On partial integral equations in the space of continuous functions
The partial integral equation under consideration is \[ x=Kx+f\equiv\left( L+M+N\right) x+f,\tag{1} \] where \[ \begin{aligned} (Lx)(t,s)&= \int_{a}^{b}l(t,s,\tau)x(\tau,s)\,d\tau,\\ (Mx)(t,s) &= \int_{c} ^{d}m(t,s,\sigma)x(t,\sigma)\,d\sigma,\\ (Nx)(t,s) &= \int_{a}^{b}\int_{c}^{d}n(t,s,\tau,\sigma)x(\tau,\sigma)\,d\sigma \,d\tau, \end{aligned} \] with integrals in the Lebesgue sense. For \(\Omega\) denoting one of \(\left[ a,b\right] ,\) \(\left[ c,d\right] \) or \(D=\left[ a,b\right] \times\left[ c,d\right] ,\) and \(\omega\in\Omega\), a function \(u(t,s,\omega)\)\ measurable on \(D\times\Omega\)\ is called {continuous on the whole} if, for each \(\varepsilon>0\), there exists \(\delta>0\)\ such that \(\left\| u(t,s,\cdot)-u(t_{0},s_{0},\cdot)\right\| _{L^{1}(\Omega)}<\varepsilon\)\ when \(\left| t-t_{0}\right| <\delta \)\ and \(\left| s-s_{0}\right| <\delta\); it is called {integrally bounded} if \(\left\| u(t,s,\cdot)\right\| _{L^{1}(\Omega)}\leq U<\infty \). The functions \(l,m,n\)\ are all integrally bounded and continuous on the whole. A first result states that the property of being Fredholm of index zero for equation (1) is equivalent in \(C(D)\)\ with the same property for the simpler equations \[ (I-L) x=f,\qquad (I-M) x=f.\tag{2} \] Then criteria for being Fredholm of index zero are given for equations (2). The paper ends with necessary and sufficient conditions for the unique solvability of equation (1); moreover, the authors give the form of the solution.
scientific article; zbMATH DE number 1916075