§ 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 ( a ≤ x ≤ b , a ≤ ξ ≤ b ) , then by

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 ≤ x ≤ 1,0 ≤ ξ ≤ 1) , let ô K ( x, ξ ) ô ≤ C ( C ⁽ C is a constant ⁾ , for all ξ on the unit square k ₀ , t is also continuous, and 

        ( 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.