Generalized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalized F-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.The authors are very grateful to the referees for their valuable comments and suggestions, and, in particular, to one of them for calling our attention on the crucial fact stated in the first part of Remark 5 and for the elegant reformulation of Theorem 13 stated in Remark 14. Salvador Romaguera acknowledges the support of the Universitat Politecnica de Valencia, Grant PAID-06-12-SP20120471.Abbas, M.; Ali, B.; Romaguera Bonilla, S. (2014). Generalized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces. Abstract and Applied Analysis. 2014:1-5. https://doi.org/10.1155/2014/391952S152014Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3, 133-181. doi:10.4064/fm-3-1-133-181Arandjelović, I., Kadelburg, Z., & Radenović, S. (2011). Boyd–Wong-type common fixed point results in cone metric spaces. Applied Mathematics and Computation, 217(17), 7167-7171. doi:10.1016/j.amc.2011.01.113Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Huang, L.-G., & Zhang, X. (2007). Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332(2), 1468-1476. doi:10.1016/j.jmaa.2005.03.087Rakotch, E. (1962). A note on contractive mappings. Proceedings of the American Mathematical Society, 13(3), 459-459. doi:10.1090/s0002-9939-1962-0148046-1Tarafdar, E. (1974). An approach to fixed-point theorems on uniform spaces. Transactions of the American Mathematical Society, 191, 209-209. doi:10.1090/s0002-9947-1974-0362283-5Dix, J. G., & Karakostas, G. L. (2009). A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems. Nonlinear Analysis: Theory, Methods & Applications, 71(9), 3872-3880. doi:10.1016/j.na.2009.02.057Latrach, K., Aziz Taoudi, M., & Zeghal, A. (2006). Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations. Journal of Differential Equations, 221(1), 256-271. doi:10.1016/j.jde.2005.04.010Meinardus, G. (1963). Invarianz bei linearen Approximationen. Archive for Rational Mechanics and Analysis, 14(1), 301-303. doi:10.1007/bf00250708Habiniak, L. (1989). Fixed point theorems and invariant approximations. Journal of Approximation Theory, 56(3), 241-244. doi:10.1016/0021-9045(89)90113-5Hicks, T. ., & Humphries, M. . (1982). A note on fixed-point theorems. Journal of Approximation Theory, 34(3), 221-225. doi:10.1016/0021-9045(82)90012-0Singh, S. . (1979). An application of a fixed-point theorem to approximation theory. Journal of Approximation Theory, 25(1), 89-90. doi:10.1016/0021-9045(79)90036-4Subrahmanyam, P. . (1977). An application of a fixed point theorem to best approximation. Journal of Approximation Theory, 20(2), 165-172. doi:10.1016/0021-9045(77)90070-3Wardowski, D. (2012). Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-94Abbas, M., Ali, B., & Romaguera, S. (2013). Fixed and periodic points of generalized contractions in metric spaces. Fixed Point Theory and Applications, 2013(1), 243. doi:10.1186/1687-1812-2013-24

Equilibria, Fixed Points, and Complexity Classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes

A mixed ℓ1\ell_1 regularization approach for sparse simultaneous approximation of parameterized PDEs

We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based $\ell_1$ regularization method that exploits both energy and sparsity. In addition, we are able to prove that, with minimal sample complexity, error estimates comparable to the best $s$-term and quasi-optimal approximations are achievable, while requiring only a priori bounds on polynomial truncation error with respect to the energy norm. Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure

Ising exponents from the functional renormalisation group

We study the 3d Ising universality class using the functional renormalisation group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and anti-symmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross-correlations of scaling exponents, and their dependence on dimensionality. We find a very good numerical convergence of the derivative expansion, also in comparison with earlier findings. Evaluating the data from all functional renormalisation group studies to date, we estimate the systematic error which is found to be small and in good agreement with findings from Monte Carlo simulations, \epsilon-expansion techniques, and resummed perturbation theory.Comment: 24 pages, 3 figures, 7 table