Abstract
AbstractIn this paper, we concerned the existence of solutions of the following nonlinear mixed fractional differential equation with the integral boundary value problem:
$$\left \{ \textstyle\begin{array}{l} {}^{C}D^{\alpha}_{1-} D^{\beta}_{0+}u(t)=f(t,u(t),D^{\beta +1}_{0+}u(t),D^{\beta}_{0+}u(t)),\quad 0< t< 1,\\ u(0)=u'(0)=0,\qquad u(1)=\int^{1}_{0}u(t)\,dA(t), \end{array}\displaystyle \right . $${D1−αCD0+βu(t)=f(t,u(t),D0+β+1u(t),D0+βu(t)),0<t<1,u(0)=u′(0)=0,u(1)=∫01u(t)dA(t), where ${}^{C}D^{\alpha}_{1-}$D1−αC is the left Caputo fractional derivative of order $\alpha\in(1,2]$α∈(1,2], and $D^{\beta}_{0+}$D0+β is the right Riemann–Liouville fractional derivative of order $\beta\in(0,1]$β∈(0,1]. The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems.
Funder
Natural Science Foundation of China
Shandong Natural Science Foundation
SDUST graduate innovation project
Tai’shan Scholar Engineering Construction Fund of Shandong Province of China
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
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