In this section, we mainly discuss upper Hölder behavior of the perturbed (weak) solution mapping for problem (2.1) where both the objective function f and the feasible set Φ are perturbed by different parameters.
Namely, this section is devoted to derive the upper bounds for the distance from the perturbed (weak) solution mapping of (2.1) to the (weak) solution mapping for problem (2.2).
end{aligned} For the convenience, we will use the abbreviated notations V, WV, S, and WS to denote the optimal value mapping, the weak optimal value mapping, the solution mapping and the weak solution mapping for problem (2.2), respectively.
For any ((lambda, mu inLambdatimes M), the optimal value mapping and weak optimal value mapping for problem (2.1) are, respectively, defined by V lambda, mu):= E bigl(f bigl(Phi(mu), lambda bigr) bigr) quadmbox{and}quad WV( lambda, mu):= WE bigl(f bigl(Phi(mu), lambda bigr) bigr).
Then the solution mapping and weak solution mapping for problem (2.1) are, respectively, defined as begin{aligned} &S lambda, mu):= bigl{ x {in}inPhi(mu):f x, lambda)in V lambda, mu ) bigr} quadmbox{and} & WS lambda, mu):= bigl{ x {in}inPhi(mu):f x, lambda)in WV lambda, mu) bigr}.
The set-valued Lagrangian map for problem (VP) is defined by (6.1).
There have been many papers to discuss the stability of solution mapping for equilibrium problems when they are perturbed by parameters (also known the parametric (generalized) equilibrium problems).
The stability analysis of the solution mapping for these problems is an important topic in vector optimization theory.
It is well known that the stability analysis of a solution mapping for equilibrium problems is an important topic in optimization theory and applications.
If where is a maximal monotone mapping for then the problem (1.1) becomes the following problem.
(I) If where is a maximal monotone mapping for then the problem (1.1) becomes the following problem. .
