Let X be a metric space. A self-map on X is said to be a...

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Let X be a metric space. A self-map Ф on X is said to be a pseudo-contraction if d(P(x), Þ(y)) ≤d(x, y) holds for all distinct x, y € X Show that if (m (x)) has a convergent subsequence, then d(x*, Þ(x)) = d($(x*), ²(x)) for some x* € X We can use the property: if DEXX is a pseudo-contraction, then (d(m+1(a), (x))) is a decreasing sequence, and hence converges. Let X be a metric space. A self-map Ф on X is said to be a pseudo-contraction if d(P(x), Þ(y)) ≤d(x, y) holds for all distinct x, y € X Show that if (m (x)) has a convergent subsequence, then d(x*, Þ(x)) = d($(x*), ²(x)) for some x* € X We can use the property: if DEXX is a pseudo-contraction, then (d(m+1(a), (x))) is a decreasing sequence, and hence converges.

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ANSWER To prove that if px has a convergent subsequence then dx x dx 2x we first need to define some ... View the full answer