Editing Integral equation (section)

=== Uniqueness and existence theorems in 1D ===
The solution to a linear Volterra integral equation of the first kind, given by the equation:<math display="block">(\mathcal{V}y)(t)=g(t)</math>can be described by the following uniqueness and existence theorem.<ref name=":2" /> Recall that the Volterra integral operator <math>\mathcal{V} : C(I) \to C(I)</math>, can be defined as follows:<ref name=":2" /><math display="block">(\mathcal{V} \varphi)(t) := \int_{t_0}^t K(t,s) \, \varphi(s) \, ds</math>where <math>t \in I = [t_0 , T]</math> and ''K''(''t'',''s'') is called the kernel and must be continuous on the interval <math>D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}</math>.<ref name=":2" /> 
{{Math theorem
| math_statement = Assume that <math> K </math> satisfies <math> K \in C(D), \, \partial K / \partial t \in C(D) </math> and <math> \vert K(t,t) \vert  \geq k_0 > 0 </math> for some <math> t \in I. </math> Then for any <math> g\in C^1(I) </math> with <math> g(0)=0 </math> the integral equation above has a unique solution in <math> y \in C(I)</math>.
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The solution to a linear Volterra integral equation of the second kind, given by the equation:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{V} y)(t)</math>can be described by the following uniqueness and existence theorem.<ref name=":2" /> 
{{Math theorem
| math_statement = Let <math> K \in C(D) </math> and let <math>R </math> denote the resolvent Kernel associated with <math> K </math>. Then, for any <math>g \in C(I) </math>, the second-kind Volterra integral equation has a unique solution  <math>y \in C(I) </math> and this solution is given by: <math>y(t)=g(t)+\int_0^t R(t,s) \, g(s) \, ds.</math>
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