_t=T − ψ(x))² dx is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence {J′(f⁽ⁿ⁾)}, where f⁽ⁿ⁾ is the nth iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given.

中文翻译：

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来自最终超确定的空间分数扩散方程中的逆源问题

_{应用了t = T} - ψ（x））² d x，表明该成本函数是Fréchet可微的，并且其导数可以通过伴随问题的求解来公式化。此外，证明了Lipschitz梯度的连续性。这些结果有助于我们证明序列{ J'（f ^（n））}的单调性和收敛性，其中f ^（n）是梯度法的第n次迭代。最后，给出了弗雷谢特导数的凸性。