Nonlocal Integral Boundary Value Problem of Bagley-Torvik Type Fractional Differential Equations\\ and Inclusions
Received:June 02, 2018  Revised:April 11, 2019
Key Word: fractional differential equations and inclusions   integral boundary conditions   Leray-Schauder theory
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11571300; 11871064).
 Author Name Affiliation Lizhen CHEN School of Applied Mathematics, Shanxi University of Finance and Economics, Shanxi 030006, P. R. China Badawi Hamza Eibadawi IBRAHIM School of Mathematical Sciences, Yangzhou University, Jiangsu 225002, P. R. China Gang LI School of Mathematical Sciences, Yangzhou University, Jiangsu 225002, P. R. China
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In this article, we consider the Bagley-Torvik type fractional differential equation $^{c}D^{\nu_1}l(t)-a^{c}D^{\nu_2}l(t)=g(t,l(t))$ and differential inclusion $^{c}D^{\nu_1}l(t)-a^{c}D^{\nu_2}l(t)\in G(t,l(t))$, $t\in (0,1)$ subjecting to $l(0)=l_0$, and $l(1)=\lambda'\int_0^{\omega}\frac{(\omega-s)^{\chi-1}l(s)}{\Gamma(\chi)}\d s$, where $1<\nu_1\leq 2$, $1\leq \nu_2<\nu_1$, $0<\omega\leq1$, $\chi=\nu_1-\nu_2>0$, $a$, $\lambda'$ are given constants. By using Leray-Schauder degree theory and fixed point theorems, we prove the existence of solutions. Our results extend the existence theorems for the classical Bagley-Torvik equation and some related models.