Initial time difference quasilinearization for Caputo Fractional Differential Equations

Abstract

This paper deals with an application of the method of quasilinearization by not demanding the Hölder continuity assumption of functions involved and by choosing upper and lower solutions with initial time difference for nonlinear Caputo fractional differential equations. Thus, we construct monotone flows that are generated by solutions of linear fractional differential equations which converge uniformly and quadratically to the unique solution of the problem. Also, necessary comparison result concerning lower and upper solutions are proved without using Hölder continuity.

Mathematics subject classification: 34A12, 34A45, 34C11.

1 Introduction

The method of quasilinearization is employed to provide an explicit analytic representation for the solution of nonlinear differential equations. In this technique, one gets monotone sequences whose iterates are the solutions of corresponding linear problems and furthermore these sequences converge uniformly and quadratically to the unique solution of the given nonlinear differential equations [1]. This is a definite advantage of this constructive technique. Also this method has been generalized, refined and extended in several directions so as to be applicable to a much larger class of nonlinear problems by not demanding convexity or concavity property. Moreover, other possibilities that have been explored make the method of generalized quasilinearization universally useful in applications [2–8].

The concept of noninteger order derivative, popularly known as fractional derivative goes back to the seventeenth century [9, 10]. Since that time the fractional calculus has drawn the attention of many famous mathematicians. It is only a few decades ago, it was realized that the derivatives of arbitrary order provide an excellent framework for modeling the real world problems in a variety of disciplines. There has been a growing interest in this new area to study the concept of fractional differential equations and fractional dynamical systems [11–18].

The application of quasilinearization for fractional differential equations is a new research area. Depending on development in fractional order differential equations, this technique is reconsidered and similar results parallel to classical theory of differential equation with integer derivatives have been obtained. Recently, only a few papers were published in this direction [19–21].

We consider the following initial value problem

$${}^c{D}^{q}x\left(t\right)=f\left(t,x\right),x\left(t_0\right)=x_0$$

(1.1)

where f ∈ C([t₀, T] × ℝ, ℝ) and ^cD^q is the Caputo's fractional derivative of order q, 0 < q < 1.

The corresponding Volterra fractional integral equation is defined as

$$x\left(t\right)=x_0+\frac{1}{\Gamma \left(q\right)}\underset{t_0}{\overset{t}{\int }}{\left(t-s\right)}^{q-1}f\left(s,x\left(s\right)\right)\mathsf{\text{d}}s$$

(1.2)

A large cycle of works have been done in the literature using local Hölder continuity assumption which is needed for comparison theorems, see [16, 17]. For instance, in [22] the solution of fractional differential equations is obtained by utilizing that condition. Obviously, of great interest is the study of the solution of nonlinear fractional differential equations without using Hölder continuity.

In this work, by not demanding Hölder continuity condition we employe the quasilinearization technique for the given nonlinear fractional order differential equation (1.1) in which upper and lower solutions will have different initial times and positions.

2 Preliminaries

In this section, some basic definitions and theorems used throughout the paper are presented. First, we begin with the definition of the class C _p [[t₀, T], ℝ].

Definition 2.1. A function σ (t) is called a C _p function if σ ∈ C [(t₀, T], ℝ] and σ(t) (t - t₀) ^p ∈ C [[t₀, T], ℝ] with p = 1 - q.

Next, we give the definition of lower and upper solutions, respectively.

Definition 2.2. A function v ∈C _p [[t₀, T], ℝ], p = 1 - q, 0 < q < 1 is said to be a lower solution of (1.1) if

$${}^c{D}^{q}v\left(t\right)\le f\left(t,v\left(t\right)\right),v\left(t_0\right)\le x\left(t_0\right).$$

It is an upper solution if the inequalities are reversed.

Now, consider the following nonhomogeneous linear fractional differential equation,

$${}^c{D}^{q}x=\lambda x+f\left(t\right),x\left(t_0\right)=x_0,$$

(2.1)

where λ is a real number and f ∈ C _p ([t₀, T] × ℝ, ℝ). The equivalent Volterra fractional integral equation for t₀ ≤ t ≤ T is

$$x\left(t\right)=x_0+\frac{\lambda }{\Gamma \left(q\right)}\underset{t_0}{\overset{t}{\int }}{\left(t-s\right)}^{q-1}x\left(s\right)\mathsf{\text{d}}s+\frac{1}{\Gamma \left(q\right)}\underset{t_0}{\overset{t}{\int }}{\left(t-s\right)}^{q-1}f\left(s\right)\mathsf{\text{d}}s,$$

(2.2)

When we apply the method of successive approximations (see [16]) to find the solution x(t) = x(t, t₀, x₀) explicitly for the given nonhomogeneous IVP (2.1), we obtain

$$x\left(t\right)=x_0{E}_q\left(\lambda {\left(t-t_0\right)}^q\right)+\underset{t_0}{\overset{t}{\int }}{\left(t-s\right)}^{q-1}{E}_{q,q}\left(\lambda {\left(t-s\right)}^q\right)f\left(s\right)\mathsf{\text{d}}s,t\in \left[t_0,T\right],$$

(2.3)

where

$$E_q\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}\frac{t^k}{\Gamma \left(qk+1\right)}\mathsf{\text{and}}{E}_{q,q}\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}\frac{t^k}{\Gamma \left(qk+q\right)}$$

are Mittag-Leffler functions of one parameter and two parameters, respectively.

If f(t) ≡ 0, we get, as the solution of the corresponding homogeneous IVP

$$x\left(t\right)=x_0{E}_q\left(\lambda {\left(t-t_0\right)}^q\right),t\in \left[t_0,T\right].$$

(2.4)

Remark 2.1. Let ^cD^qu(t) ≤ Lu(t), u(t₀) = u₀ where u ∈ C _p ([t₀, T], ℝ₊) and L is positive constant. Then we have the estimate

$$u\left(t\right)\le u_0{E}_q\left(L{\left(t-t_0\right)}^q\right)\mathsf{\text{on }}\left[t_0,T\right]$$

(2.5)

When q = 1, that result reduces to well known Gronwall's inequality. For the proof of this remark and further information about Gronwall's type inequality for fractional order differential equations, one can see [23].

If u₀ = 0, then u (t) = 0 identically on [t₀, T].

3 Comparison Theorem

In a recent study [24], the Hölder continuity assumption is relaxed to C _p continuity of the functions involved in the Riemann-Liouville fractional differential equation. In the following we also prove a comparison result by not requiring the Hölder continuity with a different argument for Caputo fractional differential equations. It is obvious that this result is essential to extend the applicability of iterative techniques such as the monotone iterative technique and the method of quasilinearization.

Theorem 3.1. Let v(t), w (t) ∈ C _p [[t₀, T], ℝ] and f ∈ C[[t₀, T] × ℝ, ℝ] and

$$\begin{array}{c}\left(\mathsf{\text{i}}\right){}^c{D}^{q}v\left(t\right)\le f\left(t,v\left(t\right)\right)\\ \left(\mathsf{\text{ii}}\right){}^c{D}^{q}w\left(t\right)\ge f\left(t,w\left(t\right)\right)\end{array}$$

Suppose further that the standard Lipschitz condition is satisfied

$$f\left(t,x\right)-f\left(t,y\right)\le L\left(x-y\right),x\ge y$$

(3.1)

and L > 0.

Then v(t₀) ≤ w(t₀) implies

$$v\left(t\right)\le w\left(t\right),t_0\le t\le T.$$

(3.2)

Proof. Suppose that v(t) ≤ w(t) for t₀ ≤ t ≤ T is not true. Then, there exists a t₁> t₀ such that v(t₁) > w(t₁) and t₀< t₁ ≤ T. Since v(t₀) ≤ w(t₀) we encounter with two cases in view of the continuity of functions involved:

(i) If v(t₀) < w(t₀), then one can find a τ₀ such that v(τ₀) = w(τ₀) and t₀< τ₀< t₁. Thus, we have v(t) > w(t) on (τ₀, t₁].

(ii) If v(t₀) = w(t₀), then two situations are possible. Namely, one can get v(t) > w(t) on (t₀, t₁] or v(t) > w(t) on (τ₀, t₁] where t₀< τ₀< t₁ and v(τ₀) = w(τ₀) as before.

In both cases, we can find an interval [t₀, t₁] or [τ₀, t₁] on which v(t) ≥ w(t).

Let us define

$$u\left(t\right)=v\left(t\right)-w\left(t\right)$$

(3.3)

and assume that u (t) is defined on [τ₀, t₁] (or on [t₀, t₁]). Note that u (t) > 0 on (τ₀, t₁] and u (τ₀) = 0. Taking Caputo's fractional derivative of both sides of (3.3), we get

$${}^c{D}^{q}u\left(t\right)={}^c{D}^{q}v\left(t\right)-{}^c{D}^{q}w\left(t\right)$$

(3.4)

By using the lower and upper properties of v (t) and w (t), we have

$${}^c{D}^{q}u\left(t\right)\le f\left(t,v\left(t\right)\right)-f\left(t,w\left(t\right)\right)$$

(3.5)

Since f is Lipschitz with L > 0 and v(t) ≥w(t) on [τ₀, t₁], we obtain

$${}^c{D}^{q}u\left(t\right)\le L\left[v\left(t\right)-w\left(t\right)\right],$$

(3.6)

Therefore we get

$${}^c{D}^{q}u\left(t\right)\le Lu\left(t\right),u\left({\tau }_0\right)=0,$$

(3.7)

which implies, in view of Remark 2.1,

$$u\left(t\right)=0\mathsf{\text{for}}{\tau }_0\le t\le t_1$$

(3.8)

which gives a contradiction. So, we have v(t) ≤ w(t) on [t₀, T].

Corollary 3.1 The function f(t, u) = σ(t)u, where σ (t) ≤ L is admissible in Theorem 3.1 to yield u(t) ≤ 0 on t₀ ≤ t ≤ T.

Observe that a dual result of corollary 2.1 is valid.

4 Quasilinearization with Initial Time Difference

The purpose of this section is to employ the quasilinearization technique for nonlinear Caputo's fractional order differential equation (1.1) by choosing lower and upper solutions with initial time difference and not imposing the Hölder continuity on functions involved. Also, we consider the function f (t, x) on the right hand side of the equation (1.1) which satisfies a weaker condition than convexity.

Theorem 4.1. Assume that

(i) α ∈ C _p [[t₀, t₀ + T], ℝ], t₀, T > 0, β ∈ C _p [[τ₀, τ₀ + T], ℝ], τ₀> 0, f ∈ C [[t₀, τ₀ + T] × ℝ, ℝ] and

$$\begin{array}{c}{}^c{D}^q\alpha \left(t\right)\le f\left(t,\alpha \left(t\right)\right),t_0\le t\le t_0+T\\ {}^c{D}^q\beta \left(t\right)\ge f\left(t,\beta \left(t\right)\right),{\tau }_0\le t\le {\tau }_0+T\end{array}$$

with α (t₀) ≤ x (s₀) ≤ β(τ₀) and t₀< s₀< τ₀ where α(t) ≤ β (t + η₁), t₀ ≤ t ≤ t₀ + T and η₁ = τ₀ - t₀;

(ii) Suppose f _x (t, x) exists and following relations hold

$$\begin{array}{c}f(t,x)\ge f(t,y)+f_x(t,y)(x-y)\text{whenever }x\ge y\text{and}\\ |f_x(t,x)-f_x(t.y)|\le L|x-y|,L>0;\end{array}$$

(iii) f (t, x) is nondecreasing in t for each x and f _x (t, x) is nondecreasing in x for each t.

Then there exists monotone sequences $\left\{{\tilde{\alpha }}_n\right\}$ and $\left\{{\tilde{\beta }}_n\right\}$ which converge uniformly and monotonically to the unique solution of (1.1) with x (s₀) = x₀ on [s₀, s₀ + T] and the convergence is quadratic.

Proof. Let ${\tilde{\beta }}_0\left(t\right)=\beta \left(t+{\eta }_1\right)$ and ${\tilde{\alpha }}_0\left(t\right)=\alpha \left(t\right)$, t₀ ≤ t ≤ t₀ + T where η₁ = τ₀ - t₀. Then we have

$${\tilde{\beta }}_0\left(t_0\right)=\beta \left({\tau }_0\right)\ge \alpha \left(t_0\right)=\tilde{\alpha }\left(t_0\right).$$

Also

$$\begin{array}{cc}\hfill {}^c{D}^q{\tilde{\beta }}_0\left(t\right)& ={}^c{D}^q\beta \left(t+{\eta }_1\right)\hfill \\ \ge f\left(t+{\eta }_1,\beta \left(t+{\eta }_1\right)\right)\hfill \\ =f\left(t+{\eta }_1,{\tilde{\beta }}_0\left(t\right)\right)\hfill \end{array}$$

since f (t, x) is nondecreasing in t for each x, we get

$${}^c{D}^q{\tilde{\beta }}_0\left(t\right)\ge f\left(t,{\tilde{\beta }}_0\left(t\right)\right)$$

Similarly, we can write

$${}^c{D}^q{\tilde{\alpha }}_0\left(t\right)={}^c{D}^q{\alpha }_0\left(t\right)\le f\left(t,{\alpha }_0\left(t\right)\right)=f\left(t,{\tilde{\alpha }}_0\left(t\right)\right)$$

which shows ${\tilde{\alpha }}_0\left(t\right)$ is a lower solution of the problem.

Consider the following linear fractional equations

$${}^c{D}^q{\tilde{\alpha }}_{n+1}\left(t\right)=f\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)\left({\tilde{\alpha }}_{n+1}-{\tilde{\alpha }}_n\right),{\tilde{\alpha }}_{n+1}\left(t_0\right)=x_0$$

(4.1)

$${}^c{D}^q{\tilde{\beta }}_{n+1}\left(t\right)=f\left(t+{\eta }_2,{\tilde{\beta }}_n\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)\left({\tilde{\beta }}_{n+1}-{\tilde{\beta }}_n\right),{\tilde{\beta }}_{n+1}\left(t_0\right)=x_0$$

(4.2)

where η₂ = s₀ - t₀. Note that unique solutions exist since the right hand side of the equations satisfy a Lipschitz condition.

We shall show that

$${\tilde{\alpha }}_0\le {\tilde{\alpha }}_1\le \cdots \le {\tilde{\alpha }}_n\le {\tilde{\beta }}_n\le \cdots \le {\tilde{\beta }}_1\le {\tilde{\beta }}_0\mathsf{\text{on}}\left[t_0,t_0+T\right].$$

(4.3)

First we must prove

$${\tilde{\alpha }}_0\le {\tilde{\alpha }}_1\le {\tilde{\beta }}_1\le {\tilde{\beta }}_0\mathsf{\text{on}}\left[t_0,t_0+T\right].$$

(4.4)

Set $p\left(t\right)={\tilde{\alpha }}_1-{\tilde{\alpha }}_0$ then

$$\begin{array}{cc}\hfill {}^c{D}^{q}p\left(t\right)& ={}^c{D}^q{\tilde{\alpha }}_1{-}^c{D}^q{\tilde{\alpha }}_0\hfill \\ \ge f\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\alpha }}_1-{\tilde{\alpha }}_0\right)-f\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\hfill \\ =f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\alpha }}_1-{\tilde{\alpha }}_0\right)\hfill \\ \hfill {}^c{D}^{q}p\left(t\right)& \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)p,p\left(t_0\right)\ge 0.\hfill \end{array}$$

Hence applying Corollary 3.1, we get ${\tilde{\alpha }}_0\le {\tilde{\alpha }}_1$ on [t₀, t₀ + T]. Similarly, one can show that ${\tilde{\beta }}_1\le {\tilde{\beta }}_0$.

Now we must prove that ${\tilde{\alpha }}_1\le {\tilde{\beta }}_1$ on [t₀, t₀ + T]. To do so, we set $p\left(t\right)={\tilde{\beta }}_1\left(t\right)-{\tilde{\alpha }}_1\left(t\right)$, then

$$\begin{array}{cc}\hfill {}^c{D}^{q}p\left(t\right)& ={}^c{D}^q{\tilde{\beta }}_1-{}^c{D}^q{\tilde{\alpha }}_1\hfill \\ =f\left(t+{\eta }_2,{\tilde{\beta }}_0\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\beta }}_1-{\tilde{\beta }}_0\right)\hfill \\ -\left[f\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\alpha }}_1-{\tilde{\alpha }}_0\right)\right]\hfill \\ =f\left(t+{\eta }_2,{\tilde{\beta }}_0\right)-f\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\beta }}_1-{\tilde{\beta }}_0-{\tilde{\alpha }}_1+{\tilde{\alpha }}_0\right)\hfill \end{array}$$

using the inequality in (ii) we get

$$\begin{array}{cc}\hfill {}^c{D}^{q}p\left(t\right)& \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\beta }}_0-{\tilde{\alpha }}_0\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\beta }}_1-{\tilde{\beta }}_0-{\tilde{\alpha }}_1+{\tilde{\alpha }}_0\right)\hfill \\ \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)\left({\tilde{\beta }}_1-{\tilde{\alpha }}_1\right)\hfill \end{array}$$

this implies that

$${}^c{D}^{q}p\left(t\right)\ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_0\right)p\left(t\right)\mathsf{\text{and}}p\left(t_0\right)=0$$

which because of Corollary 3.1 yields p (t) ≥ 0. Thus we have ${\tilde{\alpha }}_1\le {\tilde{\beta }}_1$ on [t₀, t₀ + T]. Hence (4.4) is proved.

Using mathematical induction with k > 1, we obtain

$${\tilde{\alpha }}_0\le {\tilde{\alpha }}_{k-1}\le {\tilde{\alpha }}_k\le {\tilde{\beta }}_k\le {\tilde{\beta }}_{k-1}\le {\tilde{\beta }}_0\mathsf{\text{on}}\left[t_0,t_0+T\right].$$

(4.5)

Now we need to show that

$${\tilde{\alpha }}_k\le {\tilde{\alpha }}_{k+1}\le {\tilde{\beta }}_{k+1}\le {\tilde{\beta }}_k\mathsf{\text{on }}\left[t_0,t_0+T\right].$$

(4.6)

To prove this, we set $p\left(t\right)={\tilde{\alpha }}_{k+1}-{\tilde{\alpha }}_k$ so that utilizing equations in (4.1), (4.2) and the inequality in (ii), we have

$$\begin{array}{cc}\hfill {}^c{D}^{q}p\left(t\right)& =f\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\alpha }}_{k+1}-{\tilde{\alpha }}_k\right)\hfill \\ -\left[f\left(t+{\eta }_2,{\tilde{\alpha }}_{k-1}\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_{k-1}\right)\left({\tilde{\alpha }}_k-{\tilde{\alpha }}_{k-1}\right)\right]\hfill \\ \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_{k-1}\right)\left({\tilde{\alpha }}_k-{\tilde{\alpha }}_{k-1}\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\alpha }}_{k+1}-{\tilde{\alpha }}_k\right)\hfill \\ -f_x\left(t+{\eta }_2,{\tilde{\alpha }}_{k-1}\right)({\tilde{\alpha }}_k-{\tilde{\alpha }}_{k-1})\hfill \\ \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\alpha }}_{k+1}-{\tilde{\alpha }}_k\right).\hfill \end{array}$$

Thus we obtain

$${}^c{D}^{q}p\left(t\right)\ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)p\left(t\right)\mathsf{\text{and}}p\left(t_0\right)=0.$$

Again using corollary 3.1 we get ${\tilde{\alpha }}_k\le {\tilde{\alpha }}_{k+1}$ on [t₀, t₀ + T]. In a similar way, it can be shown that ${\tilde{\beta }}_k\ge {\tilde{\beta }}_{k+1}$ on [t₀, t₀ + T]. Next we must prove ${\tilde{\alpha }}_{k+1}\le {\tilde{\beta }}_{k+1}$ on [t₀, t₀ + T]. Let $p\left(t\right)={\tilde{\beta }}_{k+1}-{\tilde{\alpha }}_{k+1}$, then

$$\begin{array}{cc}\hfill {}^c{D}^{q}p\left(t\right)& ={}^c{D}^q{\tilde{\beta }}_{k+1}{-}^c{D}^q{\tilde{\alpha }}_{k+1}\hfill \\ =f\left(t+{\eta }_2,{\tilde{\beta }}_k\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\beta }}_{k+1}-{\tilde{\beta }}_k\right)\hfill \\ -\left[f\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\alpha }}_{k+1}-{\tilde{\alpha }}_k\right)\right]\hfill \\ \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\beta }}_k-{\tilde{\alpha }}_k\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\beta }}_{k+1}-{\tilde{\beta }}_k\right)\hfill \\ -f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\alpha }}_{k+1}-{\tilde{\alpha }}_k\right)\hfill \\ =f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left[{\tilde{\beta }}_k-{\tilde{\alpha }}_k-{\tilde{\alpha }}_{k+1}+{\tilde{\alpha }}_k+{\tilde{\beta }}_{k+1}-{\tilde{\beta }}_k\right]\hfill \\ \ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)\left({\tilde{\beta }}_{k+1}-{\tilde{\alpha }}_{k+1}\right)\hfill \end{array}$$

Thus we have ${}^c{D}^{q}p\left(t\right)\ge f_x\left(t+{\eta }_2,{\tilde{\alpha }}_k\right)p\left(t\right)$ and p (t₀) = 0. It follows from the Corollary 3.1 we reach ${\tilde{\alpha }}_{k+1}\le {\tilde{\beta }}_{k+1}$ on [t₀, t₀ +T]. Hence (4.6) is proved.

Employing standard techniques (see [16]), it can be easily shown that the monotone sequences $\left\{{\tilde{\alpha }}_n\right\}$ and $\left\{{\tilde{\beta }}_n\right\}$ converge uniformly and monotonically to the unique solution $\tilde{x}\left(t\right)$ of

$${}^c{D}^q\tilde{x}\left(t\right)=f\left(t+{\eta }_2,\tilde{x}\left(t\right)\right),\tilde{x}\left(t_0\right)=x_0$$

(4.7)

Letting s = t + η₂ and changing the variable, we have

$${}^c{D}^{q}x=f\left(s,x\right),x\left(s_0\right)=x_0.$$

(4.8)

Next we will prove that the convergence is quadratic. For this purpose, consider

$$p_{n+1}=\tilde{x}-{\tilde{\alpha }}_{n+1}$$

Note that p_n+1(t₀) = 0. So we have

$$\begin{array}{cc}\hfill {}^c{D}^q{p}_{n+1}& ={}^c{D}^q\tilde{x}-{}^c{D}^q{\tilde{\alpha }}_{n+1}\hfill \\ =f\left(t+{\eta }_2,\tilde{x}\right)-\left[f\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)\left({\tilde{\alpha }}_{n+1}-{\tilde{\alpha }}_n\right)\right]\hfill \\ =\left[f\left(t+{\eta }_2,\tilde{x}\right)-f\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)\right]-f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)\left(p_n-p_{n+1}\right)\hfill \\ \le \left[f_x\left(t+{\eta }_2,\tilde{x}\right)-f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)\right]p_n+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)p_{n+1}\hfill \\ \le L{\left|p_n\right|}^2+f_x\left(t+{\eta }_2,{\tilde{\alpha }}_n\right)p_{n+1}\hfill \\ \le L{\left|p_n\right|}_0^2+N.p_{n+1}\hfill \end{array}$$

where $\left|f_x\right|\le N,{\left|p_n\right|}_0=\underset{\left[t_0,t_0+T\right]}{\mathsf{\text{max}}}\left|p_n\left(t\right)\right|$. This inequality gives the estimate

$$\begin{array}{cc}\hfill p_{n+1}& \le L{\left|p_n\right|}_0^2\underset{t_0}{\overset{t}{\int }}{\left(t-s\right)}^{q-1}{E}_{q,q}\left(N{\left(t-s\right)}^q\right)\mathsf{\text{d}}s\hfill \\ \le N_0{\left|p_n\right|}_0^2\hfill \end{array}$$

where $N_0=\frac{L{T}^q}{q}{E}_{q,q}\left(N{T}^q\right)$ and E _{q, q} is Mittag-Leffler function.

Thus we reach the desired result

$$\underset{\left[t_0,t_0+T\right]}{\mathsf{\text{max}}}|\tilde{x}-{\tilde{\alpha }}_{n+1}|\le N_0\underset{\left[t_0,t_0+T\right]}{\mathsf{\text{max}}}{\left|\tilde{x}-{\tilde{\alpha }}_n\right|}^2.$$

(4.9)

Similarly, after using suitable computation, we get the quadratic convergence of $\left\{{\tilde{\beta }}_n\right\}$ such that

$$\underset{\left[t_0,t_0+T\right]}{\mathsf{\text{max}}}\left|{\tilde{\beta }}_{n+1}-\tilde{x}\right|\le \frac{N_{0}L}{2}\underset{\left[t_0,t_0+T\right]}{\mathsf{\text{max}}}{\left|\tilde{x}-{\tilde{\alpha }}_n\right|}^2+\frac{3N_{0}L}{2}\underset{\left[t_0,t_0+T\right]}{\mathsf{\text{max}}}{\left|{\tilde{\beta }}_n-\tilde{x}\right|}^2.$$

(4.10)

The proof is complete.

Corollary 4.1. If the assumptions of Theorem 4.1 hold with s₀ = t₀, then the conclusion remains the same.

Proof. For the proof, we let ${\tilde{\beta }}_0\left(t\right)=\beta \left(t+{\eta }_1\right),{\tilde{\alpha }}_0\left(t\right)=\alpha \left(t\right),\mathsf{\text{and }}\tilde{x}\left(t\right)=x\left(t\right)$ on [t₀, t₀ + T] and we proceed as we did in Theorem 4.1.

Corollary 4.2. If the assumptions of Theorem 4.1 hold with s₀ = τ₀, then the conclusion remains the same.

Proof. This time, we must set ${\tilde{\alpha }}_0\left(t\right)=\alpha \left(t-{\eta }_1\right),{\tilde{\beta }}_0\left(t\right)=\beta \left(t\right)\mathsf{\text{ and }}\tilde{x}\left(t\right)=x\left(t\right)$ on [τ₀, τ₀ + T] proceed as we did in Theorem 4.1.

In case t₀> τ₀, a dual result of Theorem 4.1 can be proved with some suitable changes. Next result is given in this direction.

Theorem 4.2. Assume that

(i) α ∈ C _p [[t₀, t₀ + T], ℝ], t₀, T > 0, β ∈ C _p [[τ₀, τ₀ + T], ℝ], τ₀> 0, f ∈ C [[τ₀, t₀ + T] × ℝ, ℝ] and

$$\begin{array}{cc}\hfill {}^c{D}^q\alpha \left(t\right)& \le f\left(t,\alpha \left(t\right)\right),t_0\le t\le t_0+T\hfill \\ \hfill {}^c{D}^q\beta \left(t\right)& \ge f\left(t,\beta \left(t\right)\right),{\tau }_0\le t\le {\tau }_0+T\hfill \end{array}$$

with α(t₀) ≤ x (s₀) ≤ β(τ₀) and τ₀< s₀< t₀ where α(t + η₁) ≤ β(t), τ₀ ≤ t ≤ τ₀ + T and η₁ = t₀ - τ₀;

(ii) Suppose f _x (t, x) exists and following relations hold

$$\begin{array}{c}f\left(t,x\right)\ge f\left(t,y\right)+f_x\left(t,y\right)\left(x-y\right)\mathsf{\text{whenever }}x\ge y\mathsf{\text{ and}}\\ |f_x\left(t,x\right)-f_x\left(t.y\right)|\le L|x-y|,L>0;\end{array}$$

(iii) f (t, x) is nonincreasing in t for each x and f _x (t, x) is nondecreasing in x for each t.

Then the conclusion of theorem 4.1 remains valid.

Proof. The proof being similar to theorem 4.1, we omit details.

5 Conclusion

In this work, the quasilinearization technique coupled with lower and upper solutions is employed to study Caputo fractional differential equations. We have observed that this technique is convenient even though initial functions α and β are given with initial times. In this way, by not requiring Hölder continuity condition, one gets monotone sequences whose iterates are solutions of corresponding linear problems and the sequences converge uniformly and quadratically to the unique solution of the given nonlinear problem.