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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
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