Evolutionary Cooperative Spectrum Sensing Game - Signals and ...

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Evolutionary Cooperative Spectrum Sensing Game - Signals and ...

900 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 3, MARCH 2010

When x → 0, usingL’Hôpital’s rule, we know from (42)

that lim x→0 Φ(x) = (1 − τ)U 0 > 0. When x → 1,

lim x→1 Φ(x) =− τ K

< 0. SinceΦ(0) > 0, Φ(1) < 0, and

Φ(x) is a continuous function of x in (0, 1), thenΦ(x) must

have at least one intersection with the x-axis, i.e., ∃˜x, such

that Φ(˜x) =0. If there is only one such ˜x, then we can infer

that Φ(x) > 0 when x˜x. Since

Φ(x) has the same sign as f(x) when 0

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