A New Extension Theorem for Concave Operators
We present a new and interesting extension theorem for concave operators as follows. Let X be a real linear space, and let (Y,K) be a real order complete PL space. Let the set A⊂X×Y be convex. Let X0 be a real linear proper subspace of X, with θ∈(AX&#x...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2009-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/2009/571546 |
| Summary: | We present a new and interesting extension theorem for concave operators as follows. Let X be a real linear space, and let (Y,K) be a real order complete PL space. Let the set A⊂X×Y be convex. Let X0 be a real linear proper subspace of X, with θ∈(AX−X0)ri, where AX={x∣(x,y)∈A for some y∈Y}. Let g0:X0→Y be a concave operator such that g0(x)≤z whenever (x,z)∈A and x∈X0. Then there exists a concave operator g:X→Y such that (i) g is an extension of g0, that is, g(x)=g0(x) for all x∈X0, and (ii) g(x)≤z whenever (x,z)∈A. |
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| ISSN: | 1687-1820 1687-1812 |