Integral equations in measure spaces
Let \((X,A,\mu_i)\) be measure spaces and let \(s: X\to A\) be a function such that (i) \(x\not\in s(x)\) for \(x\in X\), (ii) if \(y\in s(x)\) then \(s(y)\subset s(x)\), (iii) the set \(\{(x_1, x_2)\in X^2: x_2\in s(x_1)\}\) is \(\mu_i\times \mu_j\) measurable for \(i,j= 1,2,\dots, n\). The following integral equation is considered \[ y(x)= f(x)+ \sum^n_{i=1} g_i(x) \int_{s(x)} h_i\circ yd\mu_i,\quad\text{for }x\in X,\tag{1} \] where \(f\) and \(g_i\) \((i= 1,2,\dots,n)\) are real-valued functions on \(X\) and \(h_i\) is a function from a subset \(I_i\) of \(\mathbb{R}\) into \(\mathbb{R}\) for \(i= 1,2,\dots, n\). A number of necessary and sufficient conditions is given which together with the Lipschitz condition imply the existence and uniqueness of solution of the equation (1). Some applications are also derived.
scientific article; zbMATH DE number 1925517