Fredholm-Volterra integral equation in contact problem.
The author considers the Fredholm-Volterra integral equation \[ kP(x,y,t)+q\int\limits_0^\infty\int\limits_0^\infty \frac{P(\xi,\eta,t)\,d\xi\,d\eta}{\sqrt{(x-\xi)^2+(y-\eta)^2}} +q\int\limits_0^t F(t,\tau)P(x,y,\tau) \,d\tau=f(x,y,t) \tag{1} \] in the space \(L_2(\Omega)\times C(0,T)\), under the condition \[ \int\limits_0^\infty\int\limits_0^\infty P(x,y,t)\,dx\,dy=P(t). \] Using the partial Fourier transform \[ G_\alpha(x,t)=\int\limits_{-\infty}^\infty G(x,y,t)e^{i\alpha y}\,dy \] the authors claim that equation (1) reduces to the equation \[ kP_\alpha(x,y,t)+2q\int\limits_0^\infty\int\limits_0^\infty K_0(\alpha(x-\xi))P_\alpha(\xi,t)\,d\xi +q\int\limits_0^t F(t,\tau)P_\alpha(x,\tau) \,d\tau=F_\alpha(x,t). \tag{2} \] Equation (2) is treated with series expansion and reduction to an algebraic system. Reviewer's remark: In equation (1) integration is meant, perhaps, over the half plane \(\Omega=(-\infty,\infty)\times [0,\infty)\) and not over the quarter plane \(\Omega= [0,\infty)\times [0,\infty)\) because in the latter case, after the Fourier transform, we should deal with analytic functions. The main auxiliary result, Theorem 1 on page 204, is incorrect: a convolution operator with a kernel \(k_0\in L_1(-\infty,\infty)\) is compact in \(L_2(0,\infty)\) if and only if \(k_0=0\).
scientific article; zbMATH DE number 1977997