• Title/Summary/Keyword: -Lipschitz

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OPTIMALITY AND DUALITY IN NONSMOOTH VECTOR OPTIMIZATION INVOLVING GENERALIZED INVEX FUNCTIONS

  • Kim, Moon-Hee
    • Journal of applied mathematics & informatics
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    • v.28 no.5_6
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    • pp.1527-1534
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    • 2010
  • In this paper, we consider nonsmooth optimization problem of which objective and constraint functions are locally Lipschitz. We establish sufficient optimality conditions and duality results for nonsmooth vector optimization problem given under nearly strict invexity and near invexity-infineness assumptions.

Stability of nonlinear differential system by Lyapunov method

  • An, Jeong-Hyang
    • Journal of Korea Society of Industrial Information Systems
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    • v.12 no.5
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    • pp.54-59
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    • 2007
  • We abtain some stability results for a very general differential system using the method of cone valued vector Lyapunov functions and conversely some sufficient conditions for existence of such vector Lyapunov functions.

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MULTIVALUED MIXED QUASI-VARIATIONAL-LIKE INEQUALITIES

  • Lee Byung-Soo
    • The Pure and Applied Mathematics
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    • v.13 no.3 s.33
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    • pp.197-206
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    • 2006
  • This paper introduces a class of multivalued mixed quasi-variational-like ineqcalities and shows the existence of solutions to the class of quasi-variational-like inequalities in reflexive Banach spaces.

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STUDY OF OPTIMAL EIGHTH ORDER WEIGHTED-NEWTON METHODS IN BANACH SPACES

  • Argyros, Ioannis K.;Kumar, Deepak;Sharma, Janak Raj
    • Communications of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.677-693
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    • 2018
  • In this work, we generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. The study of the local convergence based on Lipschitz constants is important because it provides the degree of difficulty for choosing initial points. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to verify the theoretical results and to show the convergence behavior.

SYSTEM OF GENERALIZED MULTI-VALUED RESOLVENT EQUATIONS: ALGORITHMIC AND ANALYTICAL APPROACH

  • Javad Balooee;Shih-sen Chang;Jinfang Tang
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.785-827
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    • 2023
  • In this paper, under some new appropriate conditions imposed on the parameter and mappings involved in the resolvent operator associated with a P-accretive mapping, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. This paper is also concerned with the construction of a new iterative algorithm using the resolvent operator technique and Nadler's technique for solving a new system of generalized multi-valued resolvent equations in a Banach space setting. The convergence analysis of the sequences generated by our proposed iterative algorithm under some appropriate conditions is studied. The final section deals with the investigation and analysis of the notion of H(·, ·)-co-accretive mapping which has been recently introduced and studied in the literature. We verify that under the conditions considered in the literature, every H(·, ·)-co-accretive mapping is actually P-accretive and is not a new one. In the meanwhile, some important comments on H(·, ·)-co-accretive mappings and the results related to them appeared in the literature are pointed out.

WEAK CONVERGENCE FOR INTERATED RANDOM MAPS

  • Lee, Oe-Sook
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.485-490
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    • 1998
  • We consider a class of discrete parameter processes on a locally compact Polish space $S$ arising from successive compositions of strictly stationary Markov random maps on $S$ into itself. Sufficient conditions for the existence of the stationary solution and the weak convergence of the distributions of $\{\Gamma_n \Gamma_{n-1} \cdots \Gamma_0x \}$ are given.

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