Convergence theorems for fixed points of demicontinuous pseudocontractive mappings

Let be an open subset of a real uniformly smooth Banach space . Suppose is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where denotes the closure of . Then, it is proved that (i) for every ; (ii) for a given , there exists a unique path , , satisfying . Moreover, if or there exists such that the set is bounded, then it is proved that, as , the path converges strongly to a fixed point of . Furthermore, explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of

Fixed-Points for Quantitative Equational Logics

We develop a fixed-point extension of quantitative equational logic and give semantics in one-bounded complete quantitative algebras. Unlike previous related work about fixed-points in metric spaces, we are working with the notion of approximate equality rather than exact equality. The result is a novel theory of fixed points which can not only provide solutions to the traditional fixed-point equations but we can also define the rate of convergence to the fixed point. We show that such a theory is the quantitative analogue of a Conway theory and also of an iteration theory; and it reflects the metric coinduction principle. We study the Bellman equation for a Markov decision process as an illustrative example

Convergence of Ishikawa iterative sequence for strongly pseudocontractive operators in arbitrary Banach spaces

Under the condition of removing the restriction any bounded, we give the convergence of the Ishikawa iteration process to a unique fixed point of a strongly pseudocontractive operator in arbitrary real Banach space. Furthermore, general convergence rate estimate is given in our results, which extend the recent results of Ciric [3] and Soltuz [12]

Function generation synthesis of planar 5R mechanism

This paper deals with the function generation problem for a planar five-bar mechanism. The inputs to the mechanism are selected as one of the fixed joints and the mid-joint, whereas the remaining fixed joint represents the output. Synthesis problem of the five-bar mechanism is analytically formulated and an objective function is expressed in polynomial form. Function generation synthesis is performed with equal spacing and Chebyshev approximation method. The four unknown construction parameters and the error are evaluated by means of five design points and the coefficients of the objective function are determined by numerical iteration using four stationary and one moving design point. Stationary points are placed at the boundaries of the motion and the moving point is re-selected at each iteration as the point corresponding to the extremum error. Iterations are repeated until the values are stabilized. The stabilization usually occurs at the third iteration. By this method, the maximum error values are approximately equated, hence the total error is bounded at certain limits. Finally the construction parameters of the mechanism are determined

On the Analysis of the Discretized Kohn-Sham Density Functional Theory

In this paper, we study a few theoretical issues in the discretized Kohn-Sham (KS) density functional theory (DFT). The equivalence between either a local or global minimizer of the KS total energy minimization problem and the solution to the KS equation is established under certain assumptions. The nonzero charge densities of a strong local minimizer are shown to be bounded below by a positive constant uniformly. We analyze the self-consistent field (SCF) iteration by formulating the KS equation as a fixed point map with respect to the potential. The Jacobian of these fixed point maps is derived explicitly. Both global and local convergence of the simple mixing scheme can be established if the gap between the occupied states and unoccupied states is sufficiently large. This assumption can be relaxed if the charge density is computed using the Fermi-Dirac distribution and it is not required if there is no exchange correlation functional in the total energy functional. Although our assumption on the gap is very stringent and is almost never satisfied in reality, our analysis is still valuable for a better understanding of the KS minimization problem, the KS equation and the SCF iteration.Comment: 29 page