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Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

The above problem was ill-defined, as the sequence $F(u_n)$ may not belong to $L^2(0,T;X)$ (see this example). Consider the same problem with added uniformly boundness, and continuity assumption:

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are uniformly bounded, and continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

The above problem was ill-defined, as the sequence $F(u_n)$ may not belong to $L^2(0,T;X)$. Consider the same problem with added uniformly boundness, and continuity assumption:

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are uniformly bounded, and continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

The above problem was ill-defined, as the sequence $F(u_n)$ may not belong to $L^2(0,T;X)$ (see this example). Consider the same problem with added uniformly boundness, and continuity assumption:

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are uniformly bounded, and continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

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Saj_Eda
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Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

The above problem was ill-defined, as the sequence $F(u_n)$ may not belong to $L^2(0,T;X)$. Consider the same problem with added uniformly boundness, and continuity assumption:

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are continuousuniformly bounded, and continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly uniformly continuous on $X$, and $u_n$ are continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

The above problem was ill-defined, as the sequence $F(u_n)$ may not belong to $L^2(0,T;X)$. Consider the same problem with added uniformly boundness, and continuity assumption:

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are uniformly bounded, and continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

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Saj_Eda
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Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly uniformly continuous on $X$, and $u_n$ are continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly uniformly continuous on $X$, and $u_n$ are continuous functions over $[0,T]$ such that $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, can we say $$F(u_n)\to F(u) \quad \text{weakly in} \quad L^2(0,T;X)$$

Note that we know
$$F(u_n(t))\to F(u(t)) \quad \text{weakly in }X\text{ for every }t\in[0,T]$$

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