An optimization problem is said to be well posed if the solution of the problem uniquely exists and the solution depends continuously on the data of the problem.
The fixed point problem (27) is said to be well posed if: (1) T has a unique fixed point (u^in X); (2) for any sequence ({x_{n}}subseteq X) with (lim_{nrightarrowinfty}q(x_{n},Tx_{n})= lim_{nrightarrow infty}q(Tx_{n},x_{n})=0), then we have (lim_{nrightarrow infty}q(x_{n},u^)= lim_{nrightarrowinfty}q u^,x_{n})=0). .
The fixed point problem of f is said to be well posed if: (i) f has a unique fixed point z ∈ X ; (ii) for any sequence { x n } n ∈ N in X such that lim n → ∞ d ( f x n, x n ) = 0, we have lim n → ∞ d ( x n, z ) = 0. .
A problem is well posed if: (1) at least one solution exists, (2) at most one solution exists, and (3) the solution is stable.
Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.
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 → ∞. .
Under the assumptions of Theorem 4, the fixed point problem for T is well posed, that is, if ({y_{n}} ) is a sequence in X satisfying (d( y_{n},Ty_{n}) rightarrow0) as (nrightarrowinfty), then (y_{n}rightarrow z) as (nrightarrow infty), where z is the unique fixed point of T (whose existence is guaranteed by Theorem 4).
Under the assumptions of Theorem 2, the fixed point problem for T is well posed, that is, if there exists a sequence ({y_{n}}) in X with (d y_{n},Ty_{n}) rightarrow0) as (nrightarrow infty), then (y_{n}rightarrow z) as (nrightarrowinfty), where z is the unique fixed point of T. Our main result is the following improvement of Theorem 2, that is, the main result from [1].
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.
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 Y 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.
Let f : Y → Y be a mapping satisfying all the requirements in the statement of Theorem 21, then the fixed point problem for f is well posed with respect to p if and only if for every sequence { x n } n ∈ N in Y such that lim n → ∞ p ( 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.
