Page 126 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 1

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 1

Thus, = / . Substituting this into (70) implies that Since ∈ , by (55), is upper‑bounded on . This, (14)
and (15) imply that > 0 for any . Then, by (55),

( , ) = 1/( (1 + ℎ /( ))). (74)

≈ 1/ for all and small . (82)
Since ∈ , summing up (74) yields that =
∑ ∈ 1/( (1 + ℎ /( ))). This and (74) yield (31), Since ∈ , summing up (82) by ∈ implies that

while (32) follows from (31) and (72).
≈ / . (83)
9.7 Proof of Proposition 5
Then, (82) and (83) imply (38).
Since ↑ ∞ there is at least one such ↑ ∞. Then, by Substituting (83) into (53), by (81), implies that for small

(54), ↓ 0 and for large we have that
the following approximation holds:
≈ /(ℎ ). (75)

≈ /(ℎ ). (84)

By (56), for large we have that

This and the fact that ∈ implies (37).
1/ ≈ + (ℎ / ) . (76)

Thus, ↓ 0 while ↑ ∞. Moreover, by (75) and (76), 9.9 Proof of Proposition 7
/ ↓ 0 for ↑ ∞. (77) Since ∈ , ↑ ∞ implies that ↑ ∞ for at least one

. Then, by (55), ↓ 0. Thus, by (55), for large we have
Then, since √1 + ≈ 1 + /2 for small , (15) implies that
that
≈ for all . (85)

ℎ
⎢ 2 + 2 + ⎥ Taking into account that ∈ summing up (85) yields
⎢ 1 ℎ ℎ ⎥
≈ ⎢ − ⎥ that = / . Substituting this into (85) implies (40).

⎢ 2 + ℎ ⎥ While substituting = / into (14) and taking into ac‑
⎣ ⎦ + count that ↑ ∞ imply (39), and the result follows.

= − . (78)
2
( / + ℎ /( )) +

9.10 Proof of Proposition 8
By (77) and (78), we have that
First prove that cannot tend to in inity while tends to
ℎ

≈ − . (79) zero. Assume that ↑ ∞. Then, by (55), ↓ 0 where

ℎ 2 2 + > 0. Thus, by (53), ↑ ∞. So, by (14), ↓ 0 also

for = 0. This contradicts the fact that ∈ . Thus,

2
Let = / . Substituting this into (79) and taking cannot tend to in inity while ↓ 0. Then, since ↓ 0
into account that ∈ imply (33) and (35).
By (14) and (75), we have that while ↓ 0, by (55), we have that
ℎ ≈ /(ℎ ) for > 0. (86)

ℎ , ≤ ,
2

≈ ℎ (80) While, by (53), since ↓ 0, we have that,

ℎ , > 2 .
≈ /(ℎ ) for > 0. (87)
Summing up (80) and taking into account that
∈ yields the following relation: =
By (14), = /(ℎ ) for = 0. This, (87) and the

∑ 2 + √ ∑ 2 . This allows to fact that ∈ implies (41).
≤ ℎ / ℎ > ℎ / ℎ
Substituting (87) into (86) implies (42), and the result fol‑
de ine / . Substituting this / into (80) implies lows.
(34) and (36), and the result follows.
9.8 Proof of Proposition 6 9.11 Proof of Proposition 9
Note that ↓ 0 since ↓ 0. Then, by (53), By (8), since ∈ , the equilibrium strategies are de ined

by ratio / , and the result follows.
↑ ∞. (81)
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