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Cone expansion and cone compression fixed point theorems for sum of two operators and their applications. (English) Zbl 1506.47096

Motivated by M. A. Krasnosel’skij’s classical fixed point theorem [Usp. Mat. Nauk 10, No. 1(63), 123–127 (1955; Zbl 0064.12002)], the authors of this interesting paper are mainly concerned with a rather detailed study of solutions to nonlinear abstract operator equations of the form \(Tx + Sx = x\), \(x \in K\), where \(K\) is a closed and convex subset of a Banach space (in particular, a closed cone), \(T\) is either a contraction or an expansion, and \(S\) is completely continuous. Then they apply their fixed point results to solving eigenvalue problems and finding positive solutions to one-parameter operator equations. Using their results, the authors are also able to establish the existence of at least one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Urysohn-Volterra type.

MSC:

47J05 Equations involving nonlinear operators (general)
47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations

Citations:

Zbl 0064.12002
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Full Text: DOI

References:

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