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<datestamp>2018-01-11</datestamp>
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<publisher>HAL CCSD</publisher>
<title lang=en>Proximal and dynamical approaches to equilibrium problems</title>
<creator>Moudafi, Abdellatif</creator>
<creator>Théra, Michel</creator>
<contributor>Département de Mathématiques et Informatique (D.M.I.) ; Université des Antilles et de la Guyane (UAG) - Université des Antilles (Pôle Guadeloupe) ; Université des Antilles (UA) - Université des Antilles (UA)</contributor>
<contributor>Laboratoire d'Arithmétique, de Calcul formel et d'Optimisation (LACO) ; Université de Limoges (UNILIM) - Centre National de la Recherche Scientifique (CNRS)</contributor>
<description>International audience</description>
<source>Ill-posed variational problems and regularization techniques</source>
<identifier>hal-00773204</identifier>
<identifier>https://hal.univ-antilles.fr/hal-00773204</identifier>
<source>https://hal.univ-antilles.fr/hal-00773204</source>
<source>Ill-posed variational problems and regularization techniques, 1998, 477, pp.187-201</source>
<language>en</language>
<subject>[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]</subject>
<type>info:eu-repo/semantics/article</type>
<type>Journal articles</type>
<description lang=en>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.</description>
<date>1998</date>
</dc>
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