Editing Integral equation (section)

== Hammerstein equations ==
A Hammerstein equation is a nonlinear first-kind Volterra integral equation of the form:<ref name=":2" /><math display="block">g(t) = \int_0^t K(t,s) \, G(s,y(s)) \, ds.</math>Under certain regularity conditions, the equation is equivalent to the implicit Volterra integral equation of the second-kind:<ref name=":2" /><math display="block">G(t, y(t)) = g_1(t) - \int_0^t K_1(t,s) \, G(s,y(s)) \, ds</math>where:<math display="block">g_1(t) := \frac{g'(t)}{K(t,t)} \,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\, K_1(t,s) := -\frac{1}{K(t,t)} \frac{\partial K(t,s)}{\partial t}.</math>The equation may however also be expressed in operator form which motivates the definition of the following operator called the nonlinear Volterra-Hammerstein operator:<ref name=":2" /><math display="block">(\mathcal{H}y)(t):= \int_0^t K(t,s) \, G(s, y(s)) \,ds</math>Here <math>G:I \times \mathbb{R} \to \mathbb{R}</math> is a smooth function while the kernel ''K'' may be continuous, i.e. bounded, or weakly singular.<ref name=":2" /> The corresponding second-kind Volterra integral equation called the Volterra-Hammerstein Integral Equation of the second kind, or simply Hammerstein equation for short, can be expressed as:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{H}y)(t) </math>In certain applications, the nonlinearity of the function ''G'' may be treated as being only semi-linear in the form of:<ref name=":2" /><math display="block">G(s,y) = y+ H(s,y)</math>In this case, we the following semi-linear Volterra integral equation:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{H}y)(t) = g(t) + \int_0^t K(t,s)[y(s)+H(s,y(s))] \, ds</math>In this form, we can state an existence and uniqueness theorem for the semi-linear Hammerstein integral equation.<ref name=":2" /> 
{{Math theorem
| math_statement = Suppose that the semi-linear Hammerstein equation has a unique solution <math> y\in C(I) </math> and <math> H:I\times \mathbb {R} \to \mathbb {R}</math> be a Lipschitz continuous function. Then the solution of this equation may be written in the form: <math> y(t)=y_{l}(t)+\int _{0}^{t}R(t,s)\,H(s,y(s))\,ds </math> where <math> y_{l}(t) </math> denotes the unique solution of the linear part of the equation above and is given by: <math> y_{l}(t)=g(t)+\int _{0}^{t}R(t,s)\,g(s)\,ds </math> with <math> R(t,s) </math> denoting the resolvent kernel.
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We can also write the Hammerstein equation using a different operator called the Niemytzki operator, or substitution operator, <math>\mathcal{N}</math> defined as follows:<ref name=":2" /><math display="block">(\mathcal{N} \varphi )(t) := G(t, \varphi(t))</math>More about this can be found on page 75 of this book.<ref name=":2" />