Publications

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    ABSTRACT: The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied.
    Fixed Point Theory and Applications 10/2014; · 1.87 Impact Factor
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    Nimit Nimana, Narin Petrot
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    ABSTRACT: The main objective of this paper is to find a common solution of split variational inclusion problem and fixed point problem of infinite family of nonexpansive operators in a setting of real Hilbert spaces. To reach this goal, the iterative algorithms which combine Moudafi's viscosity approximation method with some fixed point technically proving methods are utilized for solving the problem. We prove that the iterative schemes with some suitable control conditions converge strongly to a common solution of the considered problem. We also show that many interesting problems can be solved by using our presented results. Index Terms—Split variational inclusion problem, fixed point problem, nonexpansive operators, resolvent operators, strong convergence.
    International MultiConference of Engineers and Computer Scientists 2014, Hong Kong; 03/2014
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    ABSTRACT: In this paper, we present some fixed-point theorems that are related to a set-valued Caristi-type mapping. The main results extend the recent work which was presented by Jiang and Li (Fixed Point Theory Appl. 2013:74, 2013) from a single-valued setting to a set-valued case. Further, the presented results also improve essentially many results that have appeared, because we have removed some conditions from the auxiliary function. Meanwhile, we give some partial answers to an important problem which was raised by Kirk (Colloq. Math. 36:81-86, 1976). MSC: Primary 47H10; secondary 37C25
    Fixed Point Theory and Applications 02/2014; · 1.87 Impact Factor
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    ABSTRACT: In this paper, a multivalued variational inequality problem on uniformly prox-regular set is studied. The existence theorems for such aforementioned problem are presented and, consequently, some algorithms for finding those solutions are also constructed. The results in this paper can be viewed as an improvement of the significant result that presented in Bounkhel et al. (J Inequal Pure Appl Math 4(1), 2003, Article 14).
    Optimization Letters 01/2014; · 1.65 Impact Factor
  • Jittiporn Suwannawit, Narin Petrot
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    ABSTRACT: The concept of quasivariational inequality problem on proximally smooth sets is studied. Some sufficient conditions for solving the existence of solutions of such a problem are provided; also some interesting cases are discussed. Of course, due to the significance of proximally smooth sets, the results which are presented in this paper improve and extend many important results in the literature.
    Abstract and Applied Analysis 01/2013; 2013. · 1.10 Impact Factor
  • Narin Petrot, Javad Balooee
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    ABSTRACT: In this paper, we introduce and study a new system of general nonconvex variational inclusions involving four different nonlinear operators (SGNVI) and prove the equivalence between the SGNVI and a fixed point problem. By using this equivalent formulation, we establish the existence and uniqueness theorem for solutions of the SGNVI. We use the foregoing equivalent alternative formulation and two nearly uniformly Lipschitzian mappings S 1 and S 2 to suggest and analyze some new two-step projection iterative algorithms for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q=(S 1 ,S 2 ), which is the unique solution of the SGNVI. Further, the convergence analysis of the suggested iterative algorithms under suitable conditions is studied.
    Journal of Inequalities and Applications 01/2013; 2013. · 0.82 Impact Factor
  • Narin Petrot, Javad Balooee
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    ABSTRACT: In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings S i (i=1,2,3) to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q=(S 1 ,S 2 ,S 3 ), which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed.
    Fixed Point Theory and Applications 01/2013; 2013. · 1.87 Impact Factor
  • Jittiporn Suwannawit, Narin Petrot
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    ABSTRACT: We introduce and study a class of a system of random set-valued variational inclusionproblems. Some conditions for the existence of solutions of such problems are provided, when the operatorsare contained in the classes of generalized monotone operators, so-called ($A,m,\eta $)-monotone operator. Further,the stability of the iterative algorithm for finding a solution of the considered problem is also discussed.
    Journal of Applied Mathematics 01/2012; 2012, Special Issue. · 0.83 Impact Factor
  • Narin Petrot, Javad Balooee
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    ABSTRACT: We introduce a new system of extended general nonlinear variational inclusions with different nonlinear operators and establish the equivalence between the aforesaid system and the fixed point problem. By using this equivalent formulation, we prove the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions. We suggest and analyze a new resolvent iterative algorithm to approximate the unique solution of the system of extended general nonlinear variational inclusions which is a fixed point of a nearly uniformly Lipschitzian mapping. Subsequently, the convergence analysis of the proposed iterative algorithm under some suitable conditions is considered. Furthermore, some related works to our main problem are pointed out and discussed.
    Abstract and Applied Analysis 01/2012; 2012. · 1.10 Impact Factor
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    Narin Petrot, Javad Balooee
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    ABSTRACT: At the present article, we consider a new class of general nonlinear random A-maximal m-relaxed η-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces. By using the resolvent mapping technique for A-maximal m-relaxed η-accretive mappings due to Lan et al. and Chang's lemma, we construct a new iterative algorithm with mixed errors for finding the approximate solutions of this class of nonlinear random equations. We also verify that the approximate solutions obtained by the our proposed algorithm converge to the exact solution of the general nonlinear random A-maximal m-relaxed η-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces. Mathematical Subject Classification 2010: Primary, 47B80; Secondary, 47H40, 60H25.
    Journal of Inequalities and Applications 01/2012; 2012(1). · 0.82 Impact Factor
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    Inchan Issara, Petrot Narin
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    ABSTRACT: By using projection methods, we suggest and analyze iterative schemes for finding the approximation solvability of a system of general variational inequalities involving different nonlinear operators in the framework of Hilbert spaces. Moreover, such solutions are also fixed points of a Lipschitz mapping. Some interesting cases and examples of applying the main results are discussed and shown. The results presented in this paper are more general and include many previously known results as special cases.
    Fixed Point Theory and Applications 01/2011; · 1.87 Impact Factor
  • Suthep Suantai, Narin Petrot
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    ABSTRACT: In this paper, we consider the system of nonlinear quasi-mixed equilibrium problems. The existence theorems of solutions of such problems are provided by considering the limit point of an iterative algorithm. This means, we not only give the conditions for the existence theorems of the presented problems but also provide the algorithm to find such solutions. Moreover, the stability of such an algorithm is also discussed. The results presented in this paper are more general, and may be viewed as an extension, refinement and improvement of the previously known results in the literature.
    Applied Mathematics Letters 01/2011; 24:308-313. · 1.50 Impact Factor
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    ABSTRACT: In this paper, the concept of a set-valued contractive mapping is considered by using the idea of a generalized distance, such as the τ-distance, in metric spaces without using the concept of the Hausdorff metric. Furthermore, under some mild conditions, we provide the existence theorems for fixed-point problems of the considered mapping. Hence, our results can be viewed as a generalization and improvement of many recent results.
    Applied Mathematics Letters 01/2011; 24:1959-1967. · 1.50 Impact Factor
  • N Petrot, R Wangkeeree, P Kumam
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    ABSTRACT: In this paper, we introduce a new iterative scheme for finding solutions the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inclusion problem with a multivalued maximal monotone mapping and an alpha-inverse-strongly monotone mapping. We show that the sequence converges strongly to a common solutions for quasi variational inclusion and fixed point problems under some parameters controlling conditions. This main theorem extends a recent result of Zhang et al. [Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. Engl. Ed., 2008, 29(5)(2006), 571-581] and some other authors.
    International journal on fixed point theory computation and applications 01/2011; 12(1):165-178. · 0.78 Impact Factor
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    Yeol JE Cho, Narin Petrot
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    ABSTRACT: Without the strong monotonicity assumption of the mapping, we provide a regularization method for general variational inequality problem, when its solution set is related to a solution set of an inverse strongly monotone mapping. Consequently, an iterative algorithm for finding such a solution is constructed, and convergent theorem of the such algorithm is proved. It is worth pointing out that, since we do not assume strong monotonicity of general variational inequality problem, our results improve and extend some well-known results in the literature.
    Journal of Inequalities and Applications 01/2011; 2011(1). · 0.82 Impact Factor
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    Poom Kumam, Narin Petrot, Rabian Wangkeeree
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    ABSTRACT: In this paper, some existence theorems for the mixed quasi-variational-like inequalities problem in a reflexive Banach space are established. The auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution for the mixed quasi-variational-like inequalities problem. Consequently, not only the existence of theorems of the mixed quasi-variational-like inequalities is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results.
    Applied Mathematics and Computation 01/2011; 217:7496-7503. · 1.35 Impact Factor
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    ABSTRACT: In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.
    Journal of Global Optimization 01/2011; 51:27-46. · 1.31 Impact Factor
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    ABSTRACT: In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solution of a mixed equilibrium problem and the set of solution of the variational inequality problems for a relaxed (u, v)-cocoercive and mu-Lipschitz continuous mapping in Hilbert spaces. We show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions. Our results improve and extend the recent ones announced by [Y.J. Cho, X.Q. Qin, M. Kang, Some results for equilibrium problems and fixed point problems in Hilbert spaces, J. Comput. Anal. Appl. 11(2009) 294-316] and many others.
    Journal of Computational Analysis and Applications 01/2011; 13(3):425-449. · 0.50 Impact Factor
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    ABSTRACT: In this paper, we introduce an iterative method for finding a common element of the set of solutions of the generalized equilibrium problems, the set of solutions for the systems of nonlinear variational inequalities problems and the set of fixed points of nonexpansive mappings in Hilbert spaces. Furthermore, we apply our main result to the set of fixed points of an infinite family of strict pseudo-contraction mappings. The results obtained in this paper are viewed as a refinement and improvement of the previously known results.
    Computers & Mathematics with Applications 10/2010; · 2.07 Impact Factor
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    Yeol Je Cho, Petrot Narin
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    ABSTRACT: We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems (SMIE) in Hilbert spaces. The algorithm for finding a solution of the problem (SMIE) is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.
    Journal of Inequalities and Applications 01/2010; · 0.82 Impact Factor

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