# Unique solution for 2$\times$2 Fredholm integral equations system

Consider the following system of Fredholm integral equations with constant kernel matrix $$f(x)=K(x)\int_{0}^{1}f(s)ds$$ where $$K(x)\in C([0,1];M_{2\times 2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )).$$ For the scalar case, it is known by the Fredholm alternative that the above equation has a unique solution equals to zero if and only if $$K(x)$$ is different from $$1$$ for any $$x\in [0,1]$$.

My question is: What happens in the system case?. How I can check the spectrum condtion?. Thank you in advance.

• Your claim for the scalar case is not correct. In the scalar case, $1$ is an eigenvalue of your operator on the right if and only if $K$ has integral $1$. (By the way, this has nothing to do with the Fredholm alternative.) Mar 30, 2021 at 20:28
• Thank you sir I'm very thankful. Mar 30, 2021 at 22:49

The operator $$f\to K(x) \int_0^1 f(s) ds$$ is a compact operator from $$C([0,1];\mathbb R ^n)$$ equipped with the sub norm into itself, if $$K$$ is an $$n\times n$$ matrix valued function for any finite $$n\geq1$$, simply because bounded intervals in $$\mathbb R$$ are compact. An even simpler argument is that the range is finite dimensional.
So indeed, $$1$$ has finite multiplicity (at most $$n$$).
Now integrating over $$(0,1)$$, you find that if $$1$$ is an eigenvalue for $$K$$ with eigenvector $$f$$ then $$1$$ is an eigenvalue of $$M=\int_0^1 K(x) dx \in \mathbb R ^{n\times n}$$ with eigenvector $$\int_0^1 f(x) dx$$, and the multiplicity of $$1$$ for $$K$$ is smaller or equal to that of $$1$$ for $$M$$. If $$L$$ is an eigenvector for $$M$$ with eigenvalue $$1$$, then $$KL$$ is the corresponding eigenvector for $$K$$. So, just like in the one dimensional case, all is decided by the constant matrix $$M$$.
• Thank you sir for the answer. Is there any changes if I work in $L^2$ instead of the space of continuous functions?. Mar 30, 2021 at 22:18
• No difference. $L^1$ works as well. It is a finite dimensional problem.. Mar 31, 2021 at 5:21