Step 4. We prove (w^) and (v^) are two positive solutions of the problem (1.1).
By similar argument to (3.18), we can deduce that u2 and u3 are two positive solutions.
Hence, x 1 ( t ) and x 2 ( t ) are two positive solutions for the BVP (1.1 - 1.2 1.1 - 1.2
Hence, we have shown that (w^), (v^) are two positive solutions of problem (1.1).
As in the proof of Theorem 3.1, u 1 ∗, u 2 ∗ are two positive solutions for λ = 0.
It is easy to see that y 1, ϵ 0 ( t ) and y 2, ϵ 0 ( t ) are two positive solutions for BVP (1.1 - 1.2 1.1 - 1.2
Theorem 2 Assume that y 1 and y 2 are two positive solutions of (1) on [ t 0 − r, ∞ ) satisfying (2). Then every solution y of (1) on [ t 0 − r, ∞ ) is represented by formula (3), where t ∈ [ t 0 − r, ∞ ) and a coefficient K ∈ R depends on y. This is the reason for introducing the following definition. Definition 1 [2].
By Theorem 1.1, from (3.12), (3.13), and (3.15) we conclude that Φ has two fixed points (u^in Kcap overline{Omega}_{6}backslash Omega _{7})) and (u^in Kcap overline{Omega}_{7}backslashOmega_{5})), and (u^) and (u^) are two positive solutions of the problem (1.1).
By Theorem 1.1, we conclude that Φ has two fixed points u * * ∈ K ∩ ( Ω ̄ 6 Ω 7 ) and u * * * ∈ K ∩ ( Ω ̄ 7 Ω 5 ), and u** and u*** are two positive solution of the problem (1.1).
For with there exist two positive solutions.
