Du, Yihong Multiple solutions of condensing increasing operator equations. (Chinese. English summary) Zbl 0679.47030 Northeast. Math. J. 3, No. 3, 299-309 (1987). Summary: Let P be a cone in a Banach space E, \(R^+=[0,+\infty)\) and f: \(R^+\times P\to P\) be an operator. With the hypotheses that f is completely continuous, strongly increasing and twice continuously right differentiable, H. Amann proved that the equation \(f(\lambda,x)=x\) has two solutions for some \(\lambda\) if an a priori bound could be found [J. Funct. Anal., 17, 174-213 (1974; Zbl 0287.47037)]. We prove by a new method that this result is also true when f is condensing, and that the differentiability assumption is quite unnecessary. We also prove that similar result holds when the existence of a priori bound is replaced by the existence of a linear minorant of f, and this result is illustrated by an example of Hammerstein integral equation of superlinear type. MSC: 47J05 Equations involving nonlinear operators (general) 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47Gxx Integral, integro-differential, and pseudodifferential operators 45E99 Singular integral equations Keywords:existence of a priori bound; existence of a linear minorant; Hammerstein integral equation of superlinear type Citations:Zbl 0287.47037 PDFBibTeX XMLCite \textit{Y. Du}, Northeast. Math. J. 3, No. 3, 299--309 (1987; Zbl 0679.47030)