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Note that the fuzzy expansive and nonexpansive mapping are fuzzy isometries.
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Indeed, we will show that under some extra conditions on Theorem 3.1, the cubic mapping is fuzzy sequentially continuous.
Moreover, the paper includes a comprehensive set of examples showing that a fuzzy ψ-p-contractive mapping is fuzzy ψ-contractive and the converse is false.
Notice that F and G are two contractive maps, so they are fuzzy contractive maps in the induced fuzzy metric space.
Let be a mapping where and are fuzzy normed spaces.
Hence there exists a mapping which is fuzzy α-ψ-contractive but not fuzzy ψ-contractive, although every fuzzy ψ-contractive mapping is obviously fuzzy α-ψ-contractive.
Hence, there exists a mapping which is fuzzy ψ-p-contractive but not fuzzy ψ-contractive.
But our saliency algorithm based on background prior has a poor effect when the boundaries are fuzzy between the foreground and the background, so it is difficult to obtain some clear salient map if the boundaries are fuzzy, which leads to failure of subsequent object detection.
Suppose that conditions of Theorem 4.1 hold, and the mapping are Lipschitzian continuous fuzzy mappings with Lipschitzian constant and, respectively.
However, every fuzzy ψ-contractive mapping is obviously fuzzy ψ-p-contractive.
Since every set-valued mapping is the fuzzy mapping, hence, all results obtained in this paper are still hold for any set-valued mappings.
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