scholarly journals Fractional Euler–Lagrange Differential Equations via Caputo Derivatives

2011 ◽  
pp. 109-118 ◽  
Author(s):  
Ricardo Almeida ◽  
Agnieszka B. Malinowska ◽  
Delfim F. M. Torres
Keyword(s):  
Differential Equations ◽  
Caputo Derivatives

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

2021 ◽  
Vol 6 (12) ◽  
pp. 13119-13142
Author(s):  
Yating Li ◽  
◽  
Yansheng Liu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Ulam Stability ◽  
Caputo Derivatives ◽  
Fractional Differential ◽  
The Fixed Point Theorem

<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&lt;t&lt;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 &lt; \alpha &lt; 3 $, $ 1 &lt; \nu &lt; 2 $, $ \alpha-\nu-1 &gt; 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' &gt; 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>

Sign in / Sign up

Export Citation Format

Share Document