The existence of a fixed point for the sum of two monotone operators
The author gives sufficient conditions for the existence of an attractive fixed point of the sum \(T=A+H\) of two monotone operators in a cone in a Banach space. \(A\) is assumed to be \(\alpha\)-concave (\(A(tx)\geq t^\alpha Ax\) for all \(t\in (\eta,1)\), for some \(\alpha,\eta\in (0,1)\)), but \(H\) is only subhomogeneous (\(H(tx)\geq tHx\) for all \(t\in (\eta,1)\)). It is known that a subhomogeneous operator can have no fixed points. \textit{R.\,D.\thinspace Nussbaum} in [``Hilbert's projective metric and iterated nonlinear maps'' (Mem.\ Am.\ Math.\ Soc.\ 391) (1988; Zbl 0666.47028)] presented a sufficient condition for the existence of a solution of the problem in terms of eigenvectors. In the present paper, the following condition is assumed: there is an \(x_0\in P\) such that \(Ax_0\in C_{x_0}\) and \(Hx_0\in C_{x_0}\), and \(\lambda_m(t)=\sup_{x\in C_{x_0}}\lambda(t,x)<1\), where \(C_{x_0}\) is a component in \(P\) containing \(x_0\), and \(\lambda(t,x):(\eta,1)\times C_{x_0}\to (\alpha,1)\) is a function satisfying \[ T(tx)\geq t^{\lambda(t,x)}Tx \text{ for all } t\in (\eta,1). \] In the proof of the main Theorem 3.1, the author checks that the map \(\lambda(t,x)\) is well defined for every \((t,x)\). Under the above assumptions, the existence of a fixed point \(x_*=Tx_*\) is proved with \(T^nx_0\to x_*\). A more verifiable condition than the above one is given in Theorem 3.4. Two applications to difference equations and integral equations, respectively, are presented as illustrations in the last section.
scientific article; zbMATH DE number 5534411