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 ) ,