Zhao Daliang, Liu Yansheng
School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China.
ScientificWorldJournal. 2013 Nov 6;2013:473828. doi: 10.1155/2013/473828. eCollection 2013.
This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem DC0+αu(t)=f(t, u(t), u'(t)), 0<t<1, u(1)=u'(1)=u''(0)=0, where 2<α≤3 is a real number, DC0+α is the Caputo fractional derivative, and f:[0,1]×[0, +∞)×R→[0, +∞) is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii's fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.
本文致力于研究分数阶边值问题(DC_{0 +}^{\alpha}u(t)=f(t, u(t), u'(t)))，(0 < t < 1)，(u(1)=u'(1)=u''(0)=0)多个正解的存在性，其中(2 < \alpha\leq3)为实数，(DC_{0 +}^{\alpha})为卡普托分数阶导数，且(f:[0,1]\times[0, +\infty)\times\mathbb{R}\to[0, +\infty))连续。首先，通过构造一个特殊的锥，应用郭 - 克拉斯诺谢尔斯基不动点定理和莱格特 - 威廉姆斯不动点定理，建立了分数阶边值问题的一些新的存在性准则；其次，通过应用克拉斯诺谢尔斯基不动点定理的一个新的推广，从其辅助问题得到了所考虑边值问题存在多个正解的一个充分条件。最后，作为应用，给出了一些说明性例子以支持主要结果。
