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- W2023331372 abstract "Abstract We give several general implicit function and closed graph theorems for ser-valued functions. Let Z be a normed space, X, Y metric spaces with X complete. Let f: X ⇉ Z , F: X × Y ⇉ Z be multifunctions with z 0 ∈ f(x 0 ) ∩ F(x 0 , y 0 ) such that f is open at ( x 0 , y 0 ) and f ‘approximates’ F in an appropriate sense. Suppose that f −1 ( z ) is closed, F(x, y) is compact for each x, y and z and suppose that F ( x 0 , ·) is lower semi-continuous at y 0 . Then F (·, y ) is of closed graph ‘locally’, is open at x 0 , and there exists a function x (·) with x(y) → x 0 for y → y 0 such that z 0 ∈ F(x(y), (y)) for all y near y 0 . A more general form dealing with the non-linear rate situation is also established." @default.
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- W2023331372 date "1996-08-01" @default.
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- W2023331372 title "On closed graph and implicit function theorems for multifunctions" @default.
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- W2023331372 doi "https://doi.org/10.1017/s1446788700000124" @default.
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