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I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad.

Consider a space $X$ which we assume to be Banach. We define a linear operator $A:X\to X$ which is bounded: $ \|\mathcal A\| := \sup\limits_{x\in X}\frac{\|\mathcal Ax\|}{\|x\|}<\infty. $ I would like to discuss an existence of solution for a fixpoint equation $\mathcal Ax = x.\quad (1)$

What do I know: this is an eigenvalue problem for $\lambda = 1$ or it is a problem of finding the kernel for $(\mathcal A-\mathcal I)$, $\mathcal I$ is an identity operator. Also, if $\dim X<\infty$ then $\mathcal A$ has a correspondent finite-dimensional matrix $A$ and all properties of $(1)$ can be studied through the rang of $A$. E.g. there exists a solution of $(1)$ iff $\det (A-I)=0$ for $I$ is an identity matrix of the same size as $A$.

For the general state-space my first question is:

1.If there is a method similar to calculating $\det (A-I)$ to verify the existence of solution for $(1)$?

If the dimension of $X$ is not necessary finite, one of the main methods is to use Banach theorem based on the contraction principle - so it is only valid if $|\lambda^*(\mathcal A)<1|$ for the maximum eigenvalue in the absolute sense.

2.What can we do if the spectrum is not bounded by $1$ but just does not contain it?

There are also procedures (usually in the discrete-time setting, e.g. $X = L^2$ and $\mathcal A$ is an integral operator) connected with the iterations of operator $\mathcal A$. There are examples (if one wants, I can put it here) when for some $x\neq 0$ the limit x' = \lim\limits_{n\to\infty}\mathcal A^nx\quad (2) exists while $\mathcal A$ is not contractive.

3.Under which conditions \mathcal Ax' = x'?

Finally, there are some methods in the continuous time setting (e.g. $X = L^2$ again and we are talking about differential operators). If one would like to solve an equation $\mathcal Bx = 0$ then it's helpful to consider a function $f(t)$ such that $f(0) = x_0$ and $ \frac{df}{dt}(t) = \mathcal B f(t).\quad (3) $ Suppose, \lim\limits_{t\to\infty}f(t) = x'\in X.

4.Under which conditions \mathcal Bx' = 0?

Finally, we can put $\mathcal A = \mathcal B+\mathcal I$. Then if the conditions of 4. are satisfied, \mathcal Ax' = x'.

5. Why for the integral equations ("discrete-time") people commonly use $(2)$ while for, say PDEs they use $(3)$ rather then $(2)$?

So there are 5 questions, and I would appreciate if you can help me with answering at least one of them or referring me to a literature which covers these questions. I guess that 5. is can be an unclear question - so if it is, just tell, I will try to make it clear.

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