Corrington's Walsh function method applied to a nonlinear integral equation
The authors consider the integral equation \[ u(x) = \sum^ N_{i = 1} b_ i(a_ i(x) + \int^ x_ 0 k_ i(x,t) u(t) dt)^ i\tag{*} \] for \(x \overline{\in} I := [0,1]\), where \(b_ i \overline {\in} \mathbb{R}\), \(a_ i \overline{\in} L^ \infty(I)\) and \(k_ i \overline{\in} L^ \infty(I \times I)\). By the use of Banach's fixed point theorem the existence of a unique solution \(u \overline {\in} L^ 2\) of (*) in the ball \(B_ c := \{w \overline{\in} L^ 2 : \| w\|_ 2 \leq c\}\) with \(c\) sufficiently small is proven. The solution is approximated by finite series of Walsh-functions (special orthonormal functions). This approximated solution is obtained by projecting relation (*) onto a finite-dimensional span of Walsh-functions and applying the usual iteration method on this equation. This is the method of \textit{M. S. Corrington} [A solution of differential and integral equations with Walsh functions. IEEE Trans. Circuit Theory, CT-20, No. 5, 470-476 (1973)]. Error estimates for the approximated solution are derived on certain regularity conditions. [Remark: (1) The fixed point theorem of Banach implies existence and uniqueness of the solution. Therefore, the proof of the uniqueness in Theorem 1 is superfluous. (2) A linear Volterra integral equation with a continuous kernel has no characteristic value. Hence, the assumptions of Theorem 3 are empty, i.e., there exists no bifurcation point \((b_ 0,0)\)].
scientific article; zbMATH DE number 716120