fractional partial differential equation
Upper and lower solution method for fractional evolution equations with order \(1<\alpha<2\)
The author has used the upper and lower solution method for the investigation of a class of fractional differential equations of the form NEWLINE\[NEWLINE\begin{cases} D^{\alpha}_tu(t)Au(t)+f(t,u(t)), \quad & t\in [0,T],\\ u(0)=x_0,\; u'(0)=x_1,\quad & 1<\alpha<2.\end{cases}\tag{1}NEWLINE\]NEWLINE A function \(u\in C([0,T],X)\) is said to be a mild solution to (1) if it satisfies the operator equation NEWLINE\[NEWLINE u(t)=S_{\alpha}x_0+K_{\alpha}x_1+\int_0^t T_{\alpha}f(s,u(s))ds, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\begin{aligned} S_{\alpha} & =\frac{1}{2\pi i}\int_C e^{\lambda t}\lambda^{\alpha-1}R(\lambda^{\alpha},A)d\lambda, \\ K_{\alpha} & =\frac{1}{2\pi i}\int_C e^{\lambda t}\lambda^{\alpha-2}R(\lambda^{\alpha},A)d\lambda,\\ T_{\alpha} & =\frac{1}{2\pi i}\int_C e^{\lambda t}R(\lambda^{\alpha},A)d\lambda\end{aligned}NEWLINE\]NEWLINE where \(C\) is a suitable path, satisfying \(\lambda^{\alpha}\overline{\in}\mu+S_{\theta}\).NEWLINENEWLINEIt is supposed that the closed linear operator \(A:\mathcal{D}\subseteq X\to X\) is sectorial of the type \((M,\theta, \alpha, \mu)\) if there exist \(0<\theta<\frac{\pi}{2}\), \(M>0\) and \(\mu\in \mathbb{R}\) such that the \(\alpha\)-resolvent of \(A\) exists outside the sector \(\mu+S_{\theta}=\{\mu+\lambda^{\alpha}\mid\lambda\in \mathbb{C}\), \(\mathrm{Arg}(-\lambda^{\alpha})<\theta\}\) and \(\|(\lambda^{-\alpha}I-A)^{-1}\|\leq\frac{M}{|\lambda^{\alpha}-\mu|}\), \(\lambda^{\alpha}\overline{\in}\mu+S_{\theta}\), and the operator \(-A\) is the infinitesimal generator of an analytic semigroup.NEWLINENEWLINEAt the usage of the Mittag-Leffler function, the properties of the solution operators \(S_{\alpha},T_{\alpha}\) and \(K_{\alpha}\) are studied. Based on the theory of accretive and \(m\)-accretive operators, it is proved that the solution operators are positive. This allows prove a series of results about the solutions to (1).
scientific article; zbMATH DE number 6369559