**§ ****5 ****Nonlinear integral equations**

[ Integral and Linear Operators ] Consider the expression

_{}

For a given kernel *K* ( *x* , *ξ* ), each function *f* ( *x* ) has another function *F* ( *x* ) corresponding to it. This correspondence is called an integral operator, denoted as *K* , namely

*F=Kf*

The set of those functions *f* such that the function *F* = *Kf* exists is called the domain of operator *K.*

If operator *K* satisfies the condition

_{}_{( a}_{ is a constant)}

*Then K* is called a linear operator.

[ Bounded operator and its norm ] If there is a constant *M* , for all functions *f*

_{}

*Then K* is called a bounded operator, where it represents the norm ( modulus ) of the function *f* . *The largest lower bound of all M* that makes the above inequality true is called the norm of operator *K* , denoted as , it can also be defined as_{}_{}

_{}

Bounded operators have the following properties:

1° If *K *_{1} and *K *_{2} are bounded operators, then *K *_{1 }*K *_{2} is also a bounded operator.

2° If the function *K* ( *x* , *ξ* ) is continuous for all *x* , *ξ* , on the finite square *k _{0}* (

_{}

The defined operator *K* is a bounded operator.

3° If on the infinite interval [ *a, b* ] , the function *K* ( *x, **ξ* ) satisfies

_{}

then by

_{}

The defined operator *K* is a bounded operator.

[ Theorem of existence of solutions to nonlinear integral equations ] Consider an integral equation of the form

_{} _{ } (1)

The methods of solving linear integral equations in the previous sections are not applicable to nonlinear integral equations. The existence theorems of just a few solutions are listed below.

Theorem 1 assumes that *K* ( *x* , *ξ* ) is continuous for all *x, **ξ* on the unit square *k ** _{0}* (0 ≤

_{} ( *A* is a constant )

It is also assumed that the Lipschitz condition is satisfied_{}

_{}

where *B* is a constant independent of *ξ . *At that time , the integral equation (1) has a unique solution in *L ** _{2}* [0,1]

Theorem 2 assumes that *K* ( *x, **ξ* ) is continuous with respect to all *x, **ξ* on the unit square *k *_{0 , let}_{}* *

_{}
( *C* is a constant)

_{}satisfy

_{}
( *B* is a constant)

and for any *ε* > 0 , there is *δ* = *δ* ( *ε* ) such that

_{} ( at the time )_{}

in the formula . Then at that time , integral equation (1) has at least one solution in *L *_{2} [0,1] * ._{}_{}_{}

Theorem 3 assumes that *K* ( *x* , *ξ* ) and are both continuous functions of their independent variables, let *S* be *L *_{2} [0,1] satisfying _{}_{}

_{} ( *M* is a constant)

the whole of the function . assumed_{}

_{} ( *C* is a constant)

_{}(everything )_{}

And for any *ε* > 0 , there exists *δ* = *δ* ( *ε* ), such that

_{} ( at the time )_{}

At that time , the integral equation (1) has at least one solution in *S.*_{}

The condition of this theorem requires that *K* ( *x , **ξ* ) be continuous, and in fact it can be shown that the same result is obtained as long as the kernel *K* ( *x* , *ξ* ) is square-integrable.

Theorem 4 assumes that the conditions stated in Theorem 3 are satisfied, and let *K* ( *x, **ξ* ) satisfy _{}

_{}

At that time , the integral equation (1) has at least one solution in *S.*_{}

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