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# Green’s function

# Some general preliminary considerations

Let $(\Omega,\mu)$ be a bounded measure space and $\mathcal{F}(\Omega)$ be a linear function space of bounded functions defined on $\Omega$, i.e. $\mathcal{F}(\Omega)\subset\mathcal{L}^{\infty}(\Omega)$. We would like to note two types of functionals from the dual space $(\mathcal{F}(\Omega))^{*}$, which will be used here:

1. 2. For each $x\in\Omega$, we will consider a functional $\delta_{x}\in(\mathcal{F}(\Omega))^{*}$ defined as follows:

$\delta_{x}(f)=f(x).$ (1) Since generally, we can not speak about values at the point for functions from $\mathcal{(}L)^{\infty}$, in the following, we assume some regularity for functions from considered spaces, so that (1) is correctly defined.

# Necessary notations and motivation

Let $(\Omega_{x},\mu_{x}),\,(\Omega_{y},\mu_{y})$ be some bounded measure spaces; $\mathcal{F}(\Omega_{x}),\mathcal{G}(\Omega_{y})$ be some linear function spaces. Let $A:\mathcal{F}(\Omega_{x})\rightarrow\mathcal{G}(\Omega_{y})$ be a linear operator which has a well-defined inverse $A^{{-1}}:\mathcal{G}(\Omega_{y})\rightarrow\mathcal{F}(\Omega_{x})$.

Consider an operator equation:

$Af=g$ | (2) |

where $f\in\mathcal{F}(\Omega_{x})$ is unknown and $g\in\mathcal{G}(\Omega_{y})$ is given. We are interested to have an integral representation for solution of (2). For this purpose we write:

$f(x)=\delta_{x}(f)=\delta_{x}(A^{{-1}}(g))=[\,(A^{{-1}})^{*}\delta_{x}\,](g).$ |

# Definition of Green’s function

If $\forall x\in\Omega_{x}$ the functional $(A^{{-1}})^{*}\delta_{x}$ is regular with generator $G(\cdot,y)\in\mathcal{L}^{1}(\Omega_{y})$, then $G$ is called Green’s function of operator $A$ and solution of (2) admits the following integral representation:

$f(x)=\int\limits_{{\Omega_{y}}}G(x,y)\,g(y)\,d\mu_{y}$ |

## Mathematics Subject Classification

35C15*no label found*

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