Stability results for Ekeland's ε variational principle for set-valued mappings

In this paper, we define the Mosco convergence and Kuratowski-Painleve (P.K.) convergence for set-valued mapping sequence F-n. Under some conditions, we derive the following result: If a set-valued mapping sequence Fn, which are nonempty, compact valued, upper semicontinuous and uniformly bounded below, Mosco (or P.K.) converges to a set-valued mapping F, which is upper semicontinuous, nonempty, compact valued, then For Allepsilon > 0, lambda > 0, epsilon/lambda - ext F:= {(x) over bar is an element of X: (F(x) - (y) over bar + epsilon/lambdaparallel toe) n (-K) = theta, For Allx is an element of X\{(x) over bar}, for some (y) over bar is an element of F((x) over bar} = lim inf(n-->infinity) epsilon/lambda-ext F-n:= {(x) over bar is an element of X : (F-n(x)-E (y) over bar + epsilon/lambdaparallel tox - (x) over bar parallel toe) n (-K) = theta, For Allx is an element of X/{(x) over bar}, for some (y) over bar 5) is an element of F-n((x) over bar)}, where X is a Banach space, Y is a normed space, K subset of Y is a nonempty nontrival pointed closed convex dominating cone with nonempty interior int K, e is an element of int K, F-n : X-->2(Y) and F: X- 2(Y) are set-valued mappings.