Therefore, Q is weakly-weakly continuous.
Next, we show that Q is weakly-weakly sequentially continuous.
end{aligned} We will show that Q maps K into itself and Q restricted to the set K is weakly-weakly sequentially continuous.
Since Q is weakly-weakly sequentially continuous, then { u n − Q u n } n ≥ 1 converges weakly to u − Q u in C ( T 0, E w ).
Since K is an equicontinuous subset of (C(T_{0},E)) it follows that (Q cdot)) is weakly-weakly sequentially continuous.
Following the same reasoning as in the proof of Theorem 1 it is easy to show that Q is weakly-weakly sequentially continuous from K to K. Next, we will prove that Q has at least one fixed point (y_{0}(cdot)in K).
(f t,cdot)) is weakly-weakly sequentially continuous, for every (tin T); (f cdot,y(cdot))in P^{infty}(T,E)), for every continuous function (y(cdot):Trightarrow E); (|f t,y)|leq M), for all ((t,y in Ttimes E); for every bounded set (Asubseteq E) we have betabigl(f(Ttimes A bigr)leq gbigl(beta(A bigr), In our proof we shall use some ideas from [5] and [6].
Then by (H3), we have begin{aligned} bigl| (Qy) (t bigr| leq&Vert y_{0}Vert +int_{0}^{t} frac{ t-s)^{alpha-1}}{Gamma(alpha)}biglvert bigllangle y^{ast },frac{ t-s(s)bigr) bigrrangle bigrvert,ds leq&Vert y_{0}Vert +frac{Ma^{alpha}}{Gamma(alpha +1)}leq Vert y_{0}Vert +1, end{aligned} and using (16) it follows that Q maps K into K. Next, we show that Q is weakly-weakly sequentially continuous.
For each uniformly (mathit {ACG}_) function (x Irightarrow E), the functions (k_{1} t,cdot g(cdot,x cdot))), (k_{2} t,cdot)h cdot,x cdot ))), (f cdot,x cdot),T x)(cdot),S x)(cdot))) are HKP integrable, f, g, h are weakly-weakly continuous functions and (int _{0}^{t}g s,x s)),ds), (int_{0}^{1}h(s,x s)),ds) are bounded on I.
These places have in common an opportunity-rich supply of "weakly linked Republicans, weakly linked Democrats and independent voters whose allegiances are up for grabs," said Karl Rove, the Bush campaign's chief strategist.
When comparing the return on assets of the three most sharia-compliant economies, for example, we found that they were strongly uncorrelated with, weakly uncorrelated with and weakly correlated with the Dow Jones Industrial Average over the past five years.
