A Refinement of Quasilinearization Method for Caputo ’ s Sense Fractional-Order Differential Equations

and Applied Analysis 3 Table 1 Quasilinearization method Integer derivative Caputo’s derivative Monotone sequences Yes Yes Unique solution exists Yes Yes Uniform convergence Yes Yes Quadratic semiquadratic convergence Yes Yes Then, α0 ≤ β0 0, where α0 α0 t t − t0 |t t0 and β0 0 β0 t t − t0 |t t0 imply that α0 t ≤ β0 t , t0 ≤ t ≤ T. Corollary 2.2. The function F t, u σ t u, where σ t ≤ L, is admissible in Theorem 2.1 to yield u t ≤ 0 on t0 ≤ t ≤ T . We note that Theorem 2.1 and Corollary 2.2 also hold for Caputo’s fractional derivative; see 2 . 3. Monotone Technique and Method of Quasilinearization In monotone iterative technique that we have used an existence result of nonlinear fractionalorder differential equations with Caputo’s derivative in a sector based on theoretical considerations and described a constructive method which implies monotone sequences of functions that converge to the solution of 1.1 . Since each member of these sequences is the solution of a certain linear fractional-order differential equation with Caputo’s derivative which can be explicitly computed, the advantage and the importance of the technique need no special emphasis. Moreover, these methods can successfully be employed to generate twosided pointwise bounds on solutions of initial value problems of fractional-order differential equations with Caputo’s derivatives from which qualitative and quantitative behaviors can be investigated. The idea of relating the study of nonlinear fractional-order differential equations with Caputo’s derivative through its related linear fractional-order differential equations with Caputo’s derivative finds further extension in the method of quasilinearization. In this case, again, we obtain existence of solutions of 1.1 under certain restrictions after formulating sequences of solutions of related linear fractional-order differential equations with Caputo’s derivative. These sequences converge quadratically in the constructive methods. The method involves the formulation of upper and lower solutions. Due to some advantages of Caputo’s derivative, we have applied the quasilinearization technique to the given nonlinear fractional-order differential equations with Caputo’s derivative not Riemann-Liouville R-L derivative. Themain advantage of Caputo’s derivative is that the initial conditions for fractional-order differential equations are of the same form as those of ordinary differential equations with integer derivatives. Another difference is that Caputo’s derivative for a constant C is zero, while the RiemannLiouville fractional-order derivative for a constant C is not zero but equals to DC C t − t0 −q/Γ 1 − q , which is not zero. Table 1 depicts the correspondence between the features of quasilinearization in the context of the integer order and fractional-order with Caputo’s derivative. Therefore, under the suitable assumptions but different conditions, we have Table 1. 4 Abstract and Applied Analysis 4. Main Result In this section, wewill prove themain theorem that gives several different conditions to apply the method of generalized quasilinearization to the nonlinear fractional-order differential equations with Caputo’s derivative and state four remarks for special cases. Theorem 4.1. Assume that i f, g, h ∈ C t0, T × R,R , α0, β0 ∈ C t0, T ,R , and Dα0 ≤ F t, α0 , α0 t0 ≤ x0, Dβ0 ≥ F ( t, β0 ) , β0 t0 ≥ x0 4.1 α0 t ≤ β0 t on J, α0 t0 ≤ x0 ≤ β0 t0 , where F t, x f t, x g t, x h t, x and J t0, T . ii Assume also that fx t, x exists and fx t, x is nondecreasing in x for each t as f t, x ≥ ft, y fx ( t, y )( x − y, x ≥ y, ∣fx t, x − fx ( t, y )∣ ≤ L1 ∣x − y∣ with L1 ≥ 0. 4.2 Furthermore, gx t, x exists and gx t, x is nonincreasing in x for each t as g t, x ≥ gt, y gx t, x ( x − y, x ≥ y, ∣gx t, x − gx ( t, y )∣ ≤ L2 ∣x − y∣ with L2 ≥ 0. 4.3 iii Moreover assume that h t, x is two-sided Lipschitzian in x such that |h t, x − h t, y | ≤ K|x − y|, where K > 0 is the Lipschitz constant. Then, there exist monotone sequences {αn} and {βn} which converge uniformly and monotonically to the unique solution x t of 1.1 and the convergence is semiquadratic. Proof. Consider the following linear fractional-order initial value problems with Caputo’s derivatives order q: Dαk 1 F t, αk [ fx t, αk gx ( t, βk ) − k αk 1 − αk , αk 1 t0 x0, Dβk 1 F ( t, βk ) [ fx t, αk gx ( t, βk ) − kβk 1 − βk ) ,


Introduction
The well-known quasilinearization method 1, 2 in differential equation has been employed to obtain a sequence of lower and upper bounds which are the solutions of linear differential equations that converge quadratically.However, the convexity and concavity assumption that is demanded by the method of quasilinearization has been a stumbling block for further development of the theory.Recently, 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 and concavity property 1, 3-7 .Moreover, other possibilities that have been explored make the method of generalized quasilinearization universally useful in applications 3, 6, 7 .
The theory of nonlinear fractional-order dynamic systems has been investigated depending on the development in the theory of fractional-order differential equations.In this context, generalized quasilinearization method has been reconsidered, and similar results parallel to classical theory of differential equations have been obtained 1, 2, 8 .In this work, the quasilinearization technique coupled with lower and upper solutions is employed to study Caputo's fractional-order differential equation for which particular and general results that include several special cases are obtained.Moreover, one gets monotone sequences whose iterates are the solutions of corresponding linear problems and the sequences converge to the solutions of the original nonlinear problems.Instead of imposing the convexity assumption on the function involved, we assume weaker conditions as well as for the concave functions.This is a definite advantage of this constructive technique.Furthermore, these monotone flows are shown to converge semiquadratically.
Consider the following initial value problem: where F ∈ C t 0 , T × R, R and c D q is Caputo's sense fractional-order derivative.Let α 0 , β 0 ∈ C q t 0 , T , R and be the lower and upper solutions of 1.1 satisfying the following inequalities 1.2 and 1.3 , respectively, on J: The corresponding Volterra fractional integral equation is Caputo's sense fractional-order differential equation is given by 1.1 , and the corresponding Volterra fractional integral equation is given by 1.4 .Here, we consider the function F t, x on the right-hand side of 1.1 and split it into three parts as f t, x , g t, x , and h t, x , where f satisfies a weaker condition than convexity, g satisfies a weaker condition than concavity, and h is two-sided Lipschitzian.

Preliminaries
In this section, we state a comparison result and a corollary.For the proof, please see 2 .

Monotone Technique and Method of Quasilinearization
In monotone iterative technique that we have used an existence result of nonlinear fractionalorder differential equations with Caputo's derivative in a sector based on theoretical considerations and described a constructive method which implies monotone sequences of functions that converge to the solution of 1.1 .Since each member of these sequences is the solution of a certain linear fractional-order differential equation with Caputo's derivative which can be explicitly computed, the advantage and the importance of the technique need no special emphasis.Moreover, these methods can successfully be employed to generate twosided pointwise bounds on solutions of initial value problems of fractional-order differential equations with Caputo's derivatives from which qualitative and quantitative behaviors can be investigated.
The idea of relating the study of nonlinear fractional-order differential equations with Caputo's derivative through its related linear fractional-order differential equations with Caputo's derivative finds further extension in the method of quasilinearization.In this case, again, we obtain existence of solutions of 1.1 under certain restrictions after formulating sequences of solutions of related linear fractional-order differential equations with Caputo's derivative.These sequences converge quadratically in the constructive methods.The method involves the formulation of upper and lower solutions.
Due to some advantages of Caputo's derivative, we have applied the quasilinearization technique to the given nonlinear fractional-order differential equations with Caputo's derivative not Riemann-Liouville R-L derivative.The main advantage of Caputo's derivative is that the initial conditions for fractional-order differential equations are of the same form as those of ordinary differential equations with integer derivatives.Another difference is that Caputo's derivative for a constant C is zero, while the Riemann-Liouville fractional-order derivative for a constant C is not zero but equals to D q C C t − t 0 −q /Γ 1 − q , which is not zero.Table 1 depicts the correspondence between the features of quasilinearization in the context of the integer order and fractional-order with Caputo's derivative.Therefore, under the suitable assumptions but different conditions, we have Table 1.

Main Result
In this section, we will prove the main theorem that gives several different conditions to apply the method of generalized quasilinearization to the nonlinear fractional-order differential equations with Caputo's derivative and state four remarks for special cases.
Theorem 4.1.Assume that i f, g, h ∈ C t 0 , T × R, R , α 0 , β 0 ∈ C q t 0 , T , R , and ii Assume also that f x t, x exists and f x t, x is nondecreasing in x for each t as

4.2
Furthermore, g x t, x exists and g x t, x is nonincreasing in x for each t as g t, x ≥ g t, y g x t, x x − y , x ≥ y, g x t, x − g x t, y ≤ L 2 x − y with L 2 ≥ 0.

4.3
iii Moreover assume that h t, x is two-sided Lipschitzian in x such that |h t, x − h t, y | ≤ K|x − y|, where K > 0 is the Lipschitz constant.
Then, there exist monotone sequences {α n } and {β n } which converge uniformly and monotonically to the unique solution x t of 1.1 and the convergence is semiquadratic.
Proof.Consider the following linear fractional-order initial value problems with Caputo's derivatives order q:

4.4
Since the right-hand sides of the equations satisfy a Lipschitz condition, it is obvious that unique solutions exist.We will show that First, we will prove that and p t 0 ≤ 0. Hence, applying Corollary 2.2, we get Let us set p t α 1 t − β 0 t ; then, using ii and iii and the fact that β 0 ≥ α 0 , we have

4.9
This implies that c D q p ≤ f x t, α 0 g x t, β 0 − K p, p t 0 ≤ 0, 4.10 which because of Corollary 2.2 yields p t ≤ 0 on J. Thus, we have α 1 ≤ β 0 on J. Similarly, one can prove that α 0 ≤ β 1 ≤ β 0 on J.We now prove that α 1 t ≤ β 1 t on J.For this purpose we set p t α 1 − β 1 and note that p t 0 0.Then,

4.11
Since β 0 ≥ α 0 , by using nondecreasing property of f x and nonincreasing property of g x , we obtain 12 which shows that c D q p ≤ f x t, α 0 g x t, β 0 − K p.This proves that p t ≤ 0. Therefore, α 1 t ≤ β 1 t on J. Hence, 4.6 is proved.
Using mathematical induction with k > 1, we obtain

4.13
We must prove that

4.14
To do so, we set p t α k − α k 1 .Then, where we have used the inequalities in ii , iii and the fact that f x is nondecreasing in x and g x is nonincreasing in x.Thus, we have Again, from Corollary 2.2, we get α k ≤ α k 1 on J. Similarly, it can be shown that Next we need to show that 4.17 Thus, we have c D q p ≤ f x t, α k g x t, β k −K p and p t 0 0. Consequently, as before, it follows from Corollary 2.2 we get that α k 1 ≤ β k 1 on J.
Employing the standard arguments 2 , one can easily show that {α n } and {β n } converge uniformly and monotonically to the unique solution of 1.1 .
To prove the semiquadratic convergence, we set p n 1 x − α n 1 and r n 1 β n 1 − x.Note that p n 1 t 0 r n 1 t 0 0 and

4.18
where α n ≤ ξ, η ≤ x.Now using the nondecreasing property of f x and nonincreasing property of g x , we get

4.19
Thus, we have Then, we obtain where E q,q is the Mittag-Leffler function.
Let W 1/q T − t 0 q E q,q K * T − t 0 q ; then, where , and U 3 2kW.Thus, we reach the desired result which shows the semiquadratic convergence.Similarly, using suitable computation, we arrive at max

An Example
The following example illustrates how the main result of the theorem may be applied for the nonlinear fractional differential equation order q 1/2 and t 0 0.
Example 5.1.Let us consider the following nonlinear fractional-order initial value problem with Caputo's derivative order q 1/2: t , β 0 0 0 for 0 ≤ t ≤ 10 be lower and upper solutions of the fractional-order differential equation with Caputo's derivative order q 1/2, respectively.Then, α 0 t and β 0 t satisfy the inequalities in assumption i as

5.2
On the other hand, f x t, x 2 t/πx, g x t, x −3 t/π exist, and f x t, x is nondecreasing and g x t, x is nonincreasing in x for each t in assumption ii .Also, it can be shown that these three functions f, g, and h hold in the correspondence inequality in assumptions ii and iii with the nonnegative constants L 1 ≥ 2 10/π, L 2 ≥ 0, and K ≥ 10/π.Therefore, we can construct the monotone sequences {α k 1 } and {β k 1 } whose elements are solutions of linear fractional-order differential equations with Caputo's derivatives order q 1/2 of 5.3 and 5.4 , respectively, as β k 1 0 0 for K 10 π , 0 ≤ t ≤ 10.

5.4
Since the right-hand sides of the equations satisfy a Lipschitz condition, it is obvious that unique solutions exist such that, for all k n, Employing the standard techniques 2 , sequences {α k } and {β k } converge uniformly and monotonically to the unique solution x t 1 − 1/ √ 1 t of c D 1/2 x t t/πx 2 − 3 t/πx t/π 1 x , x 0 0, since using the fact that F satisfies a Lipschitz condition that is F x is bounded on the sector α 0 , β 0 x : 1 t for 0 ≤ t ≤ 10.