预解算子族及微分包含解的存在性和可控性

The issue is concerned with resolvent families of operators and some properties of solutions for Riemann-Liouville fractional differential inclusions in Banach spaces. It has become an active research direction of nonlinear functional analysis in recent years. We shall study the generation theorem, subordination principle of resolvent families of operators via semigroups of operators and Laplace transformation. By using the fixed point theorems and approximation techniques , we shall study the well-posedness of solutions for impulsive Riemann-Liouville fractional differential inclusions and the existence of solutions under the different topological structures of impulsive items. The methods of measure of noncompactness and optimization are adopted to study the exact controllability and approximate controllability of nonlinear fractional differential inclusions. The compactness of operators are the key factors in the discussion of fractional differential inclusions with solution operators. We shall combine the properties of resolvent operators and measures of noncompactness to discuss this problem. The topic gives sufficient consideration to three important factors of differential inclusions, fractional calculus and impulsive conditions. The expected results of our study will further develop the theories of resolvent families and differential inclusions.

本课题研究Banach空间中预解算子族及Riemann-Liouville分数阶微分包含解的有关性质，它是近年来非线性泛函分析中一个活跃的研究方向。课题利用算子半群理论和Laplace变换等方法，研究Riemann-Liouville分数阶预解算子族的生成定理和从属原则；利用不动点定理和逼近技巧，研究Banach空间中Riemann-Liouville分数阶脉冲微分包含解的适定性和脉冲项在不同拓扑下解的存在性；借助非紧性测度和最优化方法，研究非线性分数阶微分包含系统解的精确可控性与近似可控性。运用解算子讨论分数阶微分包含问题时，算子的紧性是关键因素，我们将结合预解算子性质和非紧性测度研究这一问题。本课题充分考虑微分包含、分数阶导数、脉冲三个重要因素，预期结果将进一步发展预解族理论和微分包含理论。

随着微分包含问题在物理学、材料学和工程应用中的不断深入，分数阶微分包含及其预解族理论已成为非线性分析领域的重要研究课题。由于Riemann–Liouville分数阶预解算子在零点处是无界的，且不具有半群性质，其研究具有更大的挑战性。本项目针对Riemann–Liouville分数阶预解算子族的生成定理、紧性特征以及在微分包含中的应用展开研究。我们在分数阶预解算子不具有半群性质的条件下，利用预解算子族的紧性和等度连续性得到Riemann–Liouville分数阶预解算子族一致连续的充分条件。通过重新赋范、构造新的连续函数空间的方法，我们成功克服了Riemann–Liouville分数阶微分方程的解算子在零点处无界的困难。在非紧半群条件下，利用半紧集、构造迭代的凸闭集等方法，对脉冲函数和非局部函数在不同拓扑下的情形进行讨论，给出一类非局部脉冲微分包含解的存在性，改进和推广了许多已有的结论。利用构造凸闭包的方法，通过分析非局部项在零点处的扰动，我们在非局部项不具有紧性和Lipschitz连续性条件下，得到脉冲微分包含解的存在性。