Well-Posedness and Structural Stability on a System of Simultaneous Generalized Vector Quasi-Equilibrium Problems
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Well-Posedness and Structural Stability on a System of Simultaneous Generalized Vector Quasi-Equilibrium Problems Xi-Cai Deng1 · Wen-Sheng Jia2 · Yan-Long Yang2 Received: 4 May 2018 / Revised: 27 September 2018 / Accepted: 20 October 2018 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this paper, we establish the stable results for a system of simultaneous generalized vector quasi-equilibrium problems (SSGVQEP) by using its bounded rationality model. Under the abstract frame, a unified well-posedness on Hadamard types and Tikhonov types well-posedness for SSGVQEP is introduced. Moreover, sufficient condition for the well-posedness of SSGVQEP is given. Finally, we prove that the majority (in Baire category sense) of SSGVQEP is structural stability. Keywords Hadamard well-posedness · Tikhonov well-posedness · Structural stability · Bounded rationality Mathematics Subject Classification 91A26 · 49K40 · 90C31
This research was supported by the National Natural Science Foundation of China (Nos.11761023 and 11561013), Support Plan for Science and Technology Top-notch Talents of Guizhou Higher Education Institutions (No. 2017[081]), Natural Science Foundation of Guizhou Normal College (No. 2017BS009) and Natural Science Foundation of Guizhou Province (Nos. [2017]5788, LH[2017]7723 and [2016]5609).
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Wen-Sheng Jia [email protected] Xi-Cai Deng [email protected] Yan-Long Yang [email protected]
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Department of Mathematics and Computer, Guizhou Normal College, Guiyang 550018, China
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Department of Mathematics, Guizhou University, Guiyang 550025, China
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X.-C. Deng et al.
1 Introduction As is well know, the vector equilibrium problem is an important generalization of the equilibrium problems proposed by Blum and Oettli [1], which has attracted much attention in recent years especially due to its applications within the fields of vector optimization problems and vector variational inequalities, see [2,3]. Moreover, some more general and complexity types have been investigated, see [4–7]. Recently, Zhang and Zeng [8] considered the existence and Hadamard well-posedness of a system of simultaneous generalized vector quasi-equilibrium problems (SSGVQEP) was introduced by Ansari [9]. Moreover, by suitable conditions, it contains vector equilibrium problems as special cases. However, we know that the notion of well-posedness can be divided into Hadamard types and Tykhonov types [10]. Hadamard types of wellposedness for a problem require the existence and uniqueness of the optimal solution together with continuous dependence on the problem data. Tykhonov types of wellposedness for a problem, such as Tykhonov and Levitin–Polyak well-posedness, are imposing each approximating solution sequence converges to the uniqueness optimal solution of it [11,12]. Therefore, it is necessary to study other types of well-posedness for SSGVQEP. On the other hand, it is worth noting that in the last decades ther
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