Nonlinearly weighted \(L^1\)-solvability of the first kind Fredholm integral equation (Q1977776)

This paper is devoted to the study of the first kind Fredholm integral equation \[ g(\nu)= \int^b_aK(\nu,x) f(x)dx, \tag{1} \] in which \(x\in I= [a,b]\), \(\nu\in J=[c,d]\), and \(J\times J=\Omega\). The author derives an explicit solution for (1) with kernels in a dual space which is a certain restriction of a nonlinearly weighted \(L^1\)-space \(L_C^1(\Omega, \gamma(K))\), via a perturbational auxiliary construction involving a generalized moment problem. It is illustrated how the auxiliary problem, which has a maximum a posteriori solution, can be utilized in a certain inverse-problem formulation to establish that some variational perturbation of the kernel of (1) in its dual space may yield for it that a closed-form primal solution \(\psi(x)\). The main result is that this analytical solution must satisfy an equivalent, well-posed, but nonlinear homogeneous functional equation of the second-kind represented by a Urysohn-type regularizing integral equation with a movable bifurcation point. The main result is utilized to prove the inclusion \(L^1 (I)\cap L^\infty (I;x^{-1}) \subset L^2(I;x^{-1})\) and to illustrate that this Urysohn integral equation is equivalent to a unique Fredholm integral equation of the second kind with a degenerate kernel and having the solution space \(L^1(I; \rho(\psi)) =L^2(I;x^{-1}) \cap L^1_C(\Omega; \gamma(K))\).