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Summable positive solutions of a Hammerstein integral equation with singular nonlinear term. (English) Zbl 0920.45005

The author studies the nonlinear Hammerstein integral equation \[ u(x)=\int_\Omega K(x,y)g \bigl(y,u(y)\bigr)dy, \tag{1} \] where \(K(x,y)\geq 0\) a.e. in \(\Omega\times\Omega\), \(\Omega\subset\mathbb{R}^n\) is measurable, \(g(y,s)\) is a positive Carathéodory function, defined on \(\Omega\times (0,\infty)\), bounded with respect to \(s\) as \(s\to\infty\). Equation (1) is considered as a limit equation of a family of equations \[ u(x)= \int_\Omega K(x,y) g_\varepsilon \bigl(y,u(y) \bigr)dy,\;\varepsilon>0,\tag{2} \] where \(g_\varepsilon\) are positive Carathéodory functions, defined on \(\Omega\times[0,\infty)\), such that \[ \lim_{\varepsilon\to 0}g_\varepsilon= g\text{ a.e. in }\Omega\times [0, \infty). \] In this paper the existence of a positive solution in \(L^1 (\Omega)\) of the equation (1) is proved. The proof is based on the convergence of the solutions of the equations (2).

MSC:

45G05 Singular nonlinear integral equations
45M20 Positive solutions of integral equations
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