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Proof From the continuity of f and I j, j = 1, 2, …, n, we obtain the continuity and differentiability of φ and φ ′.

end{array} (69) Then, we obtain the continuity of (int g y mathbf {T} dy|s,a)) using the assumption that g is a bounded function.

Using the continuity of f and I j, j = 1, 2, …, N, we obtain the continuity and differentiability of J and J ∈ C 1 ( H 0 1 ( 0, T ), R ).

Now we see that (frac{partial }{partial t}phi (x,t_+0)=frac{partial }{partial t}phi (x,t_-0)=0), so we obtain the continuity of the function (frac{partial phi }{partial t}), specially, (frac{partial {phi } }{partial t}|_{(x,t_=0).

In this article, we obtained the continuity results of eigenvalues and eigenfunctions and presented some new differential expressions of the eigenvalues with respect to the data.

We further study several kinds of the monotonicity conditions to obtain the Hölder continuity of the solution mapping.

To apply Theorem 3.10 for (4.1), we put a more general constraint on Ω to obtain the Hölder continuity up to boundary.

For this reason, we employ the framework based on Campanato spaces to obtain the Hölder continuity of higher order derivatives of solutions.

Finally, we recall a Gronwall-type estimate which is vital to obtain the Lipschitz continuity of solution (u alpha,beta)) with respect to coefficients α and β. [24].

In Section 3, we firstly obtain the Lipschitz continuity of the solution (u=u alpha,beta)) with respect to coefficients α and β, and then prove Theorem 1.1.

We first obtain the Lipschitz continuity of the solution (u=u alpha,beta)) with respect to coefficients α and β, and then prove that this equation is stable under the perturbation of coefficients.