AI 中文总结
研究分数阶线性方程，通过定义对称双线性形式构建分数阶变分泛函，证明方程解是泛函临界点，在积分方程中双线性形式非退化，临界点即方程解，还得出微分方程与最小二乘法的关系。
AI 中文摘要
给定分数阶线性方程\(L^\alpha u = f\)，定义一个合适的对称双线性形式，使分数阶算子\(L^\alpha\)相对于该双线性形式对称。利用此双线性形式定义分数阶变分法的泛函，证明给定分数阶方程的解是分数阶变分泛函的临界点。对于分数阶积分方程，双线性形式非退化，所有临界点都是给定方程的解；对于分数阶微分方程，得到与最小二乘法的关系。
英文摘要
Given a fractional-order linear equation $L^αu = f$, we define an appropriate symmetric bilinear form so that the fractional operator $L^α$ is symmetric with respect to that bilinear form. Using the bilinear form, we then define a functional of the fractional calculus of variations proving that the solutions of the given fractional-order equation are critical points of the fractional variational functional. In the case of fractional integral equations, the provided bilinear form is non-degenerate, and all critical points are solutions of the given equation. In the case of fractional differential equations, a relation with the least-squares method is obtained.
CommentsThis is a preprint of a paper published in 'Journal of Convex Analysis' at [https://www.heldermann.de/JCA/jcacover.htm]. Dedicated to Alexander Plakhov on the occasion of his 65th anniversary
