A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

LetHbe a real Hilbert space andC⊂H a closed convex subset. LetT:C→Cbe a nonexpansive mapping with the nonempty set of fixed pointsFix(T). Kim and Xu (2005) introduced a modified Mann iterationx0=x∈C,yn=αnxn+(1−αn)Txn,xn+1=βnu+(1−βn)yn, whereu∈Cis an arbitrary (but fixed) element, and{αn}and{βn}are two sequences in(0,1). In the case where0∈C, the minimum-norm fixed point ofTcan be obtained by takingu=0. But in the case where0∉C, this iteration process becomes invalid becausexnmay not belong toC. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of Tand prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projectionPC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.

Acoustic Sensitivity Analysis Using the Distributed Source Energy Boundary Point Method

The determination of the sensitivity of the acoustical characteristics of vibrating systems with respect to the variation of the design parameters can provide a method to low-noise design of mechanical structure objectively and quantitatively. Using the Distributed source energy boundary point method, the expressions of the change of the acoustical energy density with respect to design variable is presented in this paper. The Distributed source energy boundary point method is a speedy and precise method which can avoid the complex computing of the singularity integral in EBEM. The correctness and availability is validated by the numerical simulation.