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On the solvability of nonlinear noncompact operator equations. (English) Zbl 0623.47072
Let E and F be real Banach spaces and $$\Omega$$ be an open bounded subset of E, and p be a vector in F. A continuous mapping f:$${\bar \Omega}\to F$$ is called (p,k)-epi if f(x)$$\neq p$$ for any x in $$\partial \Omega$$ and the equation $$f(x)=h(x)+p$$ has a solution in $$\Omega$$ whenever h is a k-set contraction from $${\bar \Omega}$$ into F and $$h(x)=0$$ on $$\partial \Omega$$. The authors studied the properties of such mappings.
If f is a (p,k)-epi then it is p-epi in the sense of [M. Furi, M. Martelli, A. Vignoli, Ann. Mat. Pura Appl., IV. Ser. 124, 321-343 (1980; Zbl 0456.47051)]. However, by restricting f to be a (p,k)-epi mapping the authors can solve the equation $$f(x)=h(x)+p$$ for more general mappings h. Some applications in ordinary and partial differential equations were given.
Reviewer: Duong Minh Duc

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J05 Equations involving nonlinear operators (general) 35L15 Initial value problems for second-order hyperbolic equations 34K10 Boundary value problems for functional-differential equations