Boundary value problem for linear and nonlinear fractional differential equations

Tengteng Ma, Yu Tian, Qiang Huo, Yong Zhang

Abstract: In this paper, we are interested in the boundary conditions u(0)=au(1),u′(0)=bu′(1)<math><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mn is="true">0</mn><mo is="true">)</mo></mrow><mo is="true">=</mo><mi is="true">a</mi><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mo is="true">,</mo><mspace width="0.33em" class="nbsp" is="true"></mspace><msup is="true"><mrow is="true"><mi is="true">u</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">0</mn><mo is="true">)</mo></mrow><mo is="true">=</mo><mi is="true">b</mi><msup is="true"><mrow is="true"><mi is="true">u</mi></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></math> for linear fractional differential equation 0cDtαu(t)+λu(t)=0<math><msubsup is="true"><mrow is="true"></mrow><mrow is="true"><mn is="true">0</mn></mrow><mrow is="true"><mi is="true">c</mi></mrow></msubsup><msubsup is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">t</mi></mrow><mrow is="true"><mi is="true">α</mi></mrow></msubsup><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mi is="true">t</mi><mo is="true">)</mo></mrow><mo is="true">+</mo><mi is="true">λ</mi><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mi is="true">t</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><mn is="true">0</mn></math>. Via Laplace transform and inverse Laplace transform, we obtain eigenvalues and eigenfunctions. Furthermore, we study the same boundary conditions for nonlinear fractional differential equation 0cDtαu(t)+f(t,u(t))=0<math><msubsup is="true"><mrow is="true"></mrow><mrow is="true"><mn is="true">0</mn></mrow><mrow is="true"><mi is="true">c</mi></mrow></msubsup><msubsup is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">t</mi></mrow><mrow is="true"><mi is="true">α</mi></mrow></msubsup><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mi is="true">t</mi><mo is="true">)</mo></mrow><mo is="true">+</mo><mi is="true">f</mi><mrow is="true"><mo is="true">(</mo><mi is="true">t</mi><mo is="true">,</mo><mi is="true">u</mi><mrow is="true"><mo is="true">(</mo><mi is="true">t</mi><mo is="true">)</mo></mrow><mo is="true">)</mo></mrow><mo is="true">=</mo><mn is="true">0</mn></math>. Combining the obtained eigenvalues and eigenfunctions and the improved Leray–Schauder degree, we prove that there exists at least one nontrivial solution for nonlinear boundary value problem.