In this paper, we are concerned with the solvability for a class of nonlinear sequential fractional dynamical systems with damping infinite dimensional spaces, which involves fractional Riemann-Liouville derivatives. The solutions of the dynamical systems are obtained by utilizing the method of Laplace transform technique and are based on the formula of the Laplace transform of the Mittag-Leffler function in two parameters. Next, we present the existence and uniqueness of solutions for nonlinear sequential fractional dynamical systems with damping by using fixed point theorems under some appropriate conditions.
In this paper, we propose a new Newton-Landweber iteration for nonlinear inverse problems. We show the convergence of this method without any convergence rate under a weak nonlinearity condition. Also, this iteration will converge and inherit a certain monotonicity of the iteration error like Landweber iteration, if we restrict our inner iteration steps. Furthermore, we obtain optimal convergence rate of the new Newton-Landweber iteration under stronger nonlinearity conditions and parameter choice rules. Numerical experiments have shown some attractive results.
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