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Notes for the Annotated Bibliography:

Integral Equations:

There are also different types such as the one put forth by wazwaz:

Classification of Linear Integral Equations:


 * 1) Fredholm integral equations
 * 2) Volterra Integral equations
 * 3) Integro-differential equations
 * 4) Singular integral equations
 * 5) Volterra-Fredholm integral equations
 * 6) Volterrra-Fredholm integro-differential equations

Fredholm Integral Equations, I could add this to the main page about them as well as the general form isn't mentioned there: Fredholm integral equation

Fact 5: Fredholm Integral equations
A linear Fredholm integral equation of the second kind may be written as:$$y(t)=g(t)+\lambda(\mathcal{F}y)(t)$$

with the Fredholm integral operator

$$(\mathcal{F}y)(t) := \int_{t_0}^T K(t,s) \, y(s) \, d$$s

Fact 4: Linear Volterra Integral of the first kind
The linear Volterra integral operator $$\mathcal{V} : C(I) \to C(I)$$ may be written as follows:

$$(\mathcal{V} \phi)(t) := \int_{t_0}^t K(t,s) \, \phi(s) \, ds$$

where $$t \in I = [t_0, T]$$ and K(t,s) is called the kernel and must be continuous on the interval $$D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}$$

where the Volterra integral equation of the first kind may be written as:

$$(\mathcal{V}y)(t)=g(t)$$

and g(0)=0

The linear first kind has a unique solution if:

Theorem: Assume that K satisfies K \in C(D), \partial K / \partial t \in C(D) and |K(t,t)| \geq k_0 > 0 for some t \in I. Then for any g\in C^1(I) with g(0)=0 the integral equation above has a unique solution in y \in C(I)

Fact 1: Volterra Integral of Second kind
Linear Volterrra Integral equation of the second kind for the unknown function y(t) and a given continuous function g(t) on the interval I where t \in I:

$$y(t)=g(t)+(\mathcal{V} y)(t)$$

The solution to this equation is summarised by the theorem:

Let K \in C(D) and let R denote the resolvent Kernel associated with K. Ten for any g \in C(I) the second-kind VOlterra integral equation has a unique solution y \in C(I) and this solution is given by:

$$y(t)=g(t)+\int_0^t R(t,s) \, g(s) \, ds$$

\break\

some unique stuff:$$y(t)=g(t)+(V_\alpha y)(t)$$

where $$t \in I = [t_0, T]$$, the function g(t) is continuous on the interval I, and the Volterra integral operator $$(V_\alpha t)$$ is given by:

$$(V_\alpha t)(t) := \int_{t_0}^t (t-s)^{-\alpha} \cdot k(t,s,y(s)) \, ds $$with $$(0 \leq \alpha < 1)$$ and the kernel being k.

Fact 2: Volterra Integro-Differnetial Equations
An integro-differential equation of the Volterra type may be written as follows:

$$y'(t)=f(t, y(t))+(V_\alpha y)(t)$$(same info on t and V as above)

Fact 3: Delay Problems
There are things called "Delay Problems" which correspond to the following integro-differential equation:

$$y'(t)=f(t, y(t), y(\theta (t)))+(\mathcal{W}_{\theta, \alpha} y)(t)$$where the delay integral operator $$(\mathcal{W}_{\theta , \alpha} y)$$ is defined as follows:

$$(\mathcal{W}_{\theta, \alpha} y)(t) := \int_{\theta(t)}^t (t-s)^{-\alpha} \cdot k_2(t,s,y(s), y'(s)) \, ds $$

Fact 6: Hammerstein type (nonlinear)
nonlinear first-kind volterrra integral equation of Hammerstein type is written in the form:

$$g(t) = \int_0^t K(t,s) \, G(s,y(s)) \, ds$$

apparently under certain regularity conditions, the equation is equivalent to the implicit volterra integral equation of the second-kind:

$$G(t, y(t)) = g_1(t) - \int_0^t K_1(t,s) \, G(s,y(s)) \, ds$$where:

$$g_1(t) := \frac{g'(t)}{K(t,t)}$$

$$K_1(t,s) := -\frac{1}{K(t,t)} \frac{\partial K(t,s)}{\partial t}$$

Fact 7: Volterra-Hammerstein Operator
The Volterra-Hammerstein Operator may be written as:

$$(\mathcal{H}y)(t):= \int_0^t K(t,s) \, H(s, y(s)) \,ds$$

The following is a type of semi-linear Volterra Integral equation:

$$y(t)=g(t)+(\mathcal{V} y)(t)+(\mathcal{H}y)(t) = g(t) = \int_0^t K(t,s)[y(s)+H(s,y(s))] \, ds$$

Existence and Uniquness theorem: Suppose that the nonlinear integral equation above has a unique solution $$y \in C(I)$$ and $$H:I \times \mathbb{R} \to \mathbb{R}$$ be a Lipschitz continuous function. Then the solution of this eqution may be written in the form:

$$y(t)=y_l(t)+\int_0^t R(t,s) \, H(s, y(s)) \, ds$$

where $$y_l(t)$$ denotes the unique solution of the linear part of the equation above and is given by:

$$y_l(t) = g(t) + \int_0^t R(t,s) \, g(s) \, ds$$

and R(t,s) deniting the resolvent kernel.

The above stuff however is apparently some special case of the more general Volterra-hammerstein Intgeral Equation defined below:

Definition: The nonlinear Volterra-Hammerstein Operator may be written as:

$$(\mathcal{H}y)(t):= \int_0^t K(t,s) \, G(s, y(s)) \,ds$$

Here $$G:I \times \mathbb{R} \to \mathbb{R}$$ is smoth whilt the kernel K may be continous (bounded) or weakly singular. The coressponding second-kind Volterra integral equation called the Volterrra-Hammerstein IE of the second kind:

$$y(t)=g(t)+(\mathcal{H}y)(t) $$

In the special case above: the nonlinearity G is given by $$G(s,y) = y+ H(s,y)$$

Niemytzki Operator
We can rewrite Hammerstein stuff as above if we define the following operator:

Definition: Niemytzki Operator, also called the substitution operator, $$\mathcal{N}$$ defined as follows:

$$(\mathcal{N} \phi )(t) := G(t, \phi(t))$$

page 75 for more

Volterra-Fredholm Integral Equations
A VFIE has the form:

$$u(t,x) = g(t,x)+(\mathcal{T}u)(t,x)$$

with $$x \in \Omega$$ and $$\Omega$$ being a closed bounded region in $$\mathbb{R}^d$$ with piecewise smooth boundary

The Fredholm-Volterrra Integral Operator $$\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)$$ is defined as:

$$(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds$$

In the case where the Kernel K may be written as $$K(t,s,x,\xi) = k(t-s)H(x, \xi)$$, k is called the positive memory kernel.

Theorem (solution):

If

a) $$g \in C(I \times \Omega)$$

b) $$K \in C(D \times \Omega^2)$$ where $$D:= \{(t,s): 0 \leq s \leq t \leq T \}$$ and $$\Omega^2 = \Omega \times \Omega$$

then the linear VFIE with $$(t,x) \in I \times \Omega$$: $$u(t,x) = g(t,x)+\int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds$$

has a unique solution $$u \in C(I \times \Omega)$$ given by: $$u(t,x) = g(t,x)+\int_0^t \int_\Omega R(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds$$

where $$R \in C(D \times \Omega^2)$$ is the resolvent Kernel and is the limit of the Neumann series for the Kernel K and solves the resolvent equations:

$$R(t,s,x,\xi) = K(t,s,x,\xi)+\int_0^t \int_\Omega K(t,v,x,z) R(v,s,z,\xi) \, dz \, dv = K(t,s,x,\xi)+\int_0^t \int_\Omega R(t,v,x,z) K(v,s,z,\xi) \, dz \, dv  $$

Volterra integral equation in R^2
second-kind VIE with $$(x,y) \in \Omega := [0,X] \times [0,Y]$$, $$g \in C( \Omega)$$, $$K \in C(D_2)$$ where $$D_2 := \{(x, \xi,y,\eta): 0 \leq \xi \leq x \leq X, 0 \leq \eta \leq y \leq Y\}$$

$$u(t,x) = g(t,x)+\int_0^x \int_0^y K(x,\xi, y, \eta) \, u(\xi, \eta) \, d\eta \, d\xi$$This integral equation with the condiitions above has a unique solution $$u \in C( \Omega)$$ given by:

$$u(t,x) = g(t,x)+\int_0^x \int_0^{y} R(x,\xi, y, \eta) \, g(\xi, \eta) \, d\eta \, d\xi$$

Where R is the resolvent kernel of K.

Source 4: pdf
Basic definitions of operator and

The Benjamin-Ono equation: Benjamin–Ono equation

The article’s content is relevant and written in a neutral tone, but nothing is cited or sourced. There are two references that seem reliable, but lots of content is missing. Adding more commentary on the origin of the equation, the important/contexts in which it shows up, and solutions to the equation would be of importance to add.

Non-measurable sets: Non-measurable set

The main thing I can add to this page is expand on the examples of non-measurable sets and add a visualisation of non-measurable sets as in a paper I was reading.

Delta-Ring and Sigma-Ring pages: Delta-ring, Sigma-ring

The main thing I can add is theorems and ideas to these pages since they basically only give the definitions and no theorems or usage about these rings. I can also give more examples of delta and sigma rings, and their relation to delta and sigma algebras.

Fejer Kernel and Fejer Theorem: Fejér's theorem, Fejér kernel

I can add a recurisve derivation of the Fejer Kernel and showing some usages of the Fejer kernel. There also seems to be little commentary on both of these pages about the origin of these two concepts, specificially how they arise in Fourier Analysis.

Santiago-Bond: Josephine Santiago-Bond

Every sentence seems to be sourced and the sources seem to be reliable. However, the content is rather lacking despite being written in a formal tone. I've already reviewed this article and found that most of the content is also a little outdated and there seems to be more content online about her.