Non-linear integral equation

An integral equation containing the unknown function non-linearly. Below the basic classes of non-linear integral equations that occur frequently in the study of various applied problems are quoted; their theory is, to a certain extent, fairly well developed.

An important example is the Urysohn equation

$$\tag{1 } \phi ( x) = \lambda \int\limits _ \Omega K [ x , s , \phi ( s) ] ds ,\ \ x \in \Omega ,$$

where $\Omega$ is a closed bounded set in a finite-dimensional Euclidean space, $K [ x , s , t ]$ is a given function, the so-called kernel, which is defined for $x , s \in \Omega$, $- \infty < t < \infty$, $\lambda$ is a numerical parameter, and $\phi$ is the unknown function.

P.S. Urysohn (see [2]) made, under certain assumptions, a complete investigation of the spectrum of the eigen values of equations (1) admitting positive eigen functions. He showed that positive eigen functions $\phi ( x , \lambda )$ correspond to values $\lambda$ only in a certain interval $( \alpha , \beta )$ and that $\phi ( x , \lambda )$ is a monotone increasing function of $\lambda$ with $\phi ( x , \alpha ) = 0$ and $\phi ( x , \beta ) = \infty$.

A special case of an Urysohn equation is the Hammerstein equation

$$\tag{2 } \phi ( x) = \lambda \int\limits _ \Omega K ( x , s ) f [ s , \phi ( s) ] ds ,\ \ x \in \Omega ,$$

where $K ( x , s )$ and $f ( s , t )$ are known functions. Existence and uniqueness theorems were first established by A. Hammerstein (see [9]). He investigated equations (2) under the assumption that the real-valued function $f ( s , t )$ is jointly continuous in its arguments and that the linear integral operator generated by the kernel $K$ is self-adjoint in $L _ {2} ( \Omega )$, positive, and acts compactly from $L _ {2} ( \Omega )$ into the space of continuous functions.

Another example of a non-linear integral equation is the Lyapunov–Schmidt equation

$$\tag{3 } \sum _ {\alpha , \beta } \ \int\limits _ \Omega \dots \int\limits _ \Omega K _ {\alpha , \beta } ( x , s _ {1} \dots s _ {i} ) \times$$

$$\times \phi ^ {\alpha _ {0} } ( x ) \phi ^ {\alpha _ {1} } ( s _ {1} ) \dots \phi ^ {\alpha _ {i} } ( s _ {i} ) \times$$

$$\times v ^ {\beta _ {0} } ( x) v ^ {\beta _ {1} } ( s _ {1} ) \dots v ^ {\beta _ {i} } ( s _ {i} ) d s _ {1} \dots d s _ {i} = 0 ,\ x \in \Omega ,$$

in which $K _ {\alpha , \beta }$ and $v$ are given functions, $\phi$ is the unknown function, $i$ is fixed, and the summation is over all vectors $\alpha = ( \alpha _ {0} \dots \alpha _ {i} )$ and $\beta = ( \beta _ {0} \dots \beta _ {i} )$ with non-negative integer components. The left-hand side of (3) is called an integral power series in the two functional arguments $v$ and $\phi$.

Equations of type (3) were first considered by A.M. Lyapunov (see [1]) and later, in a more general form, by E. Schmidt (see [8]). In their research the foundations were laid of the bifurcation theory of non-linear integral equations, which aims at solving the following problem. Suppose that one searches for a solution of a non-linear problem depending on certain parameters and that for some of their values the solution may bifurcate. There arises the tasks of finding the solution itself and those parameter values for which it bifurcates (branches), the number of branches, and the representation of each branch as a function of the parameters (see [6]).

The theory of non-linear integral equations is part of the general theory of non-linear operator equations. Namely, integral equations can be regarded as specific illustrations of the corresponding operator equations. For this purpose one has to clarify general properties (continuity, compactness, etc.) of the concrete integral operators occurring in the equation.

References

 [1] A.M. Lyapunov, Zap. Akad. Nauk. St. Petersburg (1906) pp. 1–225 [2] P.S. Urysohn, Mat. Sb. , 31 (1923) pp. 236–255 [3] M.M. Vainberg, "Variational methods for the study of nonlinear operators" , Holden-Day (1964) (Translated from Russian) [4] M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian) [5] M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) [6] M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian) [7] M.M. Vainberg, "Variational methods and methods of nonlinear operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian) [8] E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen III" Math. Ann. , 65 (1908) pp. 370–399 [9] A. Hammerstein, "Nichtlineare Integralgleichungen nebst Anwendungen" Acta Math. , 54 (1930) pp. 117–176