Bulletin of the Korean Mathematical Society (대한수학회보)

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NONTRIVIAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Guo, Yingxin (Department of Mathematics, Qufu Normal University)
  • Published : 2010.01.31
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Abstract

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) $-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]$, 0 < t < 1 u(0) = u(1) = 0, where $\lambda$ > 0 is a parameter, 1 < $\alpha$ $\leq$ 2, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville differentiation, f : [0, 1] ${\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is continuous, and q(t) : (0, 1) $\rightarrow$ [0, $+\infty$] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when $\lambda$ in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.

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References

  1. O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl. 272 (2002), no. 1, 368–379. https://doi.org/10.1016/S0022-247X(02)00180-4
  2. Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), no. 2, 495–505. https://doi.org/10.1016/j.jmaa.2005.02.052
  3. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
  4. D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), no. 2, 609–625. https://doi.org/10.1006/jmaa.1996.0456
  5. B. Liu, Positive solutions of a nonlinear three-point boundary value problem, Comput. Math. Appl. 44 (2002), no. 1-2, 201–211. https://doi.org/10.1016/S0898-1221(02)00141-4
  6. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
  7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivatives, Gordon and Breach, Switzerland, 1993.
  8. S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2000), no. 2, 804–812. https://doi.org/10.1006/jmaa.2000.7123
  9. S. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), no. 1, 136–148. https://doi.org/10.1016/S0022-247X(02)00583-8

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