Bagley-Torvik型分数阶微分方程和包含非局部积分边值问题

Nonlocal Integral Boundary Value Problem of Bagley-Torvik Type Fractional Differential Equations\\ and Inclusions

中文关键词:  分数阶微分方程和包含  积分边值问题  Leray-Schauder度理论

英文关键词:fractional differential equations and inclusions  integral boundary conditions  Leray-Schauder theory

中文摘要:

      本文考虑了满足边界条件$l(0)=l_0$, $l(1)= \lambda' \int_0^{\omega}\frac{(\omega-s)^{\chi-1}l(s)}{\Gamma(\chi)}\d s$的Bagley-Torvik型分数阶微分方程$^{c}D^{\nu_1}l(t)-a^{c}D^{\nu_2}l(t)=g(t,l(t))$和微分包含$^{c}D^{\nu_1}l(t)-a^{c}D^{\nu_2}l(t)\in G(t,l(t))$, $t\in (0,1)$解的存在性, 其中$1<\nu_1\leq 2$, $1\leq \nu_2<\nu_1$, $0<\omega\leq1$, $\chi=\nu_1-\nu_2>0$, $a$, $\lambda'$为给定常数.通过使用Leray-Schauder度理论和不动点定理,我们证明了上述边值问题解的存在性. 我们的结果扩展了经典Bagley-Torvik方程和一些相关模型解的存在性定理.

英文摘要:

      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.