The fixed point problem of f is said to be well posed with respect to α if: (i) f has a unique fixed point x ∗ in X such that α ( x ∗, f ( x ∗ ) ) ≥ 1 ; (ii) for a sequence { x n } in X such that d ( x n, f ( x n ) ) → 0 as n → ∞, then x n → x ∗ as n → ∞. .
Let V^_{delta}=bigl{ mathbf{v}^_{delta}in Y^_{delta}, b^ _{delta}bigl(mathbf{v}^_{delta},q_{delta}bigr)=0, forall q_{delta}in M_{delta}bigr}, the kernel of the bilinear form (b^_{delta}(cdot,cdot)). To prove that the problem (4.2) is well posed, we need to show the following properties: (1) (mathbf{a}^_{delta}(cdot,cdot)) is continuous on (Y_{delta}^) with a norm independent of δ [6].
This means that (u_{k}) should be well posed with respect to (w_{0}), which is called the well-posed condition of iterative functions, that is, c_{1} e) biglvert nabla u_{k} e) bigrvert ^{p-2} le biglvert nabla w_{0} e) bigrvert le c_{2} e) biglvert nabla u_{k} e) bigrvert ^{p-2}, (3.6) where (c_{1} e)) and (c_{2} e)) meet the requirements of Definition 2.
This point was made beautifully clear by Professor -- excuse me, Justice -- Elena Kagan, who did us all the favor of coming up with a well-polished hypothetical posed to the able Michael Carvin, attorney for the petitioners, that demonstrated the fallacy of this latest right-wing hit on the ACA.
Then the fixed point problem for f is well posed with respect to d.
Therefore, the fixed point equation (1.1) is well posed with respect to α. □.
We introduce a condition on h which implies that the equation is subcritical, i.e., the corresponding boundary value problem is well posed with respect to data given by finite measures.
Then the fixed point problem for f is well posed with respect to p. Corollary 33 Let ( X, d ) be a complete metric space and let f : X → X be a weakly Kannan contractive mapping.
Then the fixed point problem for f is well posed with respect to d if and only if for every sequence { x n } n ∈ N in X such that lim n → ∞ d ( x n, f ( x n ) ) = 0, there exists a subsequence { x n k } k ∈ N with lim k → ∞ α ¯ ( x n k, x ∗ ) < 1, where x ∗ is the unique fixed point of f.
