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There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a contraction).

This theory uses discretization of the term $\frac{du}{dt}$ and the Crandall-Liggett theorem. The above conditions on $A$ give convergence of the discretization.

My questions:

  1. Is there an existence proof for such a theorem where time discretization is not used?

  2. Is there any some similar such existence result for equations $$F(u) + Au = f$$ where $F$ is a some operator that is not $\frac{d}{dt}$? I know this is a vague question but if I can't "put $F$ into $A$" then is there a way to solve this?

  3. Suppose that $A=A(t)$ depends on time. I have found literature (which is quite old now) that really requires the domain of $A(t)$ to be independent of $t$: $D(A(t)) = D(A(0))$ for the results to carry through. I ask if I am missing some recent work in which the conclusions of the Crandall-Liggett are valid with minor adjustments to the premises of the theorem for time-dependent operators $A(t)$ with domains not time-independent?

My first two questions stem from the fact that I don't have $\frac{d}{dt}$ but a different operator there. I don't understand fully the necessity of $\frac{d}{dt}$...

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a contraction).

This theory uses discretization of the term $\frac{du}{dt}$ and the Crandall-Liggett theorem. The above conditions on $A$ give convergence of the discretization.

My questions:

  1. Is there an existence proof for such a theorem where time discretization is not used?

  2. Is there any some similar such existence result for equations $$F(u) + Au = f$$ where $F$ is a some operator that is not $\frac{d}{dt}$? I know this is a vague question but if I can't "put $F$ into $A$" then is there a way to solve this?

My questions stem from the fact that I don't have $\frac{d}{dt}$ but a different operator there. I don't understand fully the necessity of $\frac{d}{dt}$...

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a contraction).

This theory uses discretization of the term $\frac{du}{dt}$ and the Crandall-Liggett theorem. The above conditions on $A$ give convergence of the discretization.

My questions:

  1. Is there an existence proof for such a theorem where time discretization is not used?

  2. Is there any some similar such existence result for equations $$F(u) + Au = f$$ where $F$ is a some operator that is not $\frac{d}{dt}$? I know this is a vague question but if I can't "put $F$ into $A$" then is there a way to solve this?

  3. Suppose that $A=A(t)$ depends on time. I have found literature (which is quite old now) that really requires the domain of $A(t)$ to be independent of $t$: $D(A(t)) = D(A(0))$ for the results to carry through. I ask if I am missing some recent work in which the conclusions of the Crandall-Liggett are valid with minor adjustments to the premises of the theorem for time-dependent operators $A(t)$ with domains not time-independent?

My first two questions stem from the fact that I don't have $\frac{d}{dt}$ but a different operator there. I don't understand fully the necessity of $\frac{d}{dt}$...

Source Link
TheBook
TheBook
  • 155
  • 6

Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a contraction).

This theory uses discretization of the term $\frac{du}{dt}$ and the Crandall-Liggett theorem. The above conditions on $A$ give convergence of the discretization.

My questions:

  1. Is there an existence proof for such a theorem where time discretization is not used?

  2. Is there any some similar such existence result for equations $$F(u) + Au = f$$ where $F$ is a some operator that is not $\frac{d}{dt}$? I know this is a vague question but if I can't "put $F$ into $A$" then is there a way to solve this?

My questions stem from the fact that I don't have $\frac{d}{dt}$ but a different operator there. I don't understand fully the necessity of $\frac{d}{dt}$...