Editing Integral equation (section)

=== Uniqueness and existence theorems of Fredholm–Volterra equations ===
As defined above, a VFIE has the form:<math display="block">u(t,x) = g(t,x)+(\mathcal{T}u)(t,x)</math>with <math>x \in \Omega</math> and <math>\Omega</math> being a closed bounded region in <math>\mathbb{R}^d</math> with piecewise smooth boundary.<ref name=":2" /> The Fredholm–Volterrra Integral Operator <math>\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)</math> is defined as:<ref name=":2" /><math display="block">(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds.</math>In the case where the Kernel ''K'' may be written as <math>K(t,s,x,\xi) = k(t-s)H(x, \xi)</math>, ''K'' is called the positive memory kernel.<ref name=":2" /> With this in mind, we can now introduce the following theorem:<ref name=":2" />
{{Math theorem
| math_statement = If the linear VFIE given by: <math> u(t,x) = g(t,x) + \int_0^t \int_{\Omega} K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds </math>  with  <math> (t,x) \in I \times \Omega </math> satisfies the following conditions:
* <math>g \in C(I \times \Omega)</math>, and
* <math> K \in C(D \times \Omega^2) </math> where  <math>  D:= \{(t,s): 0 \leq s \leq t \leq T \}  </math>  and  <math> \Omega^2 = \Omega \times \Omega</math>

Then the VFIE has a unique solution <math> u \in C(I \times \Omega) </math> given by <math> u(t,x) = g(t,x)+\int_0^t \int_{\Omega} R(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds </math> where <math> R \in C(D \times \Omega^2) </math> is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel <math> K  </math> and solves the resolvent equations: <math> R(t,s,x,\xi) = K(t,s,x,\xi)+\int_0^t \int_\Omega K(t,v,x,z) R(v,s,z,\xi) \, dz \, dv =  K(t,s,x,\xi)+\int_0^t \int_\Omega R(t,v,x,z) K(v,s,z,\xi) \, dz \, dv  </math>
}}