Paul Eloe, Ph.D.
In 1967, Andrzej Lasota and Zdzisław Opial proved that under sufficient conditions, uniqueness of solutions for boundary value problems for a second-order ordinary differential equation implies their existence. Lloyd Jackson and Keith Schrader then proved an extension of this result for boundary value problems of third order. In proving the third-order case, this compactness theorem is applied as a key part of the proof. It states that under sufficient conditions, uniform boundedness of a sequence of solutions on a compact domain implies existence of a subsequence which converges uniformly with respect to its zeroth, first, and second derivatives. We present an extension of this compactness theorem to a fractional differential equation of all orders in the interval (2,3].
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Mathematics | Physical Sciences and Mathematics
Masthay, Tyler, "Extending Uniqueness Implies Existence Results to Fractional Differential Equations" (2017). Honors Theses. 108.