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Multiple solutions of condensing increasing operator equations. (Chinese. English summary) Zbl 0679.47030

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

Citations:

Zbl 0287.47037
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