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Is this operator steady?
Let $I=[0,1]$ and $E$ a Banach house. We level to by $X:=mathcal {C}(I,E), $ the house of all steady features from $I$ to $E$, with $left  x capable _X=sup_{tin I }left  x(t) capable _E
$.
Let $f:Itimes Erightarrow E$ an obligation such that:

For every steady $xin X$, we maintain $f(.,x(.))$ is Pettis
integrable on $I$, 
for each $t in I,:: f_t: E rightarrow E,:u mapsto f_t(u):=f(t,u) textual content{ is steady}.$
Let $$T: X rightarrow X,:x mapsto T(x)(t):=int_{0}^{t}f(s,x(s)) ds$$
pretense: $T$ is steady.
This is how I attempted to decipher this:
For $tin I,:f_t$ is steady, that’s,
for every $uin E$, $forall epsilon>0 , exists eta_{t,u,epsilon}>0 textual content{ such that } forall vin E$ $$left uv capable  leq eta_{t,u,epsilon} Rightarrow left  f(t,u)f(t,v) capable  < epsilon
$$
Now, let $tin I$, $epsilon >0$ , and $xin X$. Let $yin X$ such that $$left  xy capable _Xleq eta_{t,x(t),epsilon};,$$
i.e. $$forall error I,:left  x(s)y(s) capable _{Etimes E}leq eta_{t,x(t),epsilon};,$$
specifically, $$left  x(t)y(t) capable _{Etimes E}leq eta_{t,x(t),epsilon};.$$
Hence, $$left  f(t,x(t))f(t,y(t)) capable  < epsilon quad(*)
$$
So, $$commence{matrix}
left  T(x)(t)T(y)(t) capable  & = &left  int_{0}^{t} f(s,x(s))f(s,y(s)) dsproper 
& leq & int_{0}^{t} left  f(s,x(s))f(s,y(s)) dsproper  quad(**)
stop{matrix}$$
sadly, I can not make use of $(*)$ in $(**)$ as a result of it $(*)$ not uniformaly on $t$.
Is our pretense correct? why?
If not, what’s the situation on $f_t$ that you just indicate as an alternative of continuity?
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