Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Based on the very recent work by Censor-Gibali-Reich (http://arxiv.org/abs/1009.3780), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel'skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.
ISSN: 0022-3239

hal-00776651
https://hal.univ-antilles.fr/hal-00776651
DOI : 10.1007/s10957-011-9814-6

Éditeur(s) : HAL CCSD
Résumé : International audience
The main purpose of this note is to propose viscosity approximation methods which amount to selecting a particular fixed-point of a given nonexpansive self mapping in a general Hilbert space. The connection with the selection principles of Attouch, in the context of convex minimization and monotone inclusion problems, is made and an application to a semi-coercive elliptic problem is then stated. Key words: nonexpansive mappings, fixed-points, monotone operators, iterative methods, convex optimization, selection, penalty ...
Lecture notes in economics and mathematical systems

hal-00776623
https://hal.univ-antilles.fr/hal-00776623

Éditeur(s) : HAL CCSD
Résumé : International audience
In the paper the following mixed equilibrium problem in a Hilbert space is considered: find x 2 C such that F(x, x)+hG(x), x−xi 0, 8x 2 C, where C is a nonempty convex closed subset of a real Hilbert space. For its solution a version of the proximal approach is studied. The general proximal method always includes a quadratic term of the kind |x−xn|2, which provides a process of transition from a point xn to a point xn+1. The authors replace this quadratic term by its linearization. Thus, they obtain a method of the kind F(xn+1, x)+hG(xn)+ −1(h 0(xn+1)−h 0(xk)), x−xn+1i 0, 8x 2 C, where h0 is the derivative of a given function on C. It is proved that if the function F(x, y) is monotone and convex in y for any x, F(x, x) = 0, and the operator G(x) is co-coercive, then the method weakly converges in a real Hilbert space to an equilibrium solution. Note that this result will also be true if the co-coercive condition is replaced by a monotonicity property. The Tikhonov regularization method (in the absence of disturbances and calculating auxiliary solutions approximately) is justified and its weak convergence in a Hilbert space is proved. It is known that the proximal method can always be interpreted as an implicit discret approximation of some differential equation or inclusion. Therefore the authors consider a differential inclusion for the extreme mapping. Permitting "-approximation for the function F(x, y) by means of a parametric family F(x, y, ") and assuming the strong monotonicity of the operator @F(x, y, ") (it is a very strong condition), the authors prove the convergence of the trajectory of the differential inclusion to an equilibrium solution.
Ill-posed variational problems and regularization techniques

hal-00773204
https://hal.univ-antilles.fr/hal-00773204

Éditeur(s) : HAL CCSD
Résumé : International audience
An iterative method for solving set-valued variational inequalities is considered and its convergence properties are studied under strong monotonicity and coercivity conditions. The results obtained in this paper include, as a special case, some known results in this field.
Mathematical Inequalities & Applications

hal-00776616
https://hal.univ-antilles.fr/hal-00776616

Éditeur(s) : HAL CCSD School of Communications and Informatics
Résumé : International audience
An approximate procedure for solving equilibrium problems is proposed and its convergence is established under natural conditions. The result obtained in this paper includes, as a special case, some known results in convex minimization and monotone inclusion fields.
ISSN: 1443-5756

hal-00780881
https://hal.univ-antilles.fr/hal-00780881

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed-point of a given nonexpansive self-mapping.
Journal of Mathematical Analysis and applications

hal-00776600
https://hal.univ-antilles.fr/hal-00776600
DOI : 10.1006/jmaa.1999.6615

Éditeur(s) : HAL CCSD Springer Verlag (Germany) Springer Verlag
Résumé : International audience
Arenal volcano is nearly unique among arc volcanoes with its 42 year long (1968–2010) continuous, small-scale activity erupting compositionally monotonous basaltic andesites that also dominate the entire, ~7000 year long, eruptive history. Only mineral zoning records reveal that basaltic andesites are the result of complex, open-system processes deriving minerals from a variety of crystallization environments and including the episodic injections of basalt. The condition of the mafic input as well as the generation of crystal-rich basaltic andesites of the recent, 1968–2010, and earlier eruptions were addressed by an experimental study at 200 MPa, 900–1,050 °C, oxidizing and fluid-saturated conditions with various fluid compositions [H2O/(H2O + CO2) = 0.3–1]. Phase equilibria were determined using a phenocryst-poor (~3 vol%) Arenal-like basalt (50.5−wt% SiO2) from a nearby scoria cone containing olivine (Fo92), plagioclase (An86), clinopyroxene (Mg# = 82) and magnetite (Xulvö = 0.13). Experimental melts generally reproduce observed compositional trends among Arenal samples. Small differences between experimental melts and natural rocks can be explained by open-system processes. At low pressure (200 MPa), the mineral assemblage as well as the mineral compositions of the natural basalt were reproduced at 1,000 °C and high water activity. The residual melt at these conditions is basaltic andesitic (55 wt% SiO2) with 5 wt% H2O. The evolution to more evolved magmas observed at Arenal occurred under fluid-saturated conditions but variable fluid compositions. At 1,000 °C and 200 MPa, a decrease of water content by approximately 1 wt% induces significant changes of the mineral assemblage from olivine + clinopyroxene + plagioclase (5 wt% H2O in the melt) to clinopyroxene + plagioclase + orthopyroxene (4 wt% H2O in the melt). Both assemblages are observed in crystal-rich basalt (15 vol%) and basaltic andesites. Experimental data indicate that the lack of orthopyroxene and the presence of amphibole, also observed in basaltic andesitic tephra units, is due to crystallization at nearly water-saturated conditions and temperatures lower than 950 °C. The enigmatic two compositional groups previously known as low- and high-Al2O3 samples at Arenal volcano may be explained by low- and high-pressure crystallization, respectively. Using high-Al as signal of deeper crystallization, first magmas of the 1968–2010 eruption evolved deep in the crust and ascent was relatively fast leaving little time for significant compositional overprint by shallower level crystallization.
ISSN: 0010-7999

hal-01115717
https://hal.archives-ouvertes.fr/hal-01115717
DOI : 10.1007/s00410-014-1040-4

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a significant extension of previously known results in this field.
ISSN: 1385-1292

hal-00780762
https://hal.univ-antilles.fr/hal-00780762
https://hal.univ-antilles.fr/hal-00780762/document
https://hal.univ-antilles.fr/hal-00780762/file/10.1.1.13.5012.pdf
DOI : 10.1023/B:POST.0000023199.34934.9f

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a significant extension of previously known results in this field.
ISSN: 1385-1292

hal-00815511
https://hal.univ-antilles.fr/hal-00815511
DOI : 10.1023/B:POST.0000023199.34934.9f