Acta Mathematica Vietnamica

For $0\leq x \leq B$, $0 < \beta <1$, we consider the integral equation $$\int_0^x (x-t)^{-\beta}K(x, t, y(y))dt=f(x)$$ under appropriate Lipschitz-like conditions on the function $K$ and some of its derivatives, the most essential condition being $$K_u(x,t,u)\geq c > 0\text{ for } 0\leq t\leq x\leq B, ~ u\in\mathbb R.$$ After a survey on theorems of existence, uniqueness and stability of the solution we generalize a numerical method, proposed and investigated 1976 by H. W. Branca for the particular case $\beta=1/2$, to all $\beta\in(0,1)$ and show it to be $O(h^2)$ convergent for all $\beta\in[0.2118, 1)$ if the solution $y$ is sufficiently smooth. The method is based on piecewise linear interpolation, one-point weighted Gauss quadrature on parition intervals of equal lenght $h$, and collocation.

Dedicated to the memory of Professor Dr. Lother Collatz

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