Showing papers on "Strongly monotone published in 2008"
TL;DR: The general existence and uniqueness result is proved which exhibits the idea of comparison principle and is also valid for fractional differential equations in a Banach space.
261 citations
TL;DR: Under classical conditions, it is proved the strong convergence of the sequences of iterates given by the considered scheme.
Abstract: This paper deals with an iterative method, in a real Hilbert space, for approximating a common element of the set of fixed points of a demicontractive operator (possibly quasi-nonexpansive or strictly pseudocontractive) and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The considered algorithm can be regarded as a combination of a variation of the hybrid steepest descent method and the so-called extragradient method. Under classical conditions, we prove the strong convergence of the sequences of iterates given by the considered scheme.
235 citations
TL;DR: In this article, the authors present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly-monotone mappings in Hilbert space.
Abstract: The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper not only improve and extend the main results in Korpelevich (Ekonomika i Matematicheskie Metody, 1976, 12(4):747–756), but also extend and replenish the corresponding results obtained by Iiduka and Takahashi (Nonlinear Anal TMA, 2005, 61(3):341–350), Takahashi and Toyoda (J Optim Theory Appl, 2003, 118(2):417–428), Nadezhkina and Takahashi (J Optim Theory Appl, 2006, 128(1):191–201), and Zeng and Yao (Taiwanese Journal of Mathematics, 2006, 10(5):1293–1303).
131 citations
TL;DR: In this paper, strong monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium.
Abstract: Strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium. This result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. As an application, it is shown that enzymatic futile cycles have a global convergence property.
83 citations
TL;DR: In this article, the authors present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, set of solutions of an equilibrium problem and set of variational inequalities for α-inverse-strongly monotone mappings in the framework of a Hilbert space.
72 citations
TL;DR: In this paper, the Tikhonov-like dynamics were studied in the case of a maximal monotone operator on a Hilbert space and the parameter function tends to 0 as t → ∞ with ∫ 0 ∞ e ( t ) d t = ∞.
61 citations
TL;DR: In this article, the authors introduce an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly-monotone mappings and obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces.
Abstract: We introduce an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. The results in this paper unify, extend, and improve some well-known results in the literature.
51 citations
TL;DR: In this paper, the uniqueness and existence of fixed point of mixed monotone operators in the partially ordered Banach space were studied and the results extend and improve recent related results.
50 citations
TL;DR: In this paper, the authors studied the problem of maximality for the sum A + B and composition L*ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Based on the Fitzpatrick function, new characterizations for the maximality of an operator were presented.
Abstract: This paper is primarily concerned with the problem of maximality for the sum A + B and composition L*ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented.
42 citations
Abstract: The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arsine distribution as well as a connection to operator-valued conditional freeness.
35 citations
TL;DR: A survey of bipotential representations of monotone and non-monotone operators can be found in this article, where the authors show that bipotentials can represent some non-associated constitutive laws in non-smooth mechanics, such as Coulomb frictional contact or Drucker-Prager model in plasticity.
Abstract: This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel's inequality, with several interesting applications related to non associated constitutive laws in non smooth mechanics, such as Coulomb frictional contact or non-associated Drucker-Prager model in plasticity.
Relations betweeen bipotentials and Fitzpatrick functions are described. Selfdual lagrangians, introduced and studied by Ghoussoub, can be seen as bipotentials representing maximal monotone operators. We show that bipotentials can represent some monotone but not maximal operators, as well as non monotone operators.
Further we describe results concerning the construction of a bipotential which represents a given non monotone operator, by using convex lagrangian covers or bipotential convex covers.
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TL;DR: In this article, it was shown that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all the Fitzpatrick functions of the operator have a function in the same family of functions.
Abstract: In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick's family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brondsted-Rockafellar property.
In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons.
We will also prove that an auxiliary condition used by the authors to prove the restricted Brondsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.
TL;DR: In this article, a new monotonicity, M -monotonicity is introduced, and the resolvent operator of an M-monotone operator is proved to be single valued and Lipschitz continuous.
Abstract: In this paper, a new monotonicity, M -monotonicity, is introduced, and the resolvent operator of an M -monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI ( C , F + G ) and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping ∏ C ( ⋅ ) is semismooth, is given for calculating e -solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.
01 Jan 2008
TL;DR: In this paper, a modified CQ iterative scheme for finding a common element of the set of solutions of an equilibrium problem was introduced and a strong convergence theorem for three sequences generated by this process was obtained.
Abstract: In this paper, we introduce a modified CQ iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of the variational inequality for an α-inverse strongly monotone mapping in a Hilbert space. We obtain a strong convergence theorem for three sequences generated by this process. Based on this result, we also get several new and interesting results which generalize and extend some well-known strong convergence theorems in the literature.
TL;DR: In this article, the Julia-Caratheodory theorem and boundary Schwarz-Wolff lemma for hyperbolically monotone mappings in the open unit ball of a complex Hilbert space were established.
Abstract: We establish a Julia-Caratheodory theorem and a boundary Schwarz-Wolff lemma for hyperbolically monotone mappings in the open unit ball of a complex Hilbert space.
TL;DR: In this paper, a necessary and sufficient condition for a bivariate inf-convolution formula was established for the enlargement of the sum of two maximal monotone operators defined on a Banach space.
Abstract: Motivated by a classical result concerning the e-subdifferential of the sum of two proper, convex and lower semicontinuous functions, we give in this paper a similar result for the enlargement of the sum of two maximal monotone operators defined on a Banach space. This is done by establishing a necessary and sufficient condition for a bivariate inf-convolution formula.
TL;DR: The results rely on calmness and inner semicontinuity and are applied to find error bounds for solutions of a strongly monotone variational inequality in which both the constraining polyhedral multifunction and themonotone operator are perturbed.
Abstract: We present some results about Lipschitzian behavior of solutions to variational conditions when the sets over which the conditions are posed, as well as the functions appearing in them, may vary. These results rely on calmness and inner semicontinuity, and we describe some conditions under which those conditions hold, especially when the sets involved in the variational conditions are convex and polyhedral. We then apply the results to find error bounds for solutions of a strongly monotone variational inequality in which both the constraining polyhedral multifunction and the monotone operator are perturbed.
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TL;DR: In this paper, it was shown that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator.
Abstract: In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Bronsted-Rockafellar type property.
We show that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator.
TL;DR: In this paper, the degree theory for monotone type f + T mappings of class S+ from a bounded open set Ω in a reflexive Banach space X into its dual X ∗ is studied.
TL;DR: In this paper, a maximal 3-cyclically monotone operator with quite bizarre properties is constructed, and it is shown that the domain is the boundary of the unit diamond in the Euclidean plane.
Abstract: Subdifferential operators of proper convex lower semicontinuous functions and, more generally, maximal monotone operators are ubiquitous in optimization and nonsmooth analysis. In between these two classes of operators are the maximal n -cyclically monotone operators. These operators were carefully studied by Asplund, who obtained a complete characterization within the class of positive semidefinite (not necessarily symmetric) matrices, and by Voisei, who presented extension theorems a la Minty. All previous explicit examples of maximal n -cyclically monotone operators are maximal monotone; thus, they inherit the known good properties of maximal monotone operators. In this paper, we construct an explicit maximal 3-cyclically monotone operator with quite bizarre properties. This construction builds upon a recent, nonconstructive and Zorn’s Lemma-based, example. Our operator possesses two striking properties that sets it far apart from both the maximal monotone operator and the subdifferential operator case: it is not maximal monotone and its domain, which is closed, fails to be convex. Indeed, the domain is the boundary of the unit diamond in the Euclidean plane. The path leading to this operator requires some new results that are interesting in their own right.
TL;DR: In this paper, an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space is introduced.
Abstract: In this paper, we introduce an general iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the two sets. Using this results, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The results of this paper extended and improved the results of Iiduka and Takahashi (Nonlinear Anal. 61:341–350, 2005).
TL;DR: In this paper, it was shown that a set-valued map M : R q R q is maximal monotone if and only if the following conditions are satisfied: (i) M is monotonous; (ii) M has a nearly convex domain; (iii) m is convex-valued; (iv) the recession cone of the values M(x) equals the normal cone to the closure of the domain of M at x; and (v) m has a closed graph.
Abstract: It is shown that a set-valued map M : R q R q is maximal monotone if and only if the following five conditions are satisfied: (i) M is monotone; (ii) M has a nearly convex domain; (iii) M is convex-valued; (iv) the recession cone of the values M(x) equals the normal cone to the closure of the domain of M at x; (v) M has a closed graph. We also show that the conditions (iii) and (v) can be replaced by Cesari’s property (Q).
TL;DR: The present paper generalizes the approach, so that it applies to a far larger class of systems, and uses it in the analysis of a nine-variable autoregulatory transcription network.
Abstract: In recent work by Angeli and the authors, it was shown that the stability and global behavior of strongly monotone dynamical systems may be profitably studied using a technique that involves feedback decompositions into “well-behaved” subsystems. The present paper generalizes the approach so that it applies to a far larger class of systems. As an illustration, the techniques are used in the analysis of a nine-variable autoregulatory transcription network. Also, extensions to delay and reaction diusion systems are considered.
TL;DR: In this paper, an implicit hybrid steepest-descent method was developed to generate an iterative sequence from an arbitrary initial point to the unique solution of the variational inequality.
Abstract: Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is strongly monotone and Lipschitzian with constants $\eta >0$ and $\kappa >0$, respectively on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed point sets of a finite number of nonexpansive mappings on $H$. We develop an implicit hybrid steepest-descent method which generates an iterative sequence $\{u_n\}$ from an arbitrary initial point $u_0\in H$. We characterize the weak convergence of $\{u_n\}$ to the unique solution $u^*$ of the variational inequality: $$\langle F(u^*),v-u^*\rangle\geq0\quad\forall v\in C.$$ Applications to constrained generalized pseudoinverse are included.
TL;DR: In this article, a new type of approximating curve for finding a particular zero of the sum of two maximal monotone operators in a Hilbert space is investigated, which consists of the zeros of perturbed problems in which one operator is replaced with its Yosida approximation and a viscosity term is added.
Abstract: A new type of approximating curve for finding a particular zero of the sum of two maximal monotone operators in a Hilbert space is investigated. This curve consists of the zeros of perturbed problems in which one operator is replaced with its Yosida approximation and a viscosity term is added. As the perturbation vanishes, the curve is shown to converge to the zero of the sum that solves a particular strictly monotone variational inequality. As an off-spring of this result, we obtain an approximating curve for finding a particular zero of the sum of several maximal monotone operators. Applications to convex optimization are discussed.
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TL;DR: In this paper, a new class of strongly representable maximal monotone operators has been introduced, and the authors study domain range properties as well as connections with other classes and calculus rules for these operators.
Abstract: Recently in [1] a new class of maximal monotone operators has been introduced. In this note we study domain range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly representable operator is locally maximal monotone, maximal monotone locally, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representative function.
TL;DR: In this paper, it was shown that the closure of the domain and the range of a maximal monotone set of type NI are convex sets, which is a positive answer to the question concerning the convexity of the closed domain and range of the maximal Monotone multifunction, posed in Simons (Minimax and Monotonicity, Lecture Notes in Mathematics 1693).
Abstract: It is shown that the closure of the domain and the closure of the range of a maximal monotone set of type NI are convex sets. This is a positive answer to the question concerning the convexity of the closure of the range of a maximal monotone multifunction, posed in Simons (Minimax and Monotonicity, Lecture Notes in Mathematics 1693. Springer-Verlag, Berlin, 1998, X Open Problems, Problem 27.7).
TL;DR: This work uses representations of maximal monotone operators for studying recession (or asymptotic) operators associated to maximal monots and shows that such a concept is useful for dealing with unboundedness.
Abstract: We use representations of maximal monotone operators for studying recession (or asymptotic) operators associated to maximal monotone operators. Such a concept is useful for dealing with unboundedness.
TL;DR: In this paper, a generalized implicit hybrid projectionproximal point algorithm for finding zeros of a maximal monotone operator in a Hilbert space setting is introduced. And the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters is established.
Abstract: In this paper, we introduce a generalized implicit hybrid projectionproximal point algorithm for finding zeros of a maximal monotone operator in a Hilbert space setting. The global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters is established.
TL;DR: In this paper, the σ-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdorff dimension zero, was investigated and it was shown that the typical function in C[0, 1] is nowhere monotone and one-to-one except on some microscopic set.
Abstract: We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the σ-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdorff dimension zero and prove that the typical function in C[0, 1] (in the sense of Baire) is nowhere monotone and one-to-one except on some microscopic set. We also give an example of a continuous nowhere monotone function of bounded variation on [0, 1], which is one-to-one except on some microscopic set, so it is not a typical function.