Integral equations convertible to fixed point equations of order-preserving operators
The paper is devoted to the problems of the form \(\varphi=KF\varphi\) where \(F\varphi=f(y,\varphi(y))\), \(K\varphi=\int_\Omega k(x,y)\varphi(y)dy\), \(x\in\Omega\subset\mathbb{R}^n\). It is assumed that \(-1\notin\sigma(K)\) and \((I+K)^{-m}K\) is order-preserving operator for some integer positive \(m\). The operator \(B\) is said to be order-preserving if \(Bx\geq By\) for \(x\geq y\). Let also \(f_m(x,u)=f(x,u)+mu\) and \(F_m\) denotes the Nemytsky operator \((F_m\varphi)(x)=f_m(x,\varphi(x))\), \(F_0\equiv F\) and let \(X=C(\Omega,\mathbb{R})\). The authors show that equation \(\varphi=KF\varphi\) is equivalent to \(\varphi=K^0F_m\varphi+K^\infty\varphi\) where \(K^0,K^\infty:X\to X\) are linear, compact and order-preserving integral operators. Some generalizations and applications of this result are also given.
scientific article; zbMATH DE number 934745