**1. Introduction**

When investigating the nonlinear equations of the form

$$A(\mathcal{Q}\_{\prime\prime}f) = f\_{\prime\prime}$$

where the operator *A*(,) *f* nonlinearly depends both on the parameter and the function *f* , the formalistic approach, which is based on linearization, is applied. The application of this approach shows, that the branching points of equation can be only those values of parameter , for which unit ( 1 ) is the eigenvalue of the corresponding linearized equation (see, eg, [20])

$$A(\mathcal{A})f = f$$

with the operator-valued function *A* : () *C XH* ( *X H*( ) is a set of linear operators, *C* is the spectral parameter), nonlinearly depending on the parameter . If the linearized equation linearly depends on the parameter , i.e. *Af f* , then its eigenvalues will be the branching points of initial equation. In a general case the curves of eigenvalues ( ) appears and then the branching points will be those values of parameter of the problem

$$A(\mathcal{X})f = \nu(\mathcal{X})f\_{\mathcal{X}}$$

for which ( ) 1.

The theory of branching solutions of nonlinear equations arose in close connection with applied problems and development of its ever-regulated by the new applied problems.

© 2012 Podlevskyi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some of these problems is reflected in the monographs [3, 4, 20], as well as in several articles (see, eg, [5, 6] and references therein)

Numerical Algorithms of Finding the Branching Lines

and Bifurcation Points of Solutions for One Class of Nonlinear Integral Equations 283

112 2

(2)

 

2

*R*

\

   

( )*I* minimum, we obtain a nonlinear

11 2 2

1 2

 

( )

*icn cm*

 

12 12 12 12 12

1 2

*M Mn*

*n Mm M n*

( )

( ) exp arg ,

 

*nm*

*i I e dd*

2 2

(3)

. Denote by *R*2 the region that corresponds to one

*c* for the variable 1

in the best way. To this

and

 

( )

*icn cm*

and denote by 1 *d* and 2 *d* , respectively, the distance between adjacent radiators along the

<sup>1</sup> , *<sup>n</sup> x dn* <sup>2</sup> , *my d m*

( )

1 12 2 *c kd c kd* , .

is periodic with a period 1 2 /

We must find such currents *nm I* on radiators that created by them directivity pattern will

end, we consider the variational statement of the problem as, for example, in [2] or [18].

Thus, the synthesis problem we formulate as a problem of minimizing the functional [18]

*I F f dd f dd*

 

( ) min , , *I nm I I H*

which characterizes the magnitude of mean-square deviation of modules of the given

*I IH*

() ( , ) ( , ) (,)

 

( ) (,) , *M Mn*

*nm*

and assume that the required amplitude directivity

is given in some region *<sup>R</sup>*2 and is described by the function that is

1 2

*n Mm M n f I e*

1 2

axes *Ox* and *Oy* . Then the coordinates of the radiators are calculated as

and the plane array factor (1) can be represented as

where

Note that the function 1 2 *f*(,)

period *R cc* 2 1 12 2 : /, / 

on the space *<sup>N</sup> H C <sup>I</sup>* , i.e.

with a period 2 2 /

pattern 1 2 *F*(,) 

1 2

 

 

 

*c* for the variable 2

continuous and nonnegative in and is equal to zero outside.

**2.1. Variational statement of the synthesis problem** 

directivity pattern and the synthesized one in the region .

system of equations for the optimum currents on radiators

*icn cm*

1 2

 

From the necessary condition of the functional

11 2 2 1 2 ( )

<sup>2</sup> 1 2 (,) (2 )

 

*c c I Fe*

*nm*

approach by the amplitude to the given directivity pattern 1 2 *F*(,)

Application of the cited above approach to the nonlinear integral operator arising at synthesis of the antenna systems according to the given amplitude directivity pattern, brings to the nonlinear two-parameter eigenvalue problem

$$T(\mathcal{A}, \mu)f = f$$

with an integral operator *T*(,) analytically depending on two spectral parameters and .

The essential difference of the two-parameter problems from the one-parameter ones is that the two-parameter problem can not have at all the solutions or, on the contrary, to have them as a continuum set, which in the case of real parameters are the curves of eigenvalues.

Such problems are still not investigated because there are still many open questions connected with this problem such as, for example, the existence of solutions and their number, and also the development of numerical methods of solving such spectral problems for algebraic, differential and integral equations.

In the given work an algorithm of finding the branching lines of the integral equation arising in the variational statement of the synthesis problem of antenna array according to the given amplitude directivity pattern as, for example, in [2] is proposed.
