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

Abstract

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions.

1. 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:^𝑐𝐷^𝑞𝑥(𝑡)=𝐹(𝑡,𝑥),𝑥(𝑡₀)=𝑥₀for𝑡≥𝑡₀, (1.1) where 𝐹∈𝐶([𝑡₀,𝑇]×ℝ,ℝ) and ^𝑐𝐷^𝑞 is Caputo’s sense fractional-order derivative. Let 𝛼₀,𝛽₀∈𝐶^𝑞([𝑡₀,𝑇],ℝ) and be the lower and upper solutions of (1.1) satisfying the following inequalities (1.2) and (1.3), respectively, on 𝐽:^𝑐𝐷^𝑞𝛼₀≤𝐹(𝑡,𝛼₀),𝛼₀(𝑡₀)≤𝑢₀for𝑡≥𝑡₀,_𝑐𝐷^𝑞𝛽₀≥𝐹(𝑡,𝛽₀),𝛽₀(𝑡₀)≥𝑢₀for𝑡≥𝑡₀. (1.2)(1.3)

The corresponding Volterra fractional integral equation is 𝑥(𝑡)=𝑥₀+1Γ(𝑞)^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝐹(𝑠,𝑥(𝑠))𝑑𝑠. (1.4)

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.

2. Preliminaries

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

Theorem 2.1. Let 𝛼₀,𝛽₀∈𝐶_𝑝([𝑡₀,𝑇],ℝ) be locally Hölder continuous for an exponent 0<𝜆<1 and 𝜆>𝑞,𝑝=1−𝑞,𝐹∈𝐶([𝑡₀,𝑇]×ℝ,ℝ), where 𝐶_𝑝([𝑡₀,𝑇],ℝ)=[𝑢∈𝐶([𝑡₀,𝑇],ℝ)and𝑢(𝑡)⋅(𝑡−𝑡₀)^𝑝∈𝐶([𝑡₀,𝑇],ℝ)], and
(i)𝐷^𝑞𝛼₀(𝑡)≤𝐹(𝑡,𝛼₀(𝑡)),(ii)𝐷^𝑞𝛽₀(𝑡)≥𝐹(𝑡,𝛽₀(𝑡)),𝑡₀<𝑡≤𝑇, where 𝐷^𝑞 is Riemann-Liouville fractional-order derivative and 𝑞 is such that 0<𝑞<1.
Suppose further that the standard Lipschitz condition is satisfied; that is, 𝐹(𝑡,𝑥)−𝐹(𝑡⋅𝑦)≤𝐿(𝑥−𝑦),𝑥≥𝑦,𝐿>0. (2.1) Then, 𝛼⁰₀≤𝛽⁰₀, where 𝛼⁰₀=𝛼₀(𝑡)(𝑡−𝑡₀)^1−𝑞|_𝑡=𝑡0 and 𝛽⁰₀=𝛽₀(𝑡)(𝑡−𝑡₀)^1−𝑞|_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 fractional-order 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 two-sided 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^𝑞𝐶=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.

Table 1 

4. 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([𝑡₀,𝑇]×ℝ,ℝ), 𝛼₀,𝛽₀∈𝐶^𝑞([𝑡₀,𝑇],ℝ), and ^𝑐𝐷^𝑞𝛼₀≤𝐹(𝑡,𝛼₀),𝛼₀(𝑡₀)≤𝑥₀,_𝑐𝐷^𝑞𝛽₀≥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 𝑓_𝑥(𝑡,𝑥) exists and 𝑓_𝑥(𝑡,𝑥) is nondecreasing in 𝑥 for each 𝑡 as 𝑓(𝑡,𝑥)≥𝑓(𝑡,𝑦)+𝑓_𝑥(𝑡,𝑦)(𝑥−𝑦),𝑥≥𝑦,|𝑓_𝑥(𝑡,𝑥)−𝑓_𝑥(𝑡,𝑦)|≤𝐿₁|𝑥−𝑦|with𝐿₁≥0. (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)|≤𝐿₂|𝑥−𝑦|with𝐿₂≥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: ^𝑐𝐷^𝑞𝛼_𝑘+1=𝐹(𝑡,𝛼_𝑘)+[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝑘](𝛼_𝑘+1−𝛼_𝑘𝛼),_𝑘+1(𝑡₀)=𝑥₀,_𝑐𝐷^𝑞𝛽_𝑘+1=𝐹(𝑡,𝛽_𝑘)+[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝑘](𝛽_𝑘+1−𝛽_𝑘𝛽),_𝑘+1(𝑡₀)=𝑥₀. (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 𝛼₀≤𝛼₁≤⋯≤𝛼_𝑘≤𝛽_𝑘≤⋯≤𝛽₁≤𝛽₀on𝐽. (4.5) First, we will prove that 𝛼₀≤𝛼₁≤𝛽₁≤𝛽₀on𝐽. (4.6) Put 𝑝(𝑡)=𝛼₀(𝑡)−𝛼₁(𝑡) on 𝐽. Then, ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼₀−^𝑐𝐷^𝑞𝛼₁≤𝐹(𝑡,𝛼₀)−𝐹(𝑡,𝛼₀)−[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛼₀)=[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾]𝑝, (4.7) and 𝑝(𝑡₀)≤0. Hence, applying Corollary 2.2, we get 𝛼₀(𝑡)≤𝛼₁(𝑡)on𝐽. (4.8) Let us set 𝑝(𝑡)=𝛼₁(𝑡)−𝛽₀(𝑡); then, using (ii) and (iii) and the fact that 𝛽₀≥𝛼₀, we have ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼₁−^𝑐𝐷^𝑞𝛽₀≤𝐹(𝑡,𝛼₀)+[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛼₀)−𝐹(𝑡,𝛽₀)≤𝐹(𝑡,𝛼₀)+[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛼₀)−𝑓(𝑡,𝛼₀)−𝑓_𝑥(𝑡,𝛼₀)(𝛽₀−𝛼₀)−𝑔(𝑡,𝛼₀)−𝑔_𝑥(𝑡,𝛽₀)(𝛽₀−𝛼₀)−ℎ(𝑡,𝛼₀)+𝐾(𝛽₀−𝛼₀)=[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛽₀). (4.9) This implies that ^𝑐𝐷^𝑞𝑝≤[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾]𝑝,𝑝(𝑡₀)≤0, (4.10) which because of Corollary 2.2 yields 𝑝(𝑡)≤0 on 𝐽. Thus, we have 𝛼₁≤𝛽₀ on 𝐽. Similarly, one can prove that 𝛼₀≤𝛽₁≤𝛽₀ on 𝐽. We now prove that 𝛼₁(𝑡)≤𝛽₁(𝑡) on 𝐽. For this purpose we set 𝑝(𝑡)=𝛼₁−𝛽₁ and note that 𝑝(𝑡₀)=0.
Then, ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼₁−^𝑐𝐷^𝑞𝛽₁=𝑓(𝑡,𝛼₀)+𝑔(𝑡,𝛼₀)+ℎ(𝑡,𝛼₀)+[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛼₀)−𝑓(𝑡,𝛽₀)−𝑔(𝑡,𝛽₀)−ℎ(𝑡,𝛽₀)−[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛽₁−𝛽₀). (4.11) Since 𝛽₀≥𝛼₀, by using nondecreasing property of 𝑓_𝑥 and nonincreasing property of 𝑔_𝑥, we obtain ^𝑐𝐷^𝑞𝑝≤𝑓_𝑥(𝑡,𝛼₀)(𝛼₀−𝛽₀)+𝑔_𝑥(𝑡,𝛽₀)(𝛼₀−𝛽₀)−𝐾(𝛼₀−𝛽₀)+[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛼₀−𝛽₁+𝛽₀)=[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾](𝛼₁−𝛽₁), (4.12) which shows that ^𝑐𝐷^𝑞𝑝≤[𝑓_𝑥(𝑡,𝛼₀)+𝑔_𝑥(𝑡,𝛽₀)−𝐾]𝑝. This proves that 𝑝(𝑡)≤0. Therefore, 𝛼₁(𝑡)≤𝛽₁(𝑡) on 𝐽. Hence, (4.6) is proved.
Using mathematical induction with 𝑘>1, we obtain 𝛼₀≤𝛼_𝑘−1≤𝛼_𝑘≤𝛽_𝑘≤𝛽_𝑘−1≤𝛽₀on𝐽. (4.13) We must prove that 𝛼_𝑘≤𝛼_𝑘+1≤𝛽_𝑘+1≤𝛽_𝑘on𝐽. (4.14) To do so, we set 𝑝(𝑡)=𝛼_𝑘−𝛼_𝑘+1. Then, ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼_𝑘−^𝑐𝐷^𝑞𝛼_𝑘+1=𝐹(𝑡,𝛼_𝑘−1)+[𝑓_𝑥(𝑡,𝛼_𝑘−1)+𝑔_𝑥(𝑡,𝛽_𝑘−1)−𝐾](𝛼_𝑘−𝛼_𝑘−1)−𝐹(𝑡,𝛼_𝑘)−[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛼_𝑘+1−𝛼_𝑘)=[𝑓(𝑡,𝛼_𝑘−1)−𝑓(𝑡,𝛼_𝑘)]+[𝑔(𝑡,𝛼_𝑘−1)−𝑔(𝑡,𝛼_𝑘)]+[ℎ(𝑡,𝛼_𝑘−1)−ℎ(𝑡,𝛼_𝑘)]+[𝑓_𝑥(𝑡,𝛼_𝑘−1)+𝑔_𝑥(𝑡,𝛽_𝑘−1)−𝐾](𝛼_𝑘−𝛼_𝑘−1)−[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛼_𝑘+1−𝛼_𝑘)≤𝑓_𝑥(𝑡,𝛼_𝑘−1)(𝛼_𝑘−1−𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘−1)(𝛼_𝑘−1−𝛼_𝑘)+𝐾(𝛼_𝑘−𝛼_𝑘−1)+𝑓_𝑥(𝑡,𝛼_𝑘−1)(𝛼_𝑘−𝛼_𝑘−1)+𝑔_𝑥(𝑡,𝛽_𝑘−1)(𝛼_𝑘−𝛼_𝑘−1)+[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)](𝛼_𝑘−𝛼_𝑘+1)+𝐾(𝛼_𝑘−1−𝛼_𝑘+𝛼_𝑘+1−𝛼_𝑘)=[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛼_𝑘−𝛼_𝑘+1), (4.15) 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 ^𝑐𝐷^𝑞𝑝≤[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾]𝑝,𝑝(𝑡₀)=0. (4.16) Again, from Corollary 2.2, we get 𝛼_𝑘≤𝛼_𝑘+1 on 𝐽. Similarly, it can be shown that 𝛽_𝑘+1≤𝛽_𝑘 on 𝐽.
Next we need to show that 𝛼_𝑘+1≤𝛽_𝑘+1 on 𝐽.
Set 𝑝(𝑡)=𝛼_𝑘+1−𝛽_𝑘+1; then, ^𝑐𝐷^𝑞𝑝=^𝑐𝐷^𝑞𝛼_𝑘+1−^𝑐𝐷^𝑞𝛽_𝑘+1=𝐹(𝑡,𝛼_𝑘)+[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛼_𝑘+1−𝛼_𝑘)−𝐹(𝑡,𝛽_𝑘)+[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛽_𝑘+1−𝛽_𝑘)≤𝑓_𝑥(𝑡,𝛼_𝑘)(𝛼_𝑘−𝛽_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)(𝛼_𝑘−𝛽_𝑘)−𝐾(𝛼_𝑘−𝛽_𝑘)+[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛼_𝑘+1−𝛼_𝑘−𝛽_𝑘+1+𝛽_𝑘)≤[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾](𝛼_𝑘+1−𝛽_𝑘+1). (4.17)
Thus, we have   ^𝑐𝐷^𝑞𝑝≤[𝑓_𝑥(𝑡,𝛼_𝑘)+𝑔_𝑥(𝑡,𝛽_𝑘)−𝐾]𝑝 and 𝑝(𝑡₀)=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(𝑡₀)=𝑟_𝑛+1(𝑡₀)=0 and ^𝑐𝐷^𝑞𝑝_𝑛+1=^𝑐𝐷^𝑞𝑥−^𝑐𝐷^𝑞𝛼_𝑛+1=𝐹(𝑡,𝑥)−𝐹(𝑡,𝛼_𝑛)−[𝑓_𝑥(𝑡,𝛼_𝑛)+𝑔_𝑥(𝑡,𝛽_𝑛)−𝐾](𝛼_𝑛+1−𝛼_𝑛)=𝑓_𝑥(𝑡,𝜉)(𝑥−𝛼_𝑛)+𝑔_𝑥(𝑡,𝜂)(𝑥−𝛼_𝑛)+𝐾(𝑥−𝛼_𝑛)+[𝑓_𝑥(𝑡,𝛼_𝑛)+𝑔_𝑥(𝑡,𝛽_𝑛)−𝐾](𝛼_𝑛−𝛼_𝑛+1), (4.18) where 𝛼_𝑛≤𝜉,𝜂≤𝑥. Now using the nondecreasing property of 𝑓_𝑥 and nonincreasing property of 𝑔_𝑥, we get ^𝑐𝐷^𝑞𝑝_𝑛+1≤[𝑓_𝑥(𝑡,𝑥)+𝑔_𝑥(𝑡,𝛼_𝑛)+𝐾]𝑝_𝑛+[𝑓_𝑥(𝑡,𝛼_𝑛)+𝑔_𝑥(𝑡,𝛽_𝑛)−𝐾](𝑝_𝑛+1−𝑝_𝑛)=[𝑓_𝑥(𝑡,𝑥)−𝑓_𝑥(𝑡,𝛼_𝑛)]𝑝_𝑛+[𝑔_𝑥(𝑡,𝛼_𝑛)−𝑔_𝑥(𝑡,𝛽_𝑛)]𝑝_𝑛+2𝐾𝑝_𝑛+[𝑓_𝑥(𝑡,𝛼_𝑛)+𝑔_𝑥(𝑡,𝛽_𝑛)−𝐾]𝑝_𝑛+1≤𝐿₁𝑝²_𝑛+𝐿₂(𝑝_𝑛+𝑟_𝑛)𝑝_𝑛+2𝐾𝑝_𝑛+[𝑓_𝑥(𝑡,𝛼_𝑛)+𝑔_𝑥(𝑡,𝛽_𝑛)−𝐾]𝑝_𝑛+1≤(𝐿₁+32𝐿₂)𝑝²_𝑛+12𝐿₂𝑟²_𝑛+2𝐾𝑝_𝑛+[𝑓_𝑥(𝑡,𝛼_𝑛)+𝑔_𝑥(𝑡,𝛽_𝑛)−𝐾]𝑝_𝑛+1. (4.19) Thus, we have ^𝑐𝐷^𝑞𝑝_𝑛+1≤(𝐿₁+32𝐿₂)|𝑝_𝑛|²₀+12𝐿₂|𝑟_𝑛|²₀+2𝐾|𝑝_𝑛|₀+𝐾^∗𝑝_𝑛+1, (4.20) where |𝑝_𝑛|₀=max_𝑡∈𝐽|𝑝_𝑛(𝑡)|, |𝑟_𝑛|₀=max_𝑡∈𝐽|𝑟_𝑛(𝑡)|, |𝑓_𝑥(𝑡,𝛼_𝑛)|≤𝐾₁, |𝑔_𝑥(𝑡,𝛽_𝑛)|≤𝐾₂, and 𝐾^∗=𝐾₁+𝐾₂−𝐾.
Then, we obtain ^𝑐𝐷^𝑞𝑝_𝑛+1≤((𝐿₁+32𝐿₂)|𝑝_𝑛|²₀+12𝐿₂|𝑟_𝑛|²₀+2𝐾|𝑝_𝑛|₀)^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝐸_𝑞,𝑞(𝐾^∗(𝑡−𝑠))𝑑𝑠, (4.21) where 𝐸_𝑞,𝑞 is the Mittag-Leffler function.
Let 𝑊=(1/𝑞)(𝑇−𝑡₀)^𝑞𝐸_𝑞,𝑞(𝐾^∗(𝑇−𝑡₀)^𝑞); then, ^𝑐𝐷^𝑞𝑝_𝑛+1≤𝑈₁|𝑝_𝑛|²₀+𝑈₂|𝑟_𝑛|²₀+𝑈₃|𝑝_𝑛|₀, (4.22) where 𝑈₁=(𝐿₁+(3/2)𝐿₂)𝑊,𝑈₂=(1/2)𝐿₂𝑊, and 𝑈₃=2𝑘𝑊.
Thus, we reach the desired result max_[𝑡0,𝑇]|𝑥−𝛼_𝑛+1|≤𝑈₁max_[𝑡0,𝑇]|𝑥−𝛼_𝑛|²+𝑈₂max_[𝑡0,𝑇]|𝛽_𝑛−𝑥|²+𝑈₃max_[𝑡0,𝑇]|𝑥−𝛼_𝑛| (4.23) which shows the semiquadratic convergence.
Similarly, using suitable computation, we arrive at max_[𝑡0,𝑇]|𝛽_𝑛+1−𝑥|≤𝑉₁max_[𝑡0,𝑇]|𝑥−𝛼_𝑛|²+𝑉₂max_[𝑡0,𝑇]|𝛽_𝑛−𝑥|²+𝑉₃max_[𝑡0,𝑇]|𝛽_𝑛−𝑥|, (4.24) where 𝑉₁=(1/2)𝐿₁𝑊, and 𝑉₂=((3/2)𝐿₁+𝐿₂)𝑊, and 𝑉₃=2𝑘𝑊.

Remark 4.2. Let 𝑓(𝑡,𝑥)+𝑔(𝑡,𝑥)=0; then, we have the monotone method, and the convergence is linear.

Remark 4.3. Let 𝑔(𝑡,𝑥)+ℎ(𝑡,𝑥)=0; then, Theorem 4.1 reduces to Theorem 3.1 of [4], and the convergence is quadratic.

Remark 4.4. Let 𝑓(𝑡,𝑥)+ℎ(𝑡,𝑥)=0 and 𝑔(𝑡,𝑥) be concave; then, Theorem 4.1 reduces to Theorem 4.1 of [4], and the convergence is quadratic.

Remark 4.5. Let ℎ(𝑡,𝑥)=0; then, Theorem 4.1 reduces to the theorem in [5].

5. 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 𝑡₀=0.

Example 5.1. Let us consider the following nonlinear fractional-order initial value problem with Caputo's derivative order 𝑞=1/2: ^𝑐𝐷^1/2𝑥(𝑡)=𝑡𝜋𝑥²−3𝑡𝜋𝑥+𝑡𝜋(1+𝑥),𝑥(0)=0,0≤𝑡≤10, (5.1) where 𝑓(𝑡,𝑥)=𝑡/𝜋𝑥²,𝑔(𝑡,𝑥)=−3𝑡/𝜋𝑥,ℎ(𝑡,𝑥)=𝑡/𝜋(1+𝑥) and 𝑓,𝑔,ℎ∈𝐶([0,10]×ℝ,ℝ).
Let 𝛼₀(𝑡)=(1/2)(1−1/√1+𝑡),𝛼₀(0)=0 and 𝛽₀(𝑡)=2(1−1/√1+𝑡),𝛽₀(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, 𝛼₀(𝑡) and 𝛽₀(𝑡) satisfy the inequalities in assumption (i) as ^𝑐𝐷^1/2𝛼₀≤𝑡𝜋𝛼²₀−3𝑡𝜋𝛼₀+𝑡𝜋(1+𝛼₀),𝛼₀(0)≤0for0≤𝑡≤10,_𝑐𝐷^1/2𝛽₀≥𝑡𝜋𝛽²₀−3𝑡𝜋𝛽₀+𝑡𝜋(1+𝛽₀),𝛽₀(0)≥0for0≤𝑡≤10. (5.2)
On the other hand, 𝑓_𝑥(𝑡,𝑥)=2𝑡/𝜋𝑥, 𝑔_𝑥(𝑡,𝑥)=−3𝑡/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 𝐿₁≥210/𝜋, 𝐿₂≥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 ^𝑐𝐷^1/2𝛼_𝑘+1=𝑡𝜋𝛼²_𝑘−3𝑡𝜋𝛼_𝑘+𝑡𝜋(1+𝛼_𝑘)+[2𝑡𝜋𝛼_𝑘−3𝑡𝜋−10𝜋](𝛼_𝑘+1−𝛼_𝑘𝛼),_𝑘+1(0)=0for𝐾=10𝜋,0≤𝑡≤10,_𝑐𝐷^1/2𝛽_𝑘+1=𝑡𝜋𝛽²_𝑘−3𝑡𝜋𝛽_𝑘+𝑡𝜋(1+𝛽_𝑘)+[2𝑡𝜋𝛼_𝑘−3𝑡𝜋−10𝜋](𝛽_𝑘+1−𝛽_𝑘𝛽),_𝑘+1(0)=0for𝐾=10𝜋,0≤𝑡≤10. (5.3)(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,𝛼₀≤𝛼₁≤⋯≤𝛼_𝑘≤𝛼_𝑘+1≤𝛽_𝑘+1≤𝛽_𝑘≤⋯≤𝛽₁≤𝛽₀on𝐽=[0,10]. (5.5)
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 ^𝑐𝐷^1/2𝑥(𝑡)=𝑡/𝜋𝑥²−3𝑡/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 [𝛼₀,𝛽₀1]=[𝑥∶21(1−√11+𝑡)≤𝑥≤2(1−√1+𝑡)]for0≤𝑡≤10. (5.6) Moreover, the convergence is semiquadratic.

Acknowledgment

This work has been supported by The Scientific and Technological Research Council of Turkey and the Department of Mathematics at Yeditepe University.