An investigation of fractional Bagley-Torvik equation

Abstract In this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.

Fractional di erential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes [15,16]. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional di erential equations are also regarded as an alternative model to nonlinear di erential equations [17]. In consequence, the subject of fractional di erential equations is gaining much importance and attention.
The Bagley-Torvik equation is a prototype fractional di erential equation which was proposed by Bagley and Torvik as an application of fractional calculus to the theory of viscoelasticity [18][19][20]. The governing equation is given by the fractional di erential equation subject to initial conditions where x(t) represents the displacement of the plate of mass M and surface area S. Furthermore, µ and ρ are the viscosity and density, respectively, of the uid in which the plate is immersed, and K is the sti ness of the spring to which the plate is attached. Finally, f (x) represents the loading force.
In the current paper we investigate the existence and uniqueness as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our governing equation is a generalization of (1.1) to an arbitrary α with < α < in the fractional derivative term. In fact, we consider the following initial value equation subject to initial conditions where D α is the Caputo fractional derivative of order α, f : [ , ] → R is a given function and a, b, A and B are real numbers. For various applications in engineering and applied sciences elds, the Bagley-Torvik equation is extensively studied in literature both from numerical and theoretical point of view [3,[21][22][23][24][25][26][27][28][29][30]. Several numerical methods have been proposed for approximate solutions of this type equations, such as successive approximation method [3,Section 8.3], Adams predictor and corrector method [21], Taylor collocation method [22], hybridizable discontinuous Galerkin method [23], Discrete spline method [24] and others [25][26][27][28]. Also, Svatoslav Staněk [29] investigate the existence and uniqueness of solutions for generalized Bagley-Torvik fractional di erential equation subject to the boundary conditions. In [30], the authors investigate the general solution of the Bagley-Torvik equation with / -order derivative or / -order derivative. Furthermore, they show that the general solution of the Bagley-Torvik equation involves actually two free constants only, and it can be determined fully by the initial displacement and initial velocity.
Our main aim is to prove the existence and uniqueness as well as construct an approximate solution for (1.3)-(1.4). This is done in Section 3. Our main tools are some applicable partially xed point theorems which are applied in the suitable partially ordered sets as well as iterative methods, whose description can be found in [31][32][33][34][35]. The advantage and importance of this method arises from the fact that it is a constructive method that yields sequences that converge to the unique solution of (1.

Auxiliary facts and results
Here, we recall several known de nitions and properties from fractional calculus theory. For details, see [1][2][3]36]. Throughout the paper provided that the integral exists. For α = , we set I α := I, the identity operator.
De nition 2.2. The Caputo fractional derivative of order α > of a function x : [ , ] → R is de ned as where n − < α < n and n ∈ N, provided the right side is pointwise de ned on [ , ]. We notice that the Caputo derivative of a constant is zero. Note that if n − < α < n and x ∈ AC n− [ , ], then Proof. This is an immediate consequence of Lemma 2.1 and Lemma 2.2, because Proof. If a = we are done, and so we henceforth assume a ≠ . Given α > , choose n ∈ N so that n α > . Now, by applying the operator I α to both sides of or equivalently, Continuing this process to the n -th step, we get . The desired result is therefore a consequence of Lemma 2.4.

Lemma 2.6. x(t) is a solution of the problem (1.3)-(1.4) if and only if it is a solution of the following integral equation
Proof. Let us note that this result is mainly proved in [29]. Let x(t) be a solution of the problem (1.3)-(1.4). Then x ∈ AC [ , ] and the equality holds almost everywhere on [ , ]. Applying the integral operator I to both sides of (2.2) and using Lemma 2.1, we deduce is a solution of the integral equation (2.1). Now, we suppose that x ∈ C [ , ] is a solution of the integral equation (2.1). It is obvious that x( ) = a and x ( ) = b. By calculations similar to those in (2.3), we establish that Di erentiating twice, we get and so Now it su ces to show that x ∈ AC [ , ]. By di erentiating both sides of (2.1), we have , the desired result is therefore a consequence of Lemma 2.5.

Main results
Before continuing our investigation, we introduce a few fundamental concepts related to the required spaces and provide some partial order on them to serve as a background for the materials to be illustrated in the later sections.
By C [ , ] we denote the class of contiuously di erentiable functions on a nite interval [ , ] with the standard norm is a Banach space. Now, we de ne an appropriate partial order on C [ , ] and prove some essential properties in this partially ordered Banach space.
De nition 3.1. We de ne the following order relation for C [ , ], Dividing by A, B (distinguish the cases A · B > and A · B < ), we will consider two cases separately.

. Investigation in the case A · B <
In this subsection we consider the case in which A · B < . We consider only the case that A > and B < . The other case is completely similar. Before continuing, we need to introduce the coupled xed point theorems which play main role in our discussion. For complete details, see [34].

De nition 3.2.
Let (X, ) be a partially ordered set and G : X × X → X. We say that G has the mixed monotone property if G (x, y) is monotone non-decreasing in x and is monotone non-increasing in y.

De nition 3.3.
We call an element (x, y) ∈ X × X a coupled xed point of the mapping G if G (x, y) = x, and G (y, x) = y. If there exist x , y ∈ X such that x G (x , y ) and y G (y , x ), then G has a coupled xed point (x * , y * ) ∈ X × X.
We de ne the following partial order on the product space X × X:

Theorem 3.2.
In addition to the hypothesis of Theorem 3.1, suppose that for every (x, y), (x,ỹ) ∈ X × X, there exists an element (u, v) ∈ X × X that is comparable to (x, y) and (x,ỹ), then G has a unique coupled xed point (x * , y * ). In view of Lemma 2.6, we transform problem (1.3)-(1.4) as the following integral equation (3.1) in the set C [ , ].

2)
and Proof. In view of (3.1), we de ne the operator G : Obviously, for any x, y ∈ C [ , ], we have G (x, y) ∈ C [ , ]. On the other hand, and so the operator G is well de ned.
Now we shall show that G has the mixed monotone property. Let x, x ∈ C [ , ] with x x. Using the monotonicity of integral operator, we have y)(t) and G (x, y) y). Similarly, let y, y ∈ C [ , ] with y y. From the monotonicity of Riemann-Liouville fractional integral operator, we have y). Thus, G (x, y) is monotone non-decreasing in x and monotone non-increasing in y. Now, for x, y, x, y ∈ C [ , ] with x x, y y, we have |y (t) − y (t)|, and Therefore, ] . On the other hand, from (3.2), we have (3.9) for every t ∈ [ , ]. Now by applying the integral operator I on both sides of inequality (3.9) and using x( ) ≤ a, we deduce for every t ∈ [ , ]. Furthermore, using (3.3) and applying similar calculation, we get y (t) ≥ G (y , x )(t) and y (t) ≥ G (y , x )(t) for every t ∈ [ , ]. Therefore, x G (x , y ) and y G (y , x ). Moreover, both sequences {xn} and {yn} converge to x * which follow from Theorem 3.3. This proves assertion (ii). Now we prove the error estimates. From (3.8) it follows that ] . (3.13) Again employing (3.8) and using (3.12) and (3.13), we deduce By a mathematical induction, we obtain 14) Then for any m ≥ n ≥ , (3.16) Letting m → ∞ in both sides of (3.16), we can obtain the error estimate (3.4). A similar argument can also be used to prove error estimate (3.5). Finally, (3.6) follows immediately from (3.4) and (3.5). x · · · xn · · · x * · · · yn · · · y y . (3.17) Example 3.1. Let us consider the following problem

. Investigation in the case A · B >
In this subsection we consider the case in which A · B > . Before continuing, we need to introduce the xed point theorem which play main role in our discussion. For complete details, see [31][32][33].
Theorem 3.5. Let (X, ) be a partially ordered set such that every pair x, y ∈ X has a lower bound and an upper bound. Furthermore, let d be a metric on X such that (X, d) is a complete metric space and G is monotone (i.e., either order-preserving or order-reversing) map from X into X such that ∃ ≤ k < : d(G (x), G (y)) ≤ kd(x, y), ∀ x y, ∃x ∈ X : x G (x ) or x G (x ).
Suppose also that either G is continuous or X is such that if xn → x is a sequence in X whose consecutive terms are comparable, then there exists a subsequence {xn k } of {xn} such that every term is comparable to the limit x. Then G has a unique xed point x * . Moreover, for every x ∈ X, limn→∞ G n (x) = x * .
A relatively simple calculation, with the help of Maple, shows that x (t) = − t is a lower solution of problem (3.21). Therefore, all the assumption of Theorem 3.6 hold and consequently, problem (3.21) has a unique solution in C [ , ]. Moreover, the unique solution of (3.21) can be obtained as limn→∞ xn where xn = G (x n− ). The graphs of xn and yn, for n = , , are shown in Figure 3. Furthermore, the graphs of x n and y n , for n = , , are shown in Figure 4.