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A functional (mathcal{F}in mathscr{C}^{1} X, mathbb{R})) is called to satisfy the ((PS _{c}) condition if each sequence ({w_{n}}) in X satisfying (mathcal{F}(w_{n})rightarrow c) in (mathbb{R}), (mathcal{F}^{prime}(w_{n})rightarrow0) in (X^{ast}) contains a strongly convergent subsequence.
A mapping T : K → X is called to satisfy Condition (E μ ) on C, if there exists μ ≥ 1 such that d ( x, T y ) ≤ μ d ( T x, x ) + d ( x, y ).
A set of controls (u_{k}in mathcal {U}_{k}, 0leq kleq N,) for (N+1) agents is called to satisfy an ε-Nash equilibrium with respect to the costs J k, 0≤k≤N, if there exists ε≥0 such that for any fixed 0≤i≤N, we have J_{i} u_{i},u_{-i})leq J_{i} u'_{i},u_{-i})+epsilon (32).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com