23 results on '"Szilárd László"'
Search Results
2. Convergence rates for an inertial algorithm of gradient type associated to a smooth non-convex minimization
- Author
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Szilárd László
- Subjects
021103 operations research ,General Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Regular polygon ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Regularization (mathematics) ,Critical point (mathematics) ,Convex optimization ,Minification ,Differentiable function ,0101 mathematics ,Gradient method ,Algorithm ,Software ,Mathematics - Abstract
We investigate an inertial algorithm of gradient type in connection with the minimization of a non-convex differentiable function. The algorithm is formulated in the spirit of Nesterov’s accelerated convex gradient method. We prove some abstract convergence results which applied to our numerical scheme allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. Further, we obtain convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent of a regularization of the objective function. Finally, some numerical experiments are presented in order to compare our numerical scheme and some algorithms well known in the literature.
- Published
- 2020
3. An Extension of the Second Order Dynamical System that Models Nesterov’s Convex Gradient Method
- Author
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Cristian Daniel Alecsa, Titus Pinţa, and Szilárd László
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0209 industrial biotechnology ,Control and Optimization ,Inertial frame of reference ,Discretization ,Applied Mathematics ,010102 general mathematics ,Fréchet derivative ,Regular polygon ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,Convex optimization ,Applied mathematics ,Ball (mathematics) ,0101 mathematics ,Gradient method ,Smoothing ,Mathematics - Abstract
In this paper we deal with a general second order continuous dynamical system associated to a convex minimization problem with a Frechet differentiable objective function. We show that inertial algorithms, such as Nesterov’s algorithm, can be obtained via the natural explicit discretization from our dynamical system. Our dynamical system can be viewed as a perturbed version of the heavy ball method with vanishing damping, however the perturbation is made in the argument of the gradient of the objective function. This perturbation seems to have a smoothing effect for the energy error and eliminates the oscillations obtained for this error in the case of the heavy ball method with vanishing damping, as some numerical experiments show. We prove that the value of the objective function in a generated trajectory converges in order $$\mathcal {O}(1/t^2)$$ to the global minimum of the objective function. Moreover, we obtain that a trajectory generated by the dynamical system converges to a minimum point of the objective function.
- Published
- 2020
4. Newton-like Inertial Dynamics and Proximal Algorithms Governed by Maximally Monotone Operators
- Author
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Szilárd László and Hedy Attouch
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Hessian matrix ,021103 operations research ,Time-dependent viscosity ,Inertial frame of reference ,Dynamics (mechanics) ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Theoretical Computer Science ,symbols.namesake ,Monotone polygon ,symbols ,0101 mathematics ,Temporal discretization ,Newton's method ,Algorithm ,Gradient method ,Software ,Mathematics - Abstract
The introduction of the Hessian damping in the continuous version of Nesterov's accelerated gradient method provides, by temporal discretization, fast proximal gradient algorithms where the oscilla...
- Published
- 2020
5. A gradient-type algorithm with backward inertial steps associated to a nonconvex minimization problem
- Author
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Szilárd László, Adrian Viorel, and Cristian Daniel Alecsa
- Subjects
Inertial frame of reference ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Regularization (mathematics) ,Critical point (mathematics) ,010101 applied mathematics ,Theory of computation ,Exponent ,Differentiable function ,Minification ,0101 mathematics ,Algorithm ,Mathematics - Abstract
We investigate an algorithm of gradient type with a backward inertial step in connection with the minimization of a nonconvex differentiable function. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. Further, we provide convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent. Finally, some numerical experiments are presented in order to compare our numerical scheme with some algorithms well known in the literature.
- Published
- 2019
6. Tikhonov regularization of a second order dynamical system with Hessian driven damping
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
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Hessian matrix ,General Mathematics ,0211 other engineering and technologies ,Dynamical Systems (math.DS) ,02 engineering and technology ,Dynamical system ,01 natural sciences ,Hessian-driven damping ,90C26 ,Tikhonov regularization ,symbols.namesake ,34G25, 47J25, 47H05, 90C26, 90C30, 65K10 ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,47J25 ,47H05 ,010102 general mathematics ,Hilbert space ,90C30 ,Function (mathematics) ,Convex optimization ,Optimization and Control (math.OC) ,Second order dynamical system ,34G25 ,symbols ,Fast convergence methods ,Convex function ,Software - Abstract
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.
- Published
- 2020
7. Second-order dynamical systems with penalty terms associated to monotone inclusions
- Author
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
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021103 operations research ,Dynamical systems theory ,Computer Science::Information Retrieval ,Applied Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,0211 other engineering and technologies ,Hilbert space ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Monotone polygon ,symbols ,Computer Science::General Literature ,Order (group theory) ,0101 mathematics ,Dynamical system (definition) ,ComputingMilieux_MISCELLANEOUS ,Analysis ,Mathematics - Abstract
In this paper, we investigate in a Hilbert space setting a second-order dynamical system of the form [Formula: see text] where [Formula: see text]image[Formula: see text] is a maximal monotone operator, [Formula: see text] is the resolvent operator of [Formula: see text] and [Formula: see text] are cocoercive operators, and [Formula: see text], and [Formula: see text] are step size, penalization and, respectively, damping functions, all depending on time. We show the existence and uniqueness of strong global solutions in the framework of the Cauchy–Lipschitz–Picard Theorem and prove ergodic asymptotic convergence for the generated trajectories to a zero of the operator [Formula: see text] where [Formula: see text] and [Formula: see text] denotes the normal cone operator of [Formula: see text]. To this end, we use Lyapunov analysis combined with the celebrated Opial Lemma in its ergodic continuous version. Furthermore, we show strong convergence for trajectories to the unique zero of [Formula: see text], provided that [Formula: see text] is a strongly monotone operator.
- Published
- 2018
8. A primal-dual dynamical approach to structured convex minimization problems
- Author
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Radu Ioan Boţ, Ernö Robert Csetnek, and Szilárd László
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Discretization ,Feasibility condition ,Applied Mathematics ,010102 general mathematics ,Numerical Analysis (math.NA) ,Dynamical Systems (math.DS) ,Dynamical system ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Optimization and Control (math.OC) ,37N40, 49N15, 90C25, 90C46 ,Saddle point ,Convex optimization ,FOS: Mathematics ,Ergodic theory ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Dynamical Systems ,Convex function ,Mathematics - Optimization and Control ,Analysis ,Mathematics - Abstract
In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm.
- Published
- 2019
9. An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions
- Author
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Szilárd László, Ernö Robert Csetnek, and Radu Ioan Bot
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T57-57.97 ,Mathematical optimization ,65K10 ,Applied mathematics. Quantitative methods ,Control and Optimization ,Optimization problem ,Inertial frame of reference ,Forward–backward algorithm ,90C30 ,QA75.5-76.95 ,Management Science and Operations Research ,Bregman divergence ,Regularization (mathematics) ,Critical point (mathematics) ,90C26 ,Computational Mathematics ,Iterated function ,Electronic computers. Computer science ,Modeling and Simulation ,Minification ,Mathematics - Abstract
We propose a forward–backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. Every sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-Łojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions. We illustrate the theoretical results by considering two numerical experiments: the first one concerns the ability of recovering the local optimal solutions of nonconvex optimization problems, while the second one refers to the restoration of a noisy blurred image.
- Published
- 2016
10. A second-order dynamical approach with variable damping to nonconvex smooth minimization
- Author
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
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Differential equation ,nonconvex optimization ,Boris Mordukhovich ,Dynamical system ,01 natural sciences ,Regularization (mathematics) ,Critical point (mathematics) ,Article ,90C26 ,convergence rate ,Second-order dynamical system ,Applied mathematics ,Differentiable function ,0101 mathematics ,Mathematics ,Variable (mathematics) ,65K10 ,Kurdyka–Łojasiewicz inequality ,Applied Mathematics ,010102 general mathematics ,90C30 ,010101 applied mathematics ,Rate of convergence ,Erratum ,Gradient method ,Analysis - Abstract
We investigate a second-order dynamical system with variable damping in connection with the minimization of a nonconvex differentiable function. The dynamical system is formulated in the spirit of the differential equation which models Nesterov's accelerated convex gradient method. We show that the generated trajectory converges to a critical point, if a regularization of the objective function satisfies the Kurdyka- Lojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Lojasiewicz exponent.
- Published
- 2018
11. Approaching nonsmooth nonconvex minimization through second-order proximal-gradient dynamical systems
- Author
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Radu Ioan Boţ, Szilárd László, and Ernö Robert Csetnek
- Subjects
021103 operations research ,Dynamical systems theory ,010102 general mathematics ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,Dynamical system ,01 natural sciences ,Regularization (mathematics) ,Critical point (mathematics) ,Limiting subdifferential ,Mathematics (miscellaneous) ,Second-order dynamical system ,Convergence (routing) ,Trajectory ,Exponent ,Applied mathematics ,Nonsmooth nonconvex optimization ,0101 mathematics ,Kurdyka–Łojasiewicz property ,Mathematics - Abstract
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.
- Published
- 2018
12. Generalized Monotone Operators on Dense Sets
- Author
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Adrian Viorel and Szilárd László
- Subjects
Discrete mathematics ,Control and Optimization ,Banach space ,Monotonic function ,Strongly monotone ,Domain (mathematical analysis) ,Convexity ,Functional Analysis (math.FA) ,47H04, 47H05, 26B25, 26E25 ,Computer Science Applications ,Mathematics - Functional Analysis ,Operator (computer programming) ,Monotone polygon ,Real-valued function ,Signal Processing ,FOS: Mathematics ,Analysis ,Mathematics - Abstract
In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets., Comment: 19 pages
- Published
- 2015
13. On Injectivity of a Class of Monotone Operators with Some Univalency Consequences
- Author
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Szilárd László
- Subjects
Discrete mathematics ,Pure mathematics ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Inverse ,Monotonic function ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Convexity ,Operator (computer programming) ,Monotone polygon ,0101 mathematics ,Mathematics ,Univalent function ,Variable (mathematics) - Abstract
In this paper, we provide sufficient conditions that ensure the convexity of the inverse images of an operator, monotone in some sense. Further, conditions that ensure the monotonicity, respectively the local injectivity of an operator, are also obtained. Combining the conditions that provide the local injectivity, respectively the convexity of the inverse images of an operator, we are able to obtain some global injectivity results. As applications, some new analytical conditions that assure the injectivity, respectively univalency of a complex function of one complex variable are obtained. We also show that some classical results, such as Alexander–Noshiro–Warschawski and Wolff theorem or Mocanu theorem, are easy consequences of our results.
- Published
- 2015
14. Multivalued variational inequalities and coincidence point results
- Author
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Szilárd László
- Subjects
Algebra ,Discrete mathematics ,Class (set theory) ,symbols.namesake ,Applied Mathematics ,Variational inequality ,Hilbert space ,symbols ,Coincidence point ,Analysis ,Mathematics - Abstract
In this paper, we establish some existence results of the solutions for several multivalued variational inequalities involving elements belonging to a class of operators that was recently introduced in the literature. As applications we obtain some new coincidence point results in Hilbert spaces.
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- 2013
15. Monotone operators and first category sets
- Author
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Szilárd László, Cornel Pintea, and Gábor Kassay
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Discrete mathematics ,Pure mathematics ,General Mathematics ,Perfect set ,Inverse ,Monotonic function ,Operator theory ,Injective function ,Convexity ,Theoretical Computer Science ,Monotone polygon ,Positive definiteness ,Analysis ,Mathematics - Abstract
In this paper we show that the local monotonicity in the sense of Minty and Browder on some residual sets assure the global monotonicity and, according to an earlier result, the convexity of the inverse images. We pay some special attention to the residual sets arising as complements of some special first Baire category sets, namely the \(\sigma \)-affine sets, the \(\sigma \)-compact sets and the \(\sigma \)-algebraic varieties. We achieve this goal gradually by showing, at first, that the continuous real valued functions of one real variable, which are locally nondecreasing on sets whose complements have no nonempty perfect subsets, are globally nondecreasing. The convexity of the inverse images combined with their discreteness, in the case of local injective operators, ensure the global injectivity. Note that the global monotonicity and the local injectivity of regular enough operators is guaranteed by the positive definiteness of the symmetric part of their Gâteaux differentials on the involved residual sets. We close this work with a short subsection on the global convexity which is obtained out of its local counterpart on some residual sets.
- Published
- 2012
16. About the Maximal Monotonicity of the Generalized Sum of Two Maximal Monotone Operators
- Author
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Szilárd László and Boglárka Burján-Mosoni
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Statistics and Probability ,Representative function ,Combinatorics ,Numerical Analysis ,Monotone polygon ,Applied Mathematics ,Monotonic function ,Geometry and Topology ,Type (model theory) ,Strongly monotone ,Analysis ,Interior point method ,Mathematics - Abstract
We give several regularity conditions, both closedness and interior point type, that ensure the maximal monotonicity of the generalized sum of two strongly-representable monotone operators, and we extend some recent results concerning on the same problem.
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- 2011
17. Monotone operators and closed countable sets
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Cornel Pintea, Gábor Kassay, and Szilárd László
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TheoryofComputation_MISCELLANEOUS ,Discrete mathematics ,Control and Optimization ,Closed set ,Applied Mathematics ,Management Science and Operations Research ,Strongly monotone ,Convexity ,Monotone polygon ,Operator (computer programming) ,Countable set ,Convex function ,Mathematics ,Complement (set theory) - Abstract
In this article we prove that the local increasing monotonicity of an operator on the complement of a certain type of closed set implies its global increasing monotonicity. As applications we obtain some global injectivity/convexity results based on local injectivity/convexity properties and some extra analytic requirements.
- Published
- 2011
18. Some Existence Results of Solutions for General Variational Inequalities
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Szilárd László
- Subjects
New class ,Algebra ,Class (set theory) ,Mathematical optimization ,Control and Optimization ,Applied Mathematics ,Variational inequality ,Theory of computation ,Management Science and Operations Research ,Mathematics - Abstract
In this paper, we introduce a new class of operators. We present some fundamental properties of the operators belonging to this class and, as applications, we establish some existence results of the solutions for several general variational inequalities involving elements belonging to this class.
- Published
- 2011
19. Vector Equilibrium Problems on Dense Sets
- Author
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Szilárd László
- Subjects
Pure mathematics ,021103 operations research ,Control and Optimization ,Vector operator ,Dual space ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Direction vector ,01 natural sciences ,Topological vector space ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Locally convex topological vector space ,Ordered vector space ,FOS: Mathematics ,0101 mathematics ,Vector potential ,Normed vector space ,Mathematics - Abstract
In this paper we provide sufficient conditions that ensure the existence of the solution of some vector equilibrium problems in Hausdorff topological vector spaces ordered by a cone. The conditions that we consider are imposed not on the whole domain of the operators involved, but rather on a self segment-dense subset of it, a special type of dense subset. We apply the results obtained to vector optimization and vector variational inequalities., Comment: arXiv admin note: substantial text overlap with arXiv:1405.2327
- Published
- 2015
- Full Text
- View/download PDF
20. Densely defined equilibrium problems
- Author
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Szilárd László and Adrian Viorel
- Subjects
Correlated equilibrium ,Sequential equilibrium ,Computer Science::Computer Science and Game Theory ,Control and Optimization ,Dense set ,Applied Mathematics ,Symmetric equilibrium ,47H04, 47H05, 26B25, 26E25, 90C33 ,Management Science and Operations Research ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Walras' law ,Nash equilibrium ,symbols ,FOS: Mathematics ,Epsilon-equilibrium ,Solution concept ,Mathematical economics ,Mathematics - Abstract
In the present work we deal with set-valued equilibrium problems for which we provide sufficient conditions for the existence of a solution. The conditions that we consider are imposed not on the whole domain, but rather on a self segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu-Gale-Nikaido-type theorem, with a considerably weakened Walras law in its hypothesis. Further, we consider a non-cooperative n-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones., Comment: 21 pages
- Published
- 2014
- Full Text
- View/download PDF
21. $\theta-$MONOTONE OPERATORS AND $\theta-$CONVEX FUNCTIONS
- Author
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Szilárd László
- Subjects
Discrete mathematics ,TheoryofComputation_MISCELLANEOUS ,Property (philosophy) ,locally monotone operator ,generalized monotone operator ,47H05 ,General Mathematics ,Monotonic function ,Convexity ,Domain (mathematical analysis) ,26B25 ,Monotone polygon ,Operator (computer programming) ,49J50 ,generalized convex function ,26A51 ,maximal monotonicity ,Convex function ,Mathematics - Abstract
In this paper we introduce a new monotonicity concept for multivalued operators, respectively, a new convexity concept for real valued functions, which generalize several monotonicity, respectively, convexity notions already known in literature. We present some fundamental properties of the operators having this monotonicity property. We show that if such a monotonicity property holds locally then the same property holds globally on the whole domain of the operator. We also show that these two new concepts are closely related. As an immediate application we furnish some surjectivity results in finite dimensional spaces.
- Published
- 2012
22. Solution existence of general variational inequalities and coincidence points
- Author
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Szilárd László and Alireza Amini-Harandi
- Subjects
General Mathematics ,Variational inequality ,Applied mathematics ,Coincidence ,Mathematics - Abstract
In this paper, by using a simple technique, we obtain several existence results of the solutions for general variational inequalities of Stampacchia type. We also show, that the existence of a coincidence point of two mappings is equivalent to the existence of the solution of a particular general variational inequality of Stampacchia type. As applications several coincidence and fixed point results are obtained.
23. Existence of solutions of inverted variational inequalities
- Author
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Szilárd László
- Subjects
Mathematics - Functional Analysis ,General Mathematics ,Variational inequality ,FOS: Mathematics ,Applied mathematics ,Functional Analysis (math.FA) ,Mathematics - Abstract
In this paper we introduce two new generalized variational inequalities and we give some existence results of the solutions for these variational inequalities involving operators belonging to a recently introduced class of operators. We show by examples, that our results fail outside of this class. Further, we establish a result that may be viewed as a generalization of Minty’s theorem, that is, we show that under some circumstances the set of solutions of these variational inequalities coincide. We also show, the condition that the operators, involved in these variational inequalities, belong to the above mentioned class, is essential in obtaining this result. As application, we show that Brouwer’s fixed point theorem is an easy consequence of our results.
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