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Journal of Function Spaces
Volume 2017 (2017), Article ID 3941084, 7 pages
https://doi.org/10.1155/2017/3941084
Research Article

A New Nonsmooth Bundle-Type Approach for a Class of Functional Equations in Hilbert Spaces

1School of Mathematics, Liaoning Normal University, Dalian 116029, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Jie Shen; moc.361@527010tt

Received 13 April 2017; Accepted 2 July 2017; Published 8 August 2017

Academic Editor: Hugo Leiva

Copyright © 2017 Jie Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new bundle-type approach for solving a class of functional equations is presented by combining bundle idea for nonsmooth optimization with common iterative process for functional equations. Our strategy is to approximate the nonsmooth function in functional equation by a sequence of convex piecewise linear functions, as in the bundle method; this makes the problem more tractable and reduces the difficulty of implementation of method. We only require the piecewise linear convex approximate functions, rather than the actual function, to satisfy the uniform boundedness condition with respect to one variable at stability centers. One example is given to demonstrate the application of the proposed method.