Integral equations with constraints and their solution methods (Q1364053)

Mathematical models of various physical phenomena and processes are described by integral equations, for instance \[ y(x)= f(x)+\int_\Omega F[x,t,y(t)]dt,\quad x\in\Omega\subset R^\ell,\tag{1} \] with additional information available about their solutions, in particular \[ \int_\Omega \Phi_s(t) y(t)dt=\alpha_s,\quad s=1,\dots,m,\tag{2} \] where \(f:\Omega\Rightarrow R\), \(F:\Omega\times\Omega\times R\Rightarrow R\) are given functions, \(y:\Omega\Rightarrow R\) is the sought function, and \(\Phi_s:\Omega\Rightarrow R\) and \(\alpha_s\in R\), \(s=1,\dots,m\), are a known system of linear independent functions and a known set of numbers. In general, problem (1) and (2) is inconsistent, but still it may be solvable for some initial values. We thus face the problem of establishing the conditions when equation (1) has a solution that satisfies the constraints (2) and developing methods to construct this solution.