【作者】 陈祥荣；

【导师】 周勇；

【作者基本信息】 湘潭大学 ， 应用数学， 2010， 硕士

【摘要】 近年来,对于分数积分方程和发展方程的研究获得了许多新的结果.但是,相对于整数阶微分方程而言,分数阶方程在理论研究方面还很不完善,有许多领域尚未涉及,需要我们进一步研究.本文讨论了一类积分方程的解的存在性、极值解的存在性问题和一类分数发展方程的非局部Cauchy问题.在第2章,我们首先利用Krasnoselskii不动点理论研究了一类特定的Volterra积分方程解的存在性问题;然后,我们利用混合不动点理论研究了这类特定积分方程极值解的存在性问题;最后,我们将结果应用到分数微分方程中并得到相应的结论.在第3章,我们采用引入概率密度函数和算子半群给出的分数发展方程适度解的定义,通过运用算子半群的不动点理论和泛函分析方法给出了适度解的存在性法则.更多还原

【Abstract】 In recent years, many new papers appeared about fractional integral equa-tions and fractional evolution equations, but, compared with the di?erential equa-tions of integer order, fractional order is far from perfect in the theory, many areasare not involved, we need to do further study.In this paper, we discuss the existence of solutions and the extremal solutionsfor integral equations and a class of fractional evolution equations with nonlocalCauchy conditions. In Chapter 2, firstly, by the use of Krasnoselskii’s fixed pointtheorem we prove the existence of positive solutions for certain Volterra integralequations; then by the use of the hybrid fixed point theorem we prove the ex-istence of extremal solutions for certain Volterra integral equations, and finally,the results are applied to a variety of fractional di?erential equations. In Chapter3, by considering probability density and semigroup, we give definitions of mildsolutions for fractional evolution equations with nonlocal conditions; by using thefunctional analysis concerning to the semigroup of operators and some fixed pointtheorems e?ectively, we give the criteria on existence of mild solutions.更多还原