Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations

The quasilinearization method is applied to a boundary value problem at resonance for a Riemann-Liouville fractional differential equation. Under suitable hypotheses, the method of upper and lower solutions is employed to establish uniqueness of solutions. A shift method, coupled with the method of upper and lower solutions, is applied to establish existence of solutions. The quasilinearization algorithm is then applied to obtain sequences of lower and upper solutions that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance.


1.
Introduction. The method of quasilinearization was introduced by Bellman and Kalba [7,8]. The method, as constructed here, is rather remarkable as both existence and uniqueness of solutions is established and a bilateral monotone iteration scheme is produced to approximate solutions of nonlinear problems with solutions of linear problems. Under suitable hypotheses, the sequences of approximate solutions converge quadratically to the unique solution.
Applications of quasilinearization are extensive. We cite [18,19,20,23] for applications to initial value problems for ordinary differential equations and we cite [1,2,10,11,16,21] for applications to boundary value problems for ordinary differential equations. More recently, researchers have successfully applied the method to fractional differential equations; see [6,25,27] for applications to initial value problems for fractional differential equations and see [9,15] for applications to boundary value problems for fractional differential equations.
Quasilinearization, coupled with a shift method, has been shown to apply to boundary value problems at resonance. In the case of ordinary differential equations, see, for example, [5,24,28]. The purpose of this study is to apply the quasilinearization method, coupled with a shift method, to a boundary value problem at resonance for a fractional differential equation of Riemann-Liouville type.
Uniqueness of solutions is essential to the algorithm, and in a recent article, [3], those authors considered a problem at resonance for an ordinary differential equation in which a new argument to obtain uniqueness of solutions was produced. In this article, we consider an analogous boundary value problem for the fractional differential equation and in doing so, produce a new argument for uniqueness of solutions. We stress that uniqueness of solutions is essential in this work and so this work differs from that in [24] or [28] where multiplicity of solutions is the motivation.
In Section 2 we provide preliminary definitions and state analogues of the second derivative test for fractional derivatives obtained in [4] and in [26]. In Section 3, we introduce the two-point fractional boundary value problem at resonance that is studied in this work. The method of upper and lower solutions is employed to obtain uniqueness of solutions. A shift method is applied and a Green's function is constructed using the Laplace transform method. Existence of solutions is then obtained through an application of the Schauder fixed point theorem. In Section 4, we apply the quasilinearization algorithm and construct a sequences of upper solutions and lower solutions that converge monotonically and quadratically to the unique solution. In Section 5, we provide a short conclusion.

Preliminaries.
Definition 2.1. [17] Let 0 < α and a ∈ R. The α th −order Riemann-Liouville fractional integral of a function y is defined by provided the right-hand side exists. For α = 0, define I α a to be the identity map. Moreover, let n denote a positive integer and assume n − 1 < α ≤ n. The α th -order Riemann-Liouville fractional derivative is defined as where D n denotes the classical n th −order derivative, if the right-hand side exists.
Definition 2.2. [17] Let m ∈ N 0 = {0, 1, 2, · · · }. We denote by C m [0, 1] a space of functions y which are m times continuously differentiable on [0, 1] with the norm In particular, for m = 0, C 0 [0, 1] = C[0, 1] is the space of continuous functions y on [0, 1] with the norm y C = max The following two theorems are analogues of the second derivative test and are proved (for a global minimum value) in [4] and [26]. These are important results for applications of upper and lower solutions to fractional differential equations.
The condition y ∈ C 2 [0, 1] is a very strong condition for applications to Riemann-Liouville fractional differential equations and so the following result has been obtained to address this difficulty.
Theorem 2.5. [26] Let 0 < ν < 1. Assume that y ∈ C(0, 1] satisfies the following conditions: Then, 3. Uniqueness of solutions and existence of solutions. Let 1 < α < 2 and assume throughout that f : [0, 1] × R → R is continuous. We consider the two point boundary value problem for a Riemann-Liouville fractional differential equation, The fractional boundary value problem (3) -(4) is at resonance because constant multiples of t α−1 satisfy the homogeneous boundary value problem Throughout, we shall assume that f is increasing in the second component. In the case of second order ordinary differential equations, this monotone assumption, coupled with the second derivative test, is standard to obtain uniqueness of solutions. Proof. Assume for the sake of contradiction that y 1 and y 2 denote two distinct solutions of the boundary value problem (3) - (4) Without loss of generality assume that u(t) has a positive maximum at t 0 ∈ [0, 1]. First, assume t 0 ∈ (0, 1). Then, u(t 0 ) > 0. Apply Theorem 2.4, and D α 0 u(t 0 ) < 0. However, y 1 and y 2 each satisfy (3), and so Thus, u(t) does not have a positive maximum at t 0 ∈ (0, 1). We shall refer to this argument as the usual contradiction.
Also, assume w and v are lower and upper solutions of the fractional boundary value problem (3) -(4). Then, Proof. The proof of this theorem is very similar to the proof of the uniqueness theorem, Theorem 3.1. Assume w is a lower solution and v is an upper solution of the fractional boundary value problem (3) -(4), respectively. Assume for the sake of contradiction that w ≤ v is false. Assume that (w − v)(t) has a positive maximum at t 0 ∈ [0, 1].
Next, we assume The proof for t 0 = 1 is similar to the proof of Theorem 3.1. It is simply a matter of replacing the second equality in each of (5) and (6) with the appropriate differential inequality.
We now address existence of solutions of the fractional boundary value problem (3) -(4). A shift argument [14] will be applied to obtain an equivalent boundary value problem that is not at resonance and then an appropriate Green's function is constructed, employing Mittag -Leffler functions. We use definitions and properties of Mittag-Leffler functions that are commonly used and refer the reader to [22] or [13]. Definition 3.4. Let α, β > 0. A two-parameter function of the Mittag-Leffler type is defined by the series expansion given by .
Lemma 3.5. The following relations hold: To obtain existence of solutions, apply a shift argument [14]. Assume K = 0 and consider the equivalent shifted equation The fractional boundary value problem (7) - (4) is not at resonance since Theorem 3.1 implies that y ≡ 0 is the only solution of the homogeneous fractional problem D α 0 y(t) = K 2 y(t) satisfying the boundary conditions, (4), for any K = 0.
Since the fractional boundary value problem (7) - (4) is not at resonance, we shall construct the corresponding Green's function of the shifted equation. To do so, apply the Laplace transform to Apply the inverse Laplace transform to obtain .
Apply D α−1 0 to (8) and obtain Note that The property (11) is observed in [17] and [22]. For the sake of self-containment, we provide some details.
Note that Substitute (10) and (12) into (11) and we obtain Now substitute (10) and (13) into (9) and we obtain Now substitute (14) into (8) to obtain Define a Green's function where and and write (8) as We derive two standard properties of the Green's function, G(K; t, s). First, we show that for K sufficiently small and lim K→0 E α,1 (K 2 ) = 1. Since Then, for K sufficiently small,   Let Then, Proof. Let K = 0 and define a truncation off (t, y(t)) = f (t, y(t)) − K 2 y(t) by 1+w(t)−y(t) , if y(t) < w(t). Define an operator T : where G(K; t, s) is given by (15). Then, y is a solution of the fractional boundary value problem Then, Then, F (t, y(t)) = f (t, y(t)) and it follows that a continuous fixed point y of T is a solution of the original fractional boundary value problem (3) -(4).
We show the details that (y−v)(t) doesn't have a positive maximum at t 0 ∈ [0, 1]. Assume for the sake of contradiction that (y − v)(t) have a positive maximum at t 0 ∈ (0, 1). Then, Since v is an upper solution of (3) -(4), it follows that This contradicts Theorem 2.4 and so, (y−v)(t) does not have a positive maximum at t 0 ∈ (0, 1).
The arguments to show (y − v)(t) does not have a positive maximum at t 0 = 0 and t 0 = 1 follow analogously to the proof in Theorem 3.1.
The uniqueness of the continuous solution, y, follows immediately since the hypotheses of Theorem 3.1 are contained in the hypotheses of Theorem 3.6.
4. The monotone method and quadratic convergence. In this section, we briefly present the monotone method and obtain a quadratic rate of convergence; Once the uniqueness and existence results from Section 3 have been obtained, the implementation of the quasilinearization algorithm is routine.
To obtain the monotone method, assume one further condition on f , that f yy exists and f yy ≥ 0.
The rate of convergence of each sequence {w n }, {v n } is quadratic.
Proof. Let w 0 , v 0 denote a lower and an upper solution of (3) -(4), respectively. So, under the assumption that f y > 0 on [0, 1] × R, we have and consider the boundary value problem for the linear non-homogeneous fractional differential equation Note that h(w 0 , v 0 ; t, w 0 (t)) = f (t, w 0 (t)), 0 ≤ t ≤ 1, and so, In particular, (26) and (27) imply w 0 , v 0 are lower and upper solutions of (25) respectively as well. Since, h satisfies the hypotheses of Theorem 3.6, there exists a continuous solution, w 1 (t), of (25) satisfying Next, we observe that w 1 is a lower solution (3) -(4). To see this, note that there exists w 0 (t) ≤ c(t) ≤ w 1 (t) ≤ v 0 (t) such that f (t, w 1 (t)) − f (t, w 0 (t)) = f y (t, c(t))(w 1 (t) − w 0 (t)) ≤ f y (t, v 0 (t))(w 1 (t) − w 0 (t)) and so, and consider the boundary value problem for the linear nonhomogeneous fractional differential equation (28). Note that there exists c(t) satisfying and so, w 0 is a lower solution of (28). Since k satisfies the hypotheses of Theorem 3.6 there exists a continuous solution, v 1 (t), of (28) satisfying An application of the mean value theorem again will give, Thus, , v 1 is an upper solution of (3) -(4). Finally, apply Theorem 3.3 to obtain Apply Theorem 3.6 with lower and upper solutions, w 1 and v 1 , respectively, and keeping in mind that the solution y obtained in Theorem 3.6 is unique, we obtain where y is the unique solution of the fractional boundary value problem, (3) - (4).
For the inductive step, assume the sequences {w k } n k=1 and {v k } n k=1 have been constructed such that for each k = 1, . . . , n, where w k is the solution of the fractional boundary value problem Moreover, w k , v k , k = 1, . . . , n denote lower and upper solutions, respectively of (3) -(4), and y is the unique solution of the fractional boundary value problem (3) -(4).
To complete the induction argument, consider the boundary value problem for the linear nonhomogeneous fractional differential equation (29) Note that h(w n , v n ; t, w n (t)) = f (t, w n (t)), 0 ≤ t ≤ 1, and h(w n , v n ; t, v n (t)) ≥ f (t, v n (t)), 0 ≤ t ≤ 1.
So, w n , v n denote a lower and an upper solution of (29) respectively as well.
Let y be the unique solution of the fractional boundary value problem (3) -(4). The arguments above to show the existence of w 1 (t) and v 1 (t) and the inequalities are readily adapted to show the existence of w n+1 (t) and v n+1 (t) and the inequalities To complete the proof of monotone convergence, {w n } and {v n } are monotone sequences of continuous functions bounded above and below, respectively, on a compact domain. So by Dini's theorem, each converges uniformly to continuous functions w and v respectively on [0, 1]. Since where the convergence is uniform on [0, 1], and where T is defined by (23), it follows that v = T v and v is the unique solution, y, of (3) -(4). Similarly, w is the unique solution, y, of (3) -(4).
We now obtain quadratic convergence and to do so, for each n, define the error e n by e n (t) = v n (t) − w n (t), 0 ≤ t ≤ 1.
So, 0 ≤ e n (t) for 0 ≤ t ≤ 1. Denote by e n C the error bound Assume without loss of generality that K > 0 is sufficiently small such that and where G(K; t, s) is defined by (15).
. Then By the mean value theorem, there exists c(t) satisfying w n (t) < c n (t) < v n (t) such that f (t, v n (t)) − f (t, w n (t)) = f y (t, c n (t))e n (t).
Employ the mean value theorem again for f y (t, c n (t)) − f y (t, v n (t)) and there existsĉ n (t) satisfying c n (t) <ĉ n (t) < v n (t) such that f y (t, c n (t)) − f y (t, v n (t)) = f yy (t,ĉ n (t))(c n (t) − v n (t)).