SOLVABILITY FOR RIEMANN-STIELTJES INTEGRAL BOUNDARY VALUE PROBLEMS OF BAGLEY-TORVIK EQUATIONS AT RESONANCE

Abstract In this paper, we study the solvability for Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equations with fractional derivative under resonant conditions. Firstly, the kernel function is presented through the Laplace transform and the properties of the kernel function are obtained. And then, some new results on the solvability for the boundary value problem are established by using Mawhin’s coincidence degree theory. Finally, two examples are presented to illustrate the applicability of our main results.


Introduction
Since the viscoelastic medium damping could not be well described through the forced vibration equation of integer order, many researchers use fractional integral or fractional derivative to describe the properties of viscoelastic materials. Therefore, fractional differential equation is playing an important role in describing viscous damping model, see [13,14,20,22,25]. In [25], Torvik and Bagley established generalized constitutive relation for viscoelastic materials in which the customary time derivatives of integer order are replaced by derivatives of fractional order and homogeneous Bagley-Torvik equation was also obtained Ay ′′ (t) + B 0 D 3 2 t y(t) + Cy(t) = 0. In [20], Podlubny studied the initial value problems for the inhomogeneous Bagley-Torvik equation As it well known that fractional differential equations have been studied for a long time. In the recent decades, a lot of research results have been published on the theory of boundary value problems of fractional differential equations, see [1-3, 5, 7-9, 15-18, 23, 26-28]. As far as we know, the resonance problem has to be considered in the theoretical study of vibration equation. In [4,10,12,19,24], the authors have studied the solvability of the fractional vibration equation under the resonant conditions. In [24], Stanek investigated the nonlocal fractional boundary value problems at resonance where α ∈ (1, 2), µ ∈ (0, 1). The existence of solutions of the problem are given by using the Leray-Schauder degree method. Since Riemann-Stieltjes integral boundary conditions not only contain the classical Riemann integral boundary conditions but also two-point boundary value and multi-point boundary conditions, Riemann-Stieltjes integral boundary value problems have much wider application.
Motivated by the above works, we study the following Riemann-Stieltjes integral boundary value problems of Bagley-Torvik equation with Caputo fractional derivative under the resonant condition is the Riemann-Stieltjes integral of x with respect to A and A(t) is a monotone increasing function and not a constant on t ∈ [0, 1], f : [0, 1] × R 2 → R is continuous.
By using the Laplace transform, the kernel function is obtained. And then, by using Mawhin's coincidence degree theory, we establish some new results of the solvability for boundary value problem (1.1) under the resonance condition g 1 (1) = 1 0 g 1 (t)dA(t), where the definition of g 1 (t) see (2.2). In order to illustrate the applicability of our main results, two examples are presented.

Preliminaries
The definitions of fractional integral, fractional derivative and Laplace transform and the related lemmas can be found in [6,11,20].
whenever the series converges is called the two-parameter Mittag-Leffler function with parameters β and δ.
(2) In view of Similar to the proof of (1), we can get (2) holds.
is monotone decreasing with respect to n ∈ N + . Then g 1 (t) is alternating series.
Furthermore, a n n! t 2n+1 is monotone decreasing with n ∈ N + . Thus, a n n! t 2n+1 E is monotone decreasing with n ∈ N + and converges to 0 as n → ∞. Therefore, according to Leibniz test for alternating series, we have g 1 (t) > 0 for t ∈ (0, 1].
Similar to the proof above, we can get g 2 (t) > 0 for t ∈ (0, 1]. By Lemma 2.5, we can get E Similar to the proof above, we can get Since Then for j ≥ 1, we can get Thus, {w j } is monotone decreasing. By Lemma 2.1, w j → 0 as j → ∞. Therefore, according to Leibniz test for alternating series, we have On the other hand, since 0 ≤ b ≤ min{1, (4) By Lemma 2.6 (2), we can show that On the other hand, by Lemma 2.5, similar to the proof of (3), we can show that Thus, {z j } is monotone decreasing. By Lemma 2.1, we have z j → 0 as j → ∞. Therefore, according to Leibniz test for alternating series, In addition,

The existence of the solutions
Throughout this paper, we always suppose that the following resonance condition is satisfied has general solution

Proof. By Lemma 2.3, we have
Apply Laplace transform to both sides of the equation (3.2), we can easily obtain By virtue of Lemma 2.4, we can show  By Lemma 3.1, we can obtain the following Lemma 3.2 holds. 1] |x(t)|. Obviously, (X, ∥ · ∥) is a Banach space.
Define the operators

7)
N : where  It is easy to see that For y ∈ ImL, there exists x ∈ DomL such that x(0) = 0 and Lx = y. By Lemma 3.1, we have Thus, If y ∈ {y ∈ X : On the other hand,
Step 2: The operator N is L-compact on any bounded open set Ω ⊂ X. Let K P : ImL → X, In views of (3.9) and (3.10), we can get K P : ImL → DomL ∩ KerP and for y ∈ ImL, Then K P is the inverse mapping of L| DomL∩KerP . Hence, and Since f is measurable, G and g 1 are continuous, we can easily get that QN : Ω → X and K P (I − Q)N : Ω → X are continuous and compact operators, that is, the operator N is L-compact on any bounded open set Ω ⊂ X.
Through calculation, we have It follows from Theorem 3.1 that boundary value problem (4.2) has at least one solution.