Necessary and sufficient conditions for the solvability and maximal regularity of abstract differential equations of mixed type in Hölder spaces (Q379117)

Let \(A\) be a closed linear operator in a complex Banach space \(X\) with the domain \(D(A)\), not necessary dense in \(X\). Suppose that for all \(\lambda\geq0\) NEWLINE\[NEWLINE\exists (A-\lambda I)^{-1}\in L(X),\quad \|(A-\lambda I)^{-1}\|\leq\frac{C}{1+\lambda}.NEWLINE\]NEWLINE Let us consider a second-order linear differential equation NEWLINE\[NEWLINE u''(x)+Au(x)=f(x),\quad x\in (0,1),NEWLINE\]NEWLINE with boundary conditions \(u(0)=d_0\), \(u'(1)=n_1\) and a function \(f\in C^\theta([0,1];X)\), \(0<\theta<1\).NEWLINENEWLINEIt is proved that the boundary value problem has a unique strict solution \(u\in C^2([0,1];X)\cap C([0,1];X)\) iff NEWLINE\[NEWLINE d_0\in D(A),\quad n_1\in D(\sqrt{-A}),\quad Ad_0-f(0)\in \overline{D(A)},\quad \sqrt{-A}n_1\in\overline{d(A)}.NEWLINE\]NEWLINE Moreover, \(u'', Au\in C^\theta([0,1];X)\) (maximum regularity property) iff NEWLINE\[NEWLINE d_0\in D(A),\quad n_1\in D(\sqrt{-A}),\quad Ad_0-f(0),\sqrt{-A}n_1\in(D(A),X)_{1-\theta/2,\infty}.NEWLINE\]NEWLINE Here \((D(A),X)_{p,q}\) for \(p\in(0,1)\), \(q\in [1,+\infty]\) is the interpolation space.