Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives

Abstract

<abstract><p>This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0<t<1;\\ x(0) = 0, \ ^{C}_{0}D^{1}_{g}x(0) = 0, \ ^{C}_{0}D^{\nu}_{g}x(1) = \int_{0}^{1}h(t)^{C}_{0}D^{\nu}_{g}x(t)g'(t)dt, \end{array}\right. $\end{document} </tex-math></disp-formula></p> <p>where $ 2 < \alpha < 3 $, $ 1 < \nu < 2 $, $ \alpha-\nu-1 > 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' > 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.</p></abstract>