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Mohamed-aziz Taoudi - One of the best experts on this subject based on the ideXlab platform.
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Sadovskii-Krasnosel'skii type fixed point theorems in Banach spaces with application to evolution equations
Journal of Applied Mathematics and Computing, 2014Co-Authors: Khalil Ezzinbi, Mohamed-aziz TaoudiAbstract:In this work, we introduce the concept of a convex-power condensing mapping $$T$$ with respect to another mapping $$S$$ as a generalization of condensing and convex-power condensing mappings. Some fixed point theorems for the sum $$T+S$$ with $$S$$ is a Strict Contraction and $$T$$ is convex-power condensing with respect to $$S$$ are established. The cases where $$S$$ is nonexpansive or expansive are also considered. Our fixed point results encompass the well known Sadovskii’s fixed point theorem and a number of its generalizations. To show the usefulness and the applicability of our fixed point results we investigate the existence of mild solutions to a broad class of neutral differential equations.
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Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations
Fixed Point Theory and Applications, 2013Co-Authors: Nawab Hussain, Mohamed-aziz TaoudiAbstract:In this paper we present some fixed point results for the sum of two mappings where S is a Strict Contraction and T is not necessarily weakly compact and satisfies a new condition formulated in terms of an axiomatic measure of weak noncompactness. Our fixed point results extend and improve several earlier results in the literature. In particular, our results encompass the analogues of Krasnosel’skii’s and Sadovskii’s fixed point theorems for sequentially weakly continuous mappings and a number of their generalizations. Finally, an application to integral equations is given to illustrate the usability of the obtained results. MSC:37C25, 40D05, 31B10.
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Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness
Journal of Differential Equations, 2012Co-Authors: Jesús Garcia-falset, Khalid Latrach, Elena Moreno-galvez, Mohamed-aziz TaoudiAbstract:Abstract We present some extension of a well-known fixed point theorem due to Burton and Kirk [T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 423–431] for the sum of two nonlinear operators one of them compact and the other one a Strict Contraction. The novelty of our results is that the involved operators need not to be weakly continuous. Finally, an example is given to illustrate our results.
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Krasnosel’skii type fixed point theorems under weak topology features
Nonlinear Analysis: Theory Methods & Applications, 2010Co-Authors: Mohamed-aziz TaoudiAbstract:Abstract In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X . Suppose that A : M → X and B : X → X are two weakly sequentially continuous mappings satisfying: (i) A M is relatively weakly compact; (ii) B is a Strict Contraction; (iii) ( x = B x + A y , y ∈ M ) ⇒ x ∈ M . Then A + B has at least one fixed point in M . This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.
Yuan Liu - One of the best experts on this subject based on the ideXlab platform.
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The Poincaré inequality and quadratic transportation-variance inequalities
Electronic Journal of Probability, 2020Co-Authors: Yuan LiuAbstract:It is known that the Poincare inequality is equivalent to the quadratic transportation-variance inequality (namely $W_{2}^{2}(f\mu ,\mu ) \leqslant C_{V} \mathrm{Var} _{\mu }(f)$), see Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_{V}$ than before, which equals the double of Poincare constant. Applying the same argument leads to more characterizations of the Poincare inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and Strict Contraction of heat flow in Wasserstein space provided that the Bakry-Emery curvature has a lower bound (here the control constants may depend on the curvature bound).Next, we present a comparison inequality between $W_{2}^{2}(f\mu ,\mu )$ and its centralization $W_{2}^{2}(f_{c}\mu ,\mu )$ for $f_{c} = \frac{|\sqrt {f} - \mu (\sqrt {f})|^{2}} {\mathrm{Var} _{\mu }(\sqrt{f} )}$, which may be viewed as some special counterpart of the Rothaus’ lemma for relative entropy. Then it yields some new bound of $W_{2}^{2}(f\mu ,\mu )$ associated to the variance of $\sqrt{f} $ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Gotze’s characterization of the Talagrand’s inequality.
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The Poincar\'e inequality and quadratic transportation-variance inequalities
arXiv: Probability, 2019Co-Authors: Yuan LiuAbstract:It is known that the Poincare inequality is equivalent to the quadratic transportation-variance inequality (namely $W_2^2(f\mu,\mu) \leqslant C_V \mathrm{Var}_\mu(f)$), see Jourdain \cite{Jourdain} and most recently Ledoux \cite{Ledoux18}. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_V$ than before, which equals the double of Poincare constant. Applying the same argument leads to more characterizations of the Poincare inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and Strict Contraction of heat flow in Wasserstein space provided that the Bakry-Emery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_2^2(f\mu,\mu)$ and its centralization $W_2^2(f_c\mu,\mu)$ for $f_c = \frac{|\sqrt{f} - \mu(\sqrt{f})|^2}{\mathrm{Var}_\mu (\sqrt{f})}$, which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of $W_2^2(f\mu,\mu)$ associated to the variance of $\sqrt{f}$ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Gotze's characterization of the Talagrand's inequality.
Jean-jacques E. Slotine - One of the best experts on this subject based on the ideXlab platform.
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Contraction and Robustness of Continuous Time Primal-Dual Dynamics.
arXiv: Optimization and Control, 2018Co-Authors: Hung D. Nguyen, Konstantin Turitsyn, Jean-jacques E. SlotineAbstract:The Primal-Dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, Strict Contraction is nontrivial to establish in most cases. This work focuses on continuous, possibly non-autonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple time-scales. We show that the PD algorithm is indeed Strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive estimates for the performance of multiple time-scale multi-layer optimization systems, and illustrate our results on a primal-dual representation of the Automatic Generation Control of power systems.
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Contraction and Robustness of Continuous Time Primal-Dual Dynamics
IEEE Control Systems Letters, 2018Co-Authors: Hung D. Nguyen, Konstantin Turitsyn, Jean-jacques E. SlotineAbstract:The Primal-dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, Strict Contraction is nontrivial to establish in most cases. This letter focuses on continuous, possibly nonautonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple timescales. We show that the PD algorithm is indeed Strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive estimates for the performance of multiple time-scale multi-layer optimization systems, and illustrate our results on a PD representation of the Automatic Generation Control of power systems.
S. Ter Horst - One of the best experts on this subject based on the ideXlab platform.
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The discrete twofold Ellis–Gohberg inverse problem
Journal of Mathematical Analysis and Applications, 2017Co-Authors: S. Ter Horst, Marinus A. Kaashoek, F. Van SchagenAbstract:In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a Strict Contraction and the case where the functions are matrix polynomials.
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Relaxed commutant lifting and a relaxed Nehari problem: redheffer state space formulas
Mathematische Nachrichten, 2009Co-Authors: S. Ter HorstAbstract:The description of all solutions to the relaxed commutant lifting problem in terms of an underlying Contraction, obtained earlier in joint work of the author with A. E. Frazho and M. A. Kaashoek, is transformed into a linear fractional Redheffer state space form. Under certain additional conditions the coefficient functions in this representation are described explicitly in terms of the original data. The main theorem is a generalization of the Redheffer description of all solutions to the classical commutant lifting problem. To illustrate the result a relaxed version of the Nehari extension problem is considered, and an explicit Redheffer description of all its solutions is given, assuming that a certain truncated Hankel operator is a Strict Contraction (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas
arXiv: Functional Analysis, 2006Co-Authors: S. Ter HorstAbstract:The description of all solutions to the relaxed commutant lifting problem in terms of an underlying Contraction, obtained earlier in joint work of the author with A.E. Frazho and M.A. Kaashoek, is transformed into a linear fractional Redheffer state space form. Under certain additional conditions the coefficient functions in this representation are described explicitly in terms of the original data. The main theorem is a generalization of the Redheffer description of all solutions to the classical commutant lifting problem. To illustrate the result a relaxed version of the Nehari extension problem is considered, and an explicit Redheffer description of all its solutions is given, assuming that a certain truncated Hankel operator is a Strict Contraction. The latter result is specified further for two special cases.
Hung D. Nguyen - One of the best experts on this subject based on the ideXlab platform.
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Contraction and Robustness of Continuous Time Primal-Dual Dynamics.
arXiv: Optimization and Control, 2018Co-Authors: Hung D. Nguyen, Konstantin Turitsyn, Jean-jacques E. SlotineAbstract:The Primal-Dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, Strict Contraction is nontrivial to establish in most cases. This work focuses on continuous, possibly non-autonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple time-scales. We show that the PD algorithm is indeed Strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive estimates for the performance of multiple time-scale multi-layer optimization systems, and illustrate our results on a primal-dual representation of the Automatic Generation Control of power systems.
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Contraction and Robustness of Continuous Time Primal-Dual Dynamics
IEEE Control Systems Letters, 2018Co-Authors: Hung D. Nguyen, Konstantin Turitsyn, Jean-jacques E. SlotineAbstract:The Primal-dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, Strict Contraction is nontrivial to establish in most cases. This letter focuses on continuous, possibly nonautonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple timescales. We show that the PD algorithm is indeed Strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive estimates for the performance of multiple time-scale multi-layer optimization systems, and illustrate our results on a PD representation of the Automatic Generation Control of power systems.