Integrated fractional resolvent operator function and fractional abstract Cauchy problem
Summary: We firstly prove that \(\beta\)-times integrated \(\alpha\)-resolvent operator function (\((\alpha, \beta)\)-ROF) satisfies a functional equation which extends that of \(\beta\)-times integrated semigroup and \(\alpha\)-resolvent operator function. Secondly, for the inhomogeneous \(\alpha\)-Cauchy problem \({}^c D_t^\alpha u(t) = A u(t) + f(t)\), \(t \in(0, T)\), \(u(0) = x_0\), \(u'(0) = x_1\), if \(A\) is the generator of an \((\alpha, \beta)\)-ROF, we give the relation between the function \(v(t) = S_{\alpha, \beta}(t) x_0 +(g_1 * S_{\alpha, \beta})(t) x_1 +(g_{\alpha - 1} * S_{\alpha, \beta} * f)(t)\) and mild solution and classical solution of it. Finally, for the problem \({}^c D_t^\alpha v(t) = A v(t) + g_{\beta + 1}(t) x\), \(t > 0\), \(v^{(k)}(0) = 0\), \(k = 0,1, \dots, N - 1\), where \(A\) is a linear closed operator. We show that \(A\) generates an exponentially bounded \((\alpha, \beta)\)-ROF on a Banach space \(X\) if and only if the problem has a unique exponentially bounded classical solution \(v_x\) and \(A v_x \in L_{\text{loc}}^1(\mathbb{R}^+, X) \). Our results extend and generalize some related results in the literature.
scientific article; zbMATH DE number 7022377