Existence of Positive Solutions for Neumann Boundary Value Problem with a Variable Coefficient

Lemma 2.1. Let (H₀) hold. Suppose φ and ψ be the solution of the linear problems

respectively. Then

• (i)

  φ(t) > 0 on [0,1], and φ^′(t) < 0 on (0,1];

• (ii)

  (ii) ψ(t) > 0 on [0,1], and ψ^′(t) > 0 on [0,1).

Proof. We will give a proof for (i) only. The proof of (ii) follows in a similar manner.

It is easy to see that the problem

has the unique solution $$ and t ∈ [0,1]. From (H₀), we know that On the other hand, for all t ∈ [0,1], we have By using comparison theorem (see [10]), we obtain Therefore, we have from (2.3) and (2.5) that Thus From the fact φ^′(0) = 0 and (2.7), we obtain φ^′(t) < 0 on (0,1].

Now, let

Theorem 2.2. Let (H₀) hold. Then for any y ∈ C[0,1], the problem

is equivalent to the integral equation

Proof. First we show that the unique solution of (2.9) can be represented by (2.10).

In fact, we know that the equation

has known two linear independent solutions φ and ψ since $$.

Now by the method of variation of constants, we can obtain that the unique solution of the problem (2.9) can be represented by

where G(t, s) is as (2.8).

Next we check that the function defined by (2.10) is a solution of (2.9).

From (2.10), we know that

So that Finally, it is easy to see that

Theorem 2.3. Let E be a Banach space, and K ⊂ E is a cone in E. Assume that Ω₁ and Ω₂ are open subsets of E with θ ∈ Ω₁ and $$. Let $$ be a completely continuous operator. In addition, suppose that either

• (i)

  ∥Tu∥ ≤ ∥u∥,   ∀ u ∈ K∩∂Ω₁ and ∥Tu∥ ≥ ∥u∥,   ∀ u ∈ K∩∂Ω₂ or

• (ii)

  ∥Tu∥ ≥ ∥u∥,   ∀ u ∈ K∩∂Ω₁ and ∥Tu∥ ≤ ∥u∥,   ∀ u ∈ K∩∂Ω₂ holds.

Then T has a fixed point in $$.

Theorem 3.1. Let (H₀) hold. Suppose that there exist a constant r > 0 such that

•  

  H₁ there exist continuous, nonnegative functions g, h, and k, such that

  $$ (3.3)

g(x) > 0 is nonincreasing, and h(x)/g(x)  is nondecreasing in x ∈ (0, r];

•  

  H₂ r − γ^*/(g(σr)(1 + h(r)/g(r))) > K^*, here $$;

•  

  H₃ there exists a continuous function ϕ_r≻0 such that

  $$ (3.4)

•  

  H₄ ϕ_r(t) + e(t)≻0 for  all  t ∈ [0,1].

Then problem (1.4) has at least one positive solution x with 0 < ∥x∥<r.

Remark 3.2. When m(t) ≡ m,   t ∈ [0,1], then (1.4) reduces to (1.1), (H₀) reduce to m ∈ (0, π/2). So Theorem 3.1 is more extensive than [5, Theorem  3.1].

Proof of Theorem 3.1. Let $$ Choose n₀ ∈ {1,2, …} such that 1/n₀ < σr₁, where 0 < r₁ < min{δ, r} is a constant. Let N₀ = {n₀ + 1, n₀ + 2, …}. Fix n ∈ N₀. Consider the boundary value problem

where We note that x is a solution of (3.1_n) if and only if Define an integral operator T_n : P → E by Then, (3.1_n) is equivalent to the fixed point equation x = T_nx. We seek a fixed point of T_n in the cone P.

Set Ω₁ = {x ∈ E∣∥x∥ < r₁}, Ω₂ = {x ∈ E∣∥x∥ < r}. If $$, then

Notice from (2.16), (H₃), and (H₄) that, for $$ on [0,1]. Also, for $$, we have so that And next, if $$, we have by (3.10), As a consequence, $$. In addition, standard arguments show that T_n is completely continuous.

If x ∈ P with ∥x∥ = r, then

and we have by (H₁), (H₂), and (H₃) Thus, ∥T_nx∥ ≤ ∥x∥. Hence,

If x ∈ P with ∥x∥ = r₁, then

and we have by (H₃) and (H₄) Thus, ∥T_nx∥ ≤ ∥x∥. Hence,

Applying (ii) of Theorem 2.3 to (3.14) and (3.17) yields that T_n has a fixed point $$, and r₁ ≤ ∥x_n∥ ≤ r. As such, x_n is a solution of (3.1_n), and

Next we prove the fact

for some constant H > 0 and for all n > n₀. To this end, integrating the first equation of (3.1_n) from 0 to 1, we obtain Then

The fact ∥x_n∥ ≤ r and (3.19) show that $$ is a bounded and equicontinuous family on [0,1]. Now the Arzela-Ascoli Theorem guarantees that $$ has a subsequence, $$, converging uniformly on [0,1] to a function x ∈ C[0,1]. From the fact ∥x_n∥ ≤ r and (3.18), x satisfies σr₁ ≤ x(t) ≤ r for all t ∈ [0,1]. Moreover, $$ satisfies the integral equation

Let k → ∞, and we arrive at where the uniform continuity of f(t, x) on [0,1] × [σr₁, r] is used. Therefore, x is a positive solution of boundary value problem (1.4). Finally, it is not difficult to show that, ∥x∥ < r.

Corollary 3.3. Let (H₀) hold. Assume that there exist continuous functions $$ and λ > 0 such that

•  

  (F) $$, for all x > 0 and t ∈ [0,1].

Then problem (1.4) has at least one positive solution if one of the following two conditions holds:

• (i)

  e_* ≥ 0;

• (ii)

  $$, where $$.

Remark 3.4. When m(t) ≡ m,     t ∈ [0,1], then (1.4) reduces to (1.1), (H₀) reduce to m ∈ (0, π/2). So Corollary 3.3 is more extensive than [5, Corollary  3.1].