Local expansions and accretive mappings
Let X and Y be complete metric spaces with Y metrically convex, let D⊂X be open, fix u0∈X, and let d(u)=d(u0,u) for all u∈D. Let f:X→2Y be a closed mapping which maps open subsets of D onto open sets in Y, and suppose f is locally expansive on D in the sense that there exists a continuous nonincreas...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1983-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S016117128300037X |
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| Summary: | Let X and Y be complete metric spaces with Y metrically convex, let D⊂X be open, fix u0∈X, and let d(u)=d(u0,u) for all u∈D. Let f:X→2Y be a closed mapping which maps open subsets of D onto open sets in Y, and suppose f is locally expansive on D in the sense that there exists a continuous nonincreasing function c:R+→R+ with ∫+∞c(s)ds=+∞ such that each point x∈D has a neighborhood N for which dist(f(u),f(v))≥c(max{d(u),d(v)})d(u,v) for all u,v∈N. Then, given y∈Y, it is shown that y∈f(D) iff there exists x0∈D such that for x∈X\D, dist(y,f(x0))≤dist(u,f(x)). This result is then applied to the study of existence of zeros of (set-valued) locally strongly accretive and ϕ-accretive mappings in Banach spaces |
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| ISSN: | 0161-1712 1687-0425 |