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  • 1
    Online Resource
    Online Resource
    Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems
    Hindawi Limited ; 2012
    In:  Abstract and Applied Analysis Vol. 2012 ( 2012), p. 1-32
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2012 ( 2012), p. 1-32
    Abstract: We consider and study the modified extragradient methods for finding a common element of the solution set Γ of a split feasibility problem (SFP) and the fixed point set Fix ( S ) of a strictly pseudocontractive mapping S in the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element of Fix ( S ) ∩ Γ where S is strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of Fix ( S ) ∩ Γ . We also propose another extragradient-like algorithm for finding an element of Fix ( S ) ∩ Γ where S : C → C is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of Fix ( S ) ∩ Γ .
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2012/975981
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2012
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 2
    Online Resource
    Online Resource
    Relaxed Iterative Algorithms for Generalized Mixed Equilibrium Problems with Constraints of Variational Inequalities and Variational Inclusions
    Hindawi Limited ; 2014
    In:  Abstract and Applied Analysis Vol. 2014 ( 2014), p. 1-25
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2014 ( 2014), p. 1-25
    Abstract: We introduce and analyze a relaxed extragradient-like viscosity iterative algorithm for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems which also solves a variational inequality problem.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2014/345212
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2014
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 3
    Online Resource
    Online Resource
    Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria
    Hindawi Limited ; 2014
    In:  Abstract and Applied Analysis Vol. 2014 ( 2014), p. 1-25
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2014 ( 2014), p. 1-25
    Abstract: We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2014/513678
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2014
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 4
    Online Resource
    Online Resource
    Generalized Outer Synchronization between Complex Networks with Unknown Parameters
    Hindawi Limited ; 2013
    In:  Abstract and Applied Analysis Vol. 2013 ( 2013), p. 1-9
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2013 ( 2013), p. 1-9
    Abstract: As is well known, complex networks are ubiquitous in the real world. One network always behaves differently from but still coexists in balance with others. This phenomenon of harmonious coexistence between different networks can be termed as “generalized outer synchronization (GOS).” This paper investigates GOS between two different complex dynamical networks with unknown parameters according to two different methods. When the exact functional relations between the two networks are previously known, a sufficient criterion for GOS is derived based on Barbalat's lemma. If the functional relations are not known, the auxiliary-system method is employed and a sufficient criterion for GOS is derived. Numerical simulations are further provided to demonstrate the feasibility and effectiveness of the theoretical results.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2013/802859
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2013
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 5
    Online Resource
    Online Resource
    Iterative Schemes for Convex Minimization Problems with Constraints
    Hindawi Limited ; 2014
    In:  Abstract and Applied Analysis Vol. 2014 ( 2014), p. 1-22
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2014 ( 2014), p. 1-22
    Abstract: We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2014/209372
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2014
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 6
    Online Resource
    Online Resource
    Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros of m -Accretive Operators in Banach Spaces
    Hindawi Limited ; 2013
    In:  Abstract and Applied Analysis Vol. 2013 ( 2013), p. 1-21
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2013 ( 2013), p. 1-21
    Abstract: The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of an m -accretive operator A in the setting of real uniformly convex and 2-uniformly smooth Banach space X . Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the set A - 1 ( 0 ) of zeros of A in X . Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2013/852760
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2013
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 7
    Online Resource
    Online Resource
    Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces
    Hindawi Limited ; 2013
    In:  Journal of Applied Mathematics Vol. 2013 ( 2013), p. 1-21
    Details
    In: Journal of Applied Mathematics, Hindawi Limited, Vol. 2013 ( 2013), p. 1-21
    Abstract: We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.
    Type of Medium: Online Resource
    ISSN: 1110-757X , 1687-0042
    URL: Article
    DOI: 10.1155/2013/761864
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2013
    detail.hit.zdb_id: 2578385-3
    SSG: 17,1
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  • 8
    Online Resource
    Online Resource
    Hybrid Iterative Scheme for Triple Hierarchical Variational Inequalities with Mixed Equilibrium, Variational Inclusion, and Minimization Constraints
    Hindawi Limited ; 2014
    In:  Abstract and Applied Analysis Vol. 2014 ( 2014), p. 1-22
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2014 ( 2014), p. 1-22
    Abstract: We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2014/767109
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2014
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 9
    Online Resource
    Online Resource
    Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems and Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense
    Hindawi Limited ; 2013
    In:  Abstract and Applied Analysis Vol. 2013 ( 2013), p. 1-14
    Details
    In: Abstract and Applied Analysis, Hindawi Limited, Vol. 2013 ( 2013), p. 1-14
    Abstract: We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping S in the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mapping S . The implicit hybrid method with regularization is based on four well-known methods: the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.
    Type of Medium: Online Resource
    ISSN: 1085-3375 , 1687-0409
    URL: Article
    DOI: 10.1155/2013/854297
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2013
    detail.hit.zdb_id: 2064801-7
    SSG: 17,1
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  • 10
    Online Resource
    Online Resource
    Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces
    Hindawi Limited ; 2012
    In:  Journal of Applied Mathematics Vol. 2012 ( 2012), p. 1-38
    Details
    In: Journal of Applied Mathematics, Hindawi Limited, Vol. 2012 ( 2012), p. 1-38
    Abstract: We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.
    Type of Medium: Online Resource
    ISSN: 1110-757X , 1687-0042
    URL: Article
    DOI: 10.1155/2012/194509
    Language: English
    Publisher: Hindawi Limited
    Publication Date: 2012
    detail.hit.zdb_id: 2578385-3
    SSG: 17,1
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