New existence theorems for Lyapunov-Schmidt integral equations (Q2388012)

The authors discuss solvability of nonlinear Lyapunov-Schmidt integral equations of the form \[ x(t)=f(t)+\sum_{\alpha_1,\dots,\alpha_\lambda=1}^\infty \int_\Omega \dots \int_\Omega k_{\alpha_1,\dots,\alpha_\lambda}(t, s_1, \dots, s_\lambda)x^{\alpha_1}(s_1)\dots x^{\alpha_\lambda}(s_\lambda) ds_1 \dots ds_\lambda, \tag{\(*\)} \] where \(\Omega\) is a set equipped with a \(\sigma\)-algebra \(A\) of measurable subsets and with a completely additive \(\sigma\)-finite measure \(\mu\) on \(A\); \(k_{\alpha_1,\dots,\alpha_\lambda} (t,s_1,\dots,s_\lambda)\) is a given family of jointly measurable functions and \(f(t)\) is a given measurable function. Writing (\(*\)) in the operator form \[ x=Ax,\tag{\(**\)} \] where \(Ax=\sum_{\alpha_1,\dots,\alpha_\lambda=1}^\infty A_{\alpha_1,\dots,\alpha_\lambda} x\), \(A_0=f(t)\) and \[ A_{\alpha_1,\dots,\alpha_\lambda} x(t)=\int_\Omega \dots \int_\Omega k_{\alpha_1,\dots,\alpha_\lambda}(t, s_1, \dots, s_\lambda)x^{\alpha_1}(s_1)\dots x^{\alpha_\lambda}(s_\lambda) ds_1 \dots ds_\lambda, \tag{\(***\)} \] they first study (\(**\)) and (\(***\)) in the general form. As an application, they establish the solvability of (\(*\)) in \(L_p\), \(1\leq p\leq\infty\), on the basis of the method of successive approximation and some topological methods. The present results generalize earlier related results in many ways. Reviewer's remark: This paper provides an almost complete answer to the problem.