝑥₁ or satisfying the initial conditions 𝑥(0)=0,𝑥^′(0)=1, where 𝒟^𝛼 denotes Caputo fractional derivative, 𝜌 is constant, 1<𝛼<2, and 0<𝛽+𝛾≤𝛼. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions on 𝑓.

1. Introduction

Fractional calculus deals with generalization of differentiation and integration to the fractional order [1, 2]. In the last few decades the fractional calculus and fractional differential equations have found applications in various disciplines [2–6]. Owing to the increasing applications, a considerable attention has been given to exact and numerical solutions of fractional differential equations [2, 6–11]. Many papers were dedicated to the existence and the uniquenesss of the fractional differential equations, to the analytic methods for solving fractional differential equations, e.g., Greens function method, the Mellin transform method, and the power series (see for example references [2, 6–26] and the references therein). On this line of taught in this manuscript we proved the existence and uniqueness of a specific nonlinear fractional order ordinary differential equations within Caputo derivatives. Very recently in [27–31], the authors and other researchers studied the existence and uniqueness of solutions of some classes of fractional differential equations with delay. The paper is organized as follows: In Section 2 we introduce some necessary definitions and mathematical preliminaries of fractional calculus. In Section 3 sufficient conditions are established for the existence and uniqueness of solutions for a class fractional order differential equations satisfying the boundary conditions or satisfying the initial conditions. In order to illustrate our results several examples are presented in Section 3.

2. Fractional Integral and Derivatives

In this section, we present some notations, definitions, and preliminary facts that will be used further in this work. The Caputo fractional derivative allows the utilization of initial and boundary conditions involving integer order derivatives, which have clear physical interpretations. Therefore, in this work we will use the Caputo fractional derivative 𝒟 proposed by Caputo in his work on the theory of viscoelasticity [32].

Let 𝛼∈ℝ, 𝑛−1<𝛼≤𝑛∈ℕ and 𝑥∈𝐶((0,∞),ℝ); then the Caputo fractional derivative of order 𝛼 defined by 𝒟^𝛼𝑥(𝑡)=ℐ^𝑛−𝛼𝑑^𝑛𝑥(𝑡)𝑑𝑡^𝑛, (2.1) where ℐ^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠, (2.2) is the Riemann-Liouville fractional integral operator of order 𝛼 and Γ is the gamma function.

The fractional integral of 𝑥(𝑡)=(𝑡−𝑎)^𝛽, 𝑎≥0, 𝛽>−1 is given as ℐ^𝛼𝑥(𝑡)=Γ𝛽+1Γ()𝛽+𝛼+1𝑡−𝑎^𝛽+𝛼. (2.3) For 𝛼,𝛽≥0, we have the following properties of fractional integrals and derivative [33].

The fractional order integral satisfies the semigroup property ℐ^𝛼ℐ^𝛽𝑥(𝑡)=ℐ^𝛽ℐ^𝛼𝑥(𝑡)=ℐ^𝛼+𝛽𝑥(𝑡). (2.4) The integer order derivative operator 𝒟^𝑚 commutes with fractional order 𝒟^𝛼, that is: 𝒟^𝑚𝒟^𝛼𝑥(𝑡)=𝒟^𝑚+𝛼𝑥(𝑡)=𝒟^𝛼𝒟^𝑚𝑥(𝑡). (2.5) The fractional operator and fractional derivative operator do not commute in general. Then the following result can be found in [33, 34].

Lemma 2.1 (see [33, 34]). For 𝛼>0, the general solution of the fractional differential equation 𝒟^𝛼𝑥(𝑡)=0 is given by 𝑥(𝑡)=^𝑟−1_𝑖=0𝑐_𝑖𝑡^𝑖,𝑐_𝑖∈ℝ,𝑖=0,1,2,…,𝑟−1,𝑟=[𝛼]+1, (2.6) where [𝛼] denotes the integer part of the real number 𝛼.
In view of Lemma 2.1 it follows that ℐ^𝛼𝒟^𝛼𝑥(𝑡)=𝑥(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑟−1𝑡^𝑟−1𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑐_𝑖∈ℝ,𝑖=0,1,…,𝑟−1. (2.7)

But in the opposite way we have 𝒟^𝛼ℐ^𝛽(𝑡)=𝒟^𝛼−𝛽𝑥(𝑡). (2.8)

Proposition 2.2. Assume that 𝑥∶[0,∞)→ℝ is continuous and 0<𝛽≤𝛼. Then (i)ℐ^𝛼(𝑡𝑥(𝑡))=𝑡ℐ^𝛼𝑥(𝑡)−𝛼ℐ^𝛼+1𝑥(𝑡),(ii)ℐ^𝛼{𝑡𝒟^𝛽𝑥(𝑡)}=𝑡ℐ^𝛼−𝛽𝑥(𝑡)−𝛼ℐ^{𝛼−𝛽+1}𝑥(𝑡).

The proof of the above proposition can be found in [9, page 53].

As a pursuit of this in this paper, we discuss the existence and uniqueness of solution for nonlinear fractional order differential equations 𝒟^𝛼−𝜌𝑡𝒟^𝛽𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡),𝑡∈(0,1), (2.9) satisfying the boundary conditions 𝑥(0)=𝑥₀,𝑥(1)=𝑥₁, (2.10) or satisfying the initial conditions 𝑥(0)=𝑥₀,𝑥^′(0)=1, (2.11) where 1<𝛼≤2 and 0<𝛽+𝛾≤𝛼.

In the following, we present the existence and the uniqueness results for fractional differential equation (2.9) with boundary conditions (2.10).

3. Existence and Uniqueness of Solutions

Lemma 3.1. Assume that 𝑓∶[0,1]×ℝ²→ℝ is continuous. Then 𝑥∈𝐶[0,1] is a solution of the boundary value problem (2.9) and (2.10) if and only if 𝑥(𝑡) is the solution of the integral equation 𝑥(𝑡)=−𝑐₀−𝑐₁𝑡+𝜌𝑡𝐼^𝛼−𝛽𝑥(𝑡)−𝜌𝛼𝐼^{𝛼−𝛽+1}𝑥(𝑡)+𝐼^𝛼𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)=𝑥₀+𝑥₁−𝑥₀𝑡+¹₀𝒢(𝑡,𝑠)𝑑𝑠, (3.1) for some constants 𝑐₀,𝑐₁ where 𝒢(𝑡,𝑠) given by 𝒢()=𝒢𝑡,𝑠¹()𝒢𝑡,𝑠0≤𝑠<𝑡,₂(𝑡,𝑠),𝑡≤𝑠≤1, (3.2) where 𝒢₁()()𝑡,𝑠=𝜌𝛼𝑡1−𝑠^𝛼−𝛽Γ−𝑡()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ+𝑡()𝛼−𝛽𝑡−𝑠^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽𝑡−𝑠^𝛼−𝛽Γ𝑥(𝑠)+()𝛼−𝛽+1𝑡−𝑠^𝛼−1Γ(𝛼)−𝑡(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠),𝒢₂()𝛼()𝑡,𝑠=𝜌𝑡1−𝑠^𝛼−𝛽Γ−()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠)−𝑡()𝛼−𝛽1−𝑠^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠). (3.3)

Proof. Assume that 𝑥∈𝐶[0,1] is a solution of the fractional differential equation (2.9) satisfying boundary conditions (2.10). Then in view of Lemma 2.1 and Proposition 2.2, we have 𝑥(𝑡)=𝜌𝑡𝐼^𝛼−𝛽𝑥(𝑡)−𝜌𝛼𝐼^{𝛼−𝛽+1}𝑥(𝑡)+𝐼^𝛼𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)−𝑐₀−𝑐₁𝑡=𝜌^𝑡₀𝑡(𝑡−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽𝑡−𝑠^𝛼−𝛽Γ𝑥(𝑠)𝛼−𝛽+1𝑑𝑠+^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠−𝑐₀−𝑐₁𝑡, (3.4) for some constants 𝑐₀ and 𝑐₁. Hence using the boundary conditions (2.10) we obtain 𝑐₀=−𝑥₀ and 𝑐₁=𝑥₀−𝑥₁+𝜌¹₀(1−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽1−𝑠^𝛼−𝛽Γ𝑥(𝑠)+𝛼−𝛽+1𝑑𝑠₁₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠. (3.5) Substituting 𝑐₀=−𝑥₀ and 𝑐₁ into (3.4) we get 𝑥(𝑡)=𝑥₀+𝑥₁−𝑥₀𝑡−𝜌𝑡¹₀(1−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽1−𝑠^𝛼−𝛽Γ𝑥(𝑠)𝛼−𝛽+1𝑑𝑠−𝑡¹₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)+𝑑𝑠_𝑡0𝑡(𝑡−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽𝑡−𝑠^𝛼−𝛽Γ𝑥(𝑠)𝛼−𝛽+1𝑑𝑠+𝜌^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠,=𝑥₀+𝑥₁−𝑥₀𝑡+𝜌^𝑡₀𝛼𝑡(1−𝑠)^𝛼−𝛽Γ−𝑡()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ+𝑡()𝛼−𝛽𝑡−𝑠^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽𝑡−𝑠^𝛼−𝛽Γ𝑥(𝑠)+𝛼−𝛽+1𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)−𝑡(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠+𝜌¹_𝑡𝛼𝑡(1−𝑠)^𝛼−𝛽Γ−𝑡()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠)−𝛼−𝛽𝑑𝑠_1𝑡𝑡(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠=𝑥₀+𝑥₁−𝑥₀𝑡+¹₀𝒢(𝑡,𝑠)𝑑𝑠. (3.6)(3.7)(3.8)(3.9)
We consider the space ℬ=𝑥(𝑡)∶𝑥(𝑡)∈𝐶[0,1],𝒟^𝛾𝑥(𝑡)∈𝐶[0,1], (3.10) furnished with the norm ‖𝑥(𝑡)‖=max_[]𝑡∈0,1|𝑥(𝑡)|+max_[]𝑡∈0,1𝒟^𝛾𝑥(𝑡). (3.11) The space ℬ is a Banach space [35].

Theorem 3.2. Let 𝑓∶[0,1]×ℝ²→ℝ be continuous, and there exists a function 𝜂∶[0,1]→[0,∞], such that 𝑓(𝑡,𝑥,𝑦)≤𝜂(𝑡)+𝑎|𝑥|+𝑏|𝑦|, 𝑎,𝑏≥0, 2𝑎+2𝑏+𝛼|𝜌|≤2𝛿 where 𝛿=min{Γ(𝛼−𝛽−𝛾+2),Γ(𝛼−𝛽−𝛾+1),Γ(𝛼−𝛾+1),Γ(𝛼+1)}. Then, the boundary value problem (2.9), (2.10) has a solution.

Proof. Define an operator ℱ∶ℬ→ℬ by ℱ𝑥(𝑡)=𝑥₀+𝑥₁−𝑥₀𝑡−𝑡¹₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠+𝜌𝑡¹₀𝛼(1−𝑠)^𝛼−𝛽Γ−()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠)𝛼−𝛽𝑑𝑠+𝜌𝑡𝐼^𝛼−𝛽𝑥(𝑡)−𝜌𝛼𝐼^{𝛼−𝛽+1}𝑥(𝑡)+𝐼^𝛼𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)=𝑥₀+𝑥₁−𝑥₀𝑡+¹₀𝒢(𝑡,𝑠)𝑑𝑠. (3.12)(3.13)(3.14)(3.15) In order to show that the boundary value problem (2.9), (2.10) has a solution, it is sufficient to prove that the operator ℱ has a fixed point. For 𝑠≤𝑡, from (3.2), we have |𝒢(≤𝜌()𝑡,𝑠)|2𝛼1−𝑠^𝛼−𝛽Γ+2()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠+2()𝛼−𝛽|)|1−𝑠^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)≤𝑚₁𝛼(1−𝑠)^𝛼−𝛽+(1−𝑠)^{𝛼−𝛽−1}|𝑥(𝑠)|+𝑚₁(1−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)≤𝑚₁(𝛼+1)(1−𝑠)^{𝛼−𝛽−1}|𝑥(𝑠)|+(1−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)≤𝑚₁(1−𝑠)^{𝛼−𝛽−1}3|𝑥(𝑠)|+𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)≤𝑚₁(1−𝑠)^{𝛼−𝛽−1}(3+𝑎)|𝑥(𝑠)|+𝜂(𝑠)+𝑏𝒟^𝛾𝑥(𝑠)≤𝑚₁𝑚₂(1−𝑠)^{𝛼−𝛽−1}, (3.16) where 𝑚₁2𝜌=maxΓ,2𝜌𝛼−𝛽+1Γ,2𝛼−𝛽Γ(𝛼),𝑚₂=max(3+𝑎)|𝑥(𝑠)|,𝜂(𝑠),𝑏𝒟^𝛾𝑥(𝑠),0≤𝑠≤1. (3.17) On the other hand, for 𝑠>𝑡, we arrive at same conclusion. Therefore, ¹₀|𝒢(𝑡,𝑠)|𝑑𝑠≤𝑚₁𝑚₂¹₀(1−𝑠)^{𝛼−𝛽−1}𝑚𝑑𝑠=¹𝑚₂𝛼−𝛽. (3.18) Choose ℛ≥max{ℛ₁,ℛ₂}, where ℛ₁=max{𝑚₁𝑚₂/2(𝛼−𝛽),(1/2)(2|𝑥₀|+|𝑥₁|)} and ℛ₂5𝑥=max¹−𝑥₀,5𝜂2Γ1−𝛾,5𝜂2Γ𝛼−𝛾+1(),5𝜌𝛼2Γ𝛼+1,5𝜌2Γ𝛼−𝛽+22Γ𝛼−𝛽+1. (3.19) Define the set Ω={𝑥∈ℬ∶‖𝑥‖≤8ℛ}. For 𝑥∈Ω, using (3.15) and (3.18), we obtain |ℱ𝑥(𝑡)|≤𝑥₀+𝑥₁−𝑥₀𝑡+¹₀|𝒢(𝑡,𝑠)|𝑑𝑠≤2𝑥₀+𝑥₁+𝑚₁𝑚₂𝛼−𝛽≤2ℛ+2ℛ=4ℛ. (3.20) From the Caputo derivative and with using (3.12)–(3.14), we have 𝒟^𝛾(ℱ𝑥(𝑡))=𝐼^1−𝛾𝑑ℱ𝑥(𝑡)𝑑𝑡=−𝐼^1−𝛾𝑑𝑡𝑑𝑡¹₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠+𝐼^1−𝛾𝑑𝑑𝑡𝜌𝑡¹₀𝛼(1−𝑠)^𝛼−𝛽Γ−()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠)𝛼−𝛽𝑑𝑠+𝐼^1−𝛾𝑑𝑥𝑑𝑡⁰+𝑥₁−𝑥₀𝑡+𝜌𝑡𝐼^𝛼−𝛽𝑥(𝑡)−𝜌𝛼𝐼^{𝛼−𝛽+1}𝑥(𝑡)+𝐼^1−𝛾𝑑𝐼𝑑𝑡^𝛼𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)=−𝐼^1−𝛾1₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)𝑑𝑠+𝜌𝐼^1−𝛾1₀𝛼(1−𝑠)^𝛼−𝛽Γ−()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠)𝛼−𝛽𝑑𝑠+𝐼^1−𝛾𝑥₁−𝑥₀+𝜌(1−𝛼)𝐼^𝛼−𝛽𝑥(𝑡)+𝜌𝑡𝐼^{𝛼−𝛽−1}𝑥(𝑡)+𝐼^𝛼−1𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡). (3.21) Then, (2.3) yields 𝒟^𝛾((𝑡𝑡ℱ𝑥))=−^1−𝛾Γ1−𝛾¹₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)+𝑑𝑠𝜌𝑡^1−𝛾Γ1−𝛾¹₀𝛼(1−𝑠)^𝛼−𝛽Γ−()𝛼−𝛽+11−𝑠^{𝛼−𝛽−1}Γ𝑥(𝑠)+𝑥𝛼−𝛽𝑑𝑠₁−𝑥₀𝑡^1−𝛾Γ()𝐼1−𝛾+𝜌1−𝛼^{𝛼−𝛽−𝛾+1}𝑥(𝑡)+𝜌𝑡𝐼^{𝛼−𝛽−𝛾}𝑥(𝑡)+𝐼^𝛼−𝛾𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡). (3.22) Hence, 𝒟^𝛾((𝑡≤𝑡ℱ𝑥))^1−𝛾Γ()𝜂(𝑡)|𝑥(𝑡𝒟𝛼+1+𝑎)|+𝑏^𝛾𝑥(𝑡+𝜌𝛼)Γ+1𝛼−𝛽+2Γ𝑡𝛼−𝛽+1^1−𝛾+𝑥₁−𝑥₀Γ+𝜌(𝑥‖1−𝛾1−𝛼)‖Γ𝛼−𝛽−𝛾+1^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−𝛾}+𝜌𝑥‖𝑑𝑠‖Γ𝛼−𝛽−𝛾^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−𝛾−1}𝑑𝑠+𝐼^𝛼−𝛾𝜂(𝑡)+𝑎|𝑥(𝑡)|+𝑏𝒟^𝛾𝑥(𝑡≤𝑡_1−𝛾)Γ()𝜂(𝑡)|𝑥(𝑡𝒟𝛼+1+𝑎)|+𝑏^𝛾𝑥(𝑡+𝜌𝛼)Γ+1𝛼−𝛽+2Γ𝑡𝛼−𝛽+1^1−𝛾+𝑥₁−𝑥₀Γ+𝜌(𝑥‖𝑡1−𝛾1−𝛼)‖^{𝛼−𝛽−𝛾+1}Γ+𝜌𝑥‖𝑡𝛼−𝛽−𝛾+2‖^{𝛼−𝛽−𝛾}Γ+𝜂+(𝑥‖𝛼−𝛽−𝛾+1𝑎+𝑏)‖Γ𝑡𝛼−𝛾+1^𝛼−𝛾. (3.23) Thus, 𝒟^𝛾((𝑡≤𝑥ℱ𝑥))¹−𝑥₀Γ+𝜂1−𝛾Γ+𝜂𝛼−𝛾+1Γ()𝜌(𝛼+1+ℛ1−𝛼)Γ,𝜌𝛼−𝛽−𝛾+2Γ+𝛼−𝛽−𝛾+1𝑎+𝑏Γ+𝛼−𝛾+1𝑎+𝑏Γ()+𝛼+1𝜌𝛼Γ+𝜌𝛼−𝛽+2Γ𝜌)𝛼−𝛽+1≤2ℛ+ℛ(𝛼−1𝛿+𝜌𝛿+2𝑎+2𝑏𝛿𝜌=2ℛ+2𝑎+2𝑏+𝛼𝛿ℛ≤2ℛ+2ℛ=4ℛ. (3.24) Therefore, ‖ℱ𝑥(𝑡)‖≤4ℛ+4ℛ=8ℛ. Thus, ℱ∶Ω→Ω. Finally, it remains to show that ℱ is completely continuous. For any 𝑥∈Ω, let ℓ=max_𝑡∈[0,1]|𝑓(𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡))|; then for 0≤𝑡₁≤𝑡₂≤1 and using (3.12)–(3.14), we have ℱ𝑥𝑡₂−ℱ𝑥𝑡₁≤𝑥₁−𝑥₀𝑡₂−𝑡₁+ℓ𝑡₂−𝑡₁¹₀(1−𝑠)^𝛼−1Γ(𝛼)+𝜌𝑡𝑑𝑠₂−𝑡₁¹₀(1−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽1−𝑠^𝛼−𝛽Γ𝑥(𝑠)𝛼−𝛽+1𝑑𝑠+ℓ^𝑡2₀𝑡₂−𝑠^𝛼−1Γ(𝛼)−𝑡₁−𝑠^𝛼−1Γ(𝛼)+𝜌𝑥𝑑𝑠‖‖^𝑡2₀𝑡₂𝑡₂−𝑠^{𝛼−𝛽−1}Γ−𝛼𝑡𝛼−𝛽²−𝑠^𝛼−𝛽Γ−𝑡𝛼−𝛽+1𝑑𝑠₁𝑡₁−𝑠^{𝛼−𝛽−1}Γ−𝛼𝑡𝛼−𝛽¹−𝑠^𝛼−𝛽Γ≤𝑥𝛼−𝛽+1𝑑𝑠₁−𝑥₀𝑡₂−𝑡₁+ℓ𝑡₂−𝑡₁Γ()+𝜌𝑥𝑡𝛼+1‖‖₂−𝑡₁1Γ−𝛼𝛼−𝛽+1Γ𝑡𝛼−𝛽+2+ℓ¹−𝑡₂^𝛼+𝑡^𝛼₁−𝑡^𝛼₂Γ()‖𝑥‖𝑡𝛼+1+𝛼¹−𝑡₂^{𝛼−𝛽+1}+𝑡₁^{𝛼−𝛽+1}−𝑡₂^{𝛼−𝛽+1}Γ+‖𝑥𝑡𝛼−𝛽+2‖₁−𝑡₂𝑡₁^𝛼−𝛽+𝑡₁−𝑡₂^𝛼−𝛽Γ.𝛼−𝛽+1 (3.25) Hence, it follows that ‖ℱ𝑥(𝑡₂)−ℱ𝑥(𝑡₁)‖→0, as 𝑡₂→𝑡₁. By the Arzela-Ascoli theorem, ℱ∶Ω→Ω is completely continuous. Thus by using the Schauder fixed-point theorem, it was proved that the boundary value problem (2.9), (2.10) has a solution.

Theorem 3.3. Let 𝑓∶[0,1]×ℝ²→ℝ be continuous. If there exists a constant 𝜇 such that |𝑓(𝑡,𝑥,𝑦)−𝑓(𝑡,𝑥,𝑦)|≤𝜇(|𝑥−𝑥|+|𝑦−𝑦|) for each 𝑡∈[0,1] and all 𝑥,𝑥,𝑦,𝑦∈ℝ and 4ℳ+3𝜇≤1, where 2𝜌ℳ=maxΓ,𝜌)𝛼−𝛽+1(1+𝛼Γ,𝜌(𝛼−𝛽+21−𝛼)Γ,𝜌𝛼−𝛽−𝛾+2Γ𝛼−𝛽−𝛾+1. (3.26) Then the boundary value problem (2.9) with boundary conditions (2.10) has a unique solution.

Proof. Under condition on 𝑓, we have |ℱ𝑥(𝑡)−ℱ𝑥(𝑡)|≤𝑡𝜌¹₀(1−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽1−𝑠^𝛼−𝛽Γ(𝑠)(𝑠𝛼−𝛽+1[𝑥−𝑥)]𝑑𝑠+𝑡¹₀(1−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)−𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)+𝜌𝑑𝑠_𝑡0𝑡(𝑡−𝑠)^{𝛼−𝛽−1}Γ−𝛼()𝛼−𝛽𝑡−𝑠^𝛼−𝛽Γ𝑥(𝑠)(𝑠+𝛼−𝛽+1[−𝑥)]𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓𝑠,𝑥(𝑠),𝒟^𝛾𝑥(𝑠)−𝑓𝑠,𝑥(𝑠),𝒟^𝛾(𝑠)≤𝜌𝑥𝑑𝑠Γ−𝛼−𝛽+1𝛼𝜌Γ‖+‖‖𝛼−𝛽+2‖𝑥−𝑥2𝜇𝑥−𝑥Γ()+𝛼+1𝜌𝑡^𝛼−𝛽Γ−𝛼−𝛽+1𝜌𝛼𝑡Γ‖+‖‖𝑡𝛼−𝛽+2‖𝑥−𝑥2𝜇𝑥−𝑥^𝛼Γ()≤𝜌𝛼+11+𝑡^𝛼−𝛽Γ+𝜌)𝑡𝛼−𝛽+1(1+𝛼Γ+𝛼−𝛽+22𝜇1+𝑡^𝛼Γ()‖‖.𝛼+1𝑥−𝑥 (3.27) Using (3.22) we conclude 𝒟^𝛾(ℱ𝑥)(𝑡)−𝒟^𝛾(ℱ𝑥)(𝑡)≤𝜌(1−𝛼)𝐼^{𝛼−𝛽−𝛾+1}(𝑥(𝑡)−𝑥(𝑡))+𝑡𝜌𝐼^{𝛼−𝛽−𝛾}(𝑥(𝑡)(𝑡+𝐼−𝑥))_𝛼−𝛾𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)−𝑓𝑡,𝑥(𝑡),𝒟^𝛾(𝑡≤𝜌(‖𝑥)1−𝛼)‖𝑥−𝑥Γ𝛼−𝛽−𝛾+1^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−𝛾}+𝜌‖𝑑𝑠‖𝑥−𝑥Γ𝛼−𝛽−𝛾^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−𝛾−1}+‖‖𝑑𝑠2𝜇𝑥−𝑥Γ𝛼−𝛾^𝑡₀(𝑡−𝑠)^{𝛼−𝛾−1}≤𝜌||𝑡𝑑𝑠1−𝛼^{𝛼−𝛽−𝛾+1}Γ+𝛼−𝛽−𝛾+2𝜌𝑡^{𝛼−𝛽−𝛾}Γ+𝛼−𝛽−𝛾+12𝜇𝑡^𝛼−𝛾Γ‖‖.𝛼−𝛾+1𝑥−𝑥 (3.28) Thus, we have ‖ℱ𝑥(𝑡)−ℱ𝑥(𝑡)‖≤4ℳ+6𝜇Γ()<‖‖𝛼+14ℳ+3𝜇𝑥−𝑥. (3.29) Therefore, by the contraction mapping theorem, the boundary value problem (2.9), (2.10) has a unique solution.

Theorem 3.4. Let 𝑓∶[0,1]→[0,∞], such that 𝑓(𝑡,𝑥,𝑦)≤𝜂(𝑡)+𝑎|𝑥|+𝑏|𝑦|, 𝑎,𝑏≥0 with 𝑎+𝑏+𝛼|𝜌≤𝛿| where 𝛿=min{Γ(𝛼−𝛽−𝛾+1),Γ(𝛼−𝛽−𝛾+2),Γ(𝛼−𝛽−𝛾+3)}. Then the initial value problem (2.9), (2.10) has a solution.

Proof. In view of Lemma 2.1 and Proposition 2.2, we have 𝑥(𝑡)=𝜌𝑡𝐼^𝛼−𝛽𝑥(𝑡)−𝜌𝛼𝐼^{𝛼−𝛽+1}𝑥(𝑡)+𝐼^𝛼𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)−𝑐₀−𝑐₁𝑡. (3.30) Then, 𝑥^′(𝑡)=𝜌(1−𝛼)𝐼^𝛼−𝛽𝑥(𝑡)+𝜌𝑡𝐼^{𝛼−𝛽−1}𝑥(𝑡)+𝐼^𝛼=1𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡)−𝑐₀−𝑐₁𝑡. (3.31) By initial conditions we have 𝑐₀=−𝑥₀ and 𝑐₁=−1. Define an operator 𝒯∶Ω→Ω by 𝒯𝑥(𝑡)=𝑥₀+𝑡+𝜌𝑡𝐼^𝛼−𝛽𝑥(𝑡)−𝜌𝛼𝐼^{𝛼−𝛽+1}𝑥(𝑡)+𝐼^𝛼𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡). (3.32) Can be easily to prove that 𝒯∶Ω→Ω is completely continuous as operator ℱ.

Theorem 3.5. Let 𝑓∶[0,1]×ℝ²→ℝ be continuous. If there exists a constant 𝜇 such that |𝑓(𝑡,𝑥,𝑦)−𝑓(𝑡,𝑥,𝑦)|≤𝜇(|𝑥−𝑥|+|𝑦−𝑦|) for each 𝑡∈[0,1] and all 𝑥,𝑥,𝑦,𝑦∈ℝ and 3(ℳ+𝜇)≤1, where 𝜌ℳ=maxΓ,𝜌𝛼𝛼−𝛽+1Γ,𝜌𝛼−𝛽+2Γ𝛼−𝛽−𝛾+1, (3.33) then the initial value value problem (2.9), (2.11) has a unique solution.

The proof of the Theorem 3.5 is similar to the proof of Theorem 3.3. Note that 𝑑𝒯𝑥(𝑡)()ℐ𝑑𝑡=1+𝜌1−𝛼^𝛼−𝛽𝑥(𝑡)+𝜌𝑡ℐ^{𝛼−𝛽−1}𝑥(𝑡)+ℐ^𝛼−1𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡). (3.34)

Then using Proposition 2.2 we have,𝒟^𝛾(𝒯𝑥(𝑡))=ℐ^1−𝛾𝑑𝒯𝑥(𝑡)=𝑡𝑑𝒯_1−𝛾Γ+𝛼()ℐ1−𝛾1−𝜌−𝜌1−𝛼^{𝛼−𝛽−𝛾+1}𝑥(𝑡)+𝜌𝑡ℐ^{𝛼−𝛽−𝛾}𝑥(𝑡)+ℐ^𝛼−𝛾𝑓𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡) (3.35)

Example 3.6. Consider the following boundary value problem for nonlinear fractional order differential equation: 𝒟^3/2−𝑡𝒟^1/2𝑥(𝑡)=3𝑒^𝑡+1𝑥(𝑡)+110𝒟10^1/2𝑥(𝑡)^1/3(),𝑥(0),𝑡∈0,1=𝑥₀,𝑥(1)=𝑥₁. (3.36) Then, (3.36) with assumed boundary conditions has a solution in Ω.

In Example 3.6??𝑓(𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡))=³3𝑒^𝑡+(1/10)𝑥(𝑡)+(1/10)𝒟^1/2𝑥(𝑡) satisfies the conditions required in Theorem 3.2, that is 𝑓𝑡,𝑥(𝑡),𝒟^1/2𝑥(𝑡)≤𝑒^𝑡+1|𝑥(𝑡+130)|𝒟30^1/2𝑥(𝑡) (3.37) and 𝛿=min{Γ(3/2),Γ(2),Γ(5/2)}=Γ(3/2)=√𝜋/2 and 2𝑎+2𝑏+𝛼𝜌=47/30<2𝛿=√𝜋.

Example 3.7. Consider the following boundary value problem for nonlinear fractional order differential equation: 𝒟^3/2−(1/8)𝑡𝒟^1/2𝑥(𝑡)=1𝑥(𝑡)+121𝒟21^1/2𝑥(𝑡)(),𝑥(0),𝑡∈0,1=𝑥₀,𝑥(1)=𝑥₁. (3.38) Then, (3.38) with assumed boundary conditions has unique solution in Ω.

In Example 3.7??𝑓(𝑡,𝑥(𝑡),𝒟^𝛾𝑥(𝑡))=(1/21)𝑥(𝑡)+(1/21)𝒟^1/2𝑥(𝑡) satisfies the conditions required in Theorem 3.3. 𝐿=max{1/3√𝜋,1/8√𝜋,1/12√𝜋,1/4√𝜋}=1/3√𝜋 and 4ℳ+3𝜇=4/3√𝜋+1/7<1.

4. Conclusion

We considered a class of nonlinear fractional order differential equations involving Caputo fractional derivative with lower terminal at 0 in order to study the existence solution satisfying the boundary conditions or satisfying the initial conditions. The unique solution under Lipschitz condition is also derived. In order to illustrate our results several examples are presented. The presented research work can be generalized to multiterm nonlinear fractional order differential equations with polynomial coefficients.