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Existence and multiplicity of positive solutions for a nonlocal differential equation
Boundary Value Problems volume 2011, Article number: 5 (2011)
Abstract
In this paper, the existence and multiplicity results of positive solutions for a nonlocal differential equation are mainly considered.
Introduction
In this paper, we are concerned with the existence and multiplicity of positive solutions for the following nonlinear differential equation with nonlocal boundary value condition
where α, β, γ, δ are nonnegative constants, ρ = αγ + αδ + βγ > 0, q ≥ 1; , denote the RiemannStieltjes integrals.
Many authors consider the problem
because of the importance in numerous physical models: system of particles in thermodynamical equilibrium interacting via gravitational potential, 2D fully turbulent behavior of a real flow, onedimensional fluid flows with rate of strain proportional to a power of stress multiplied by a function of temperature, etc. In [1, 2], the authors use the Krasnoselskii fixed point theorem to obtain one positive solution for the following nonlocal equation with zero Dirichlet boundary condition
when the nonlinearity f is a sublinear or superlinear function in a sense to be established when necessary. Nonlocal BVPs of ordinary differential equations or system arise in a variety of areas of applied mathematics and physics. In recent years, more and more papers were devoted to deal with the existence of positive solutions of nonlocal BVPs (see [3–9] and references therein). Inspired by the above references, our aim in the present paper is to investigate the existence and multiplicity of positive solutions to Equation 1 using the Krasnosel'skii fixed point theorem and LeggettWilliams fixed point theorem.
This paper is organized as follows: In Section 2, some preliminaries are given; In Section 3, we give the existence results.
Preliminaries
Lemma 2.1[3]. Let y(t) ∈ C([0, 1]), then the problem
has a unique solution
where the Green function G(t, s) is
It is easy to see that
and there exists a such that G(t, s) ≥ θ G(s, s), θ ≤ t ≤ 1  θ, 0 ≤ s ≤ 1.
For convenience, we assume the following conditions hold throughout this paper:
(H1) f, g, Φ: R^{+} → R^{+} are continuous and nondecreasing functions, and Φ (0) > 0;
(H2) φ(t) is an increasing nonconstant function defined on [0, 1] with φ(0) = 0;
(H3) h(t) does not vanish identically on any subinterval of (0, 1) and satisfies
Obviously, u ∈ C^{2}(0, 1) is a solution of Equation 1 if and only if u ∈ C(0, 1) satisfies the following nonlinear integral equation
At the end of this section, we state the fixed point theorems, which will be used in Section 3.
Let E be a real Banach space with norm  ·  and P ⊂ E be a cone in E, P_{ r } = {x ∈ P : x < r}(r > 0). Then, . A map α is said to be a nonnegative continuous concave functional on P if α: P → [0, +∞) is continuous and
for all x, y ∈ P and t ∈ [0, 1]. For numbers a, b such that 0 < a < b and α is a nonnegative continuous concave functional on P, we define the convex set
Lemma 2.2[10]. Let be completely continuous and α be a nonnegative continuous concave functional on P such that α (x) = x for all . Suppose there exists 0 < d < a < b = c such that

(i)
{x ∈ P (α, a, b): α (x) > a} ≠ ∅ and α (Ax) > a for x ∈ P (α, a, b);

(ii)
Ax < d for x ≤ d;
(iii) α(Ax) > a for x ∈ P (α, a, c) with Ax > b.
Then, A has at least three fixed points x_{1}, x_{2}, x_{3} satisfying
Lemma 2.3[10]. Let E be a Banach space, and let P ⊂ E be a closed, convex cone in E, assume Ω_{1}, Ω_{2} are bounded open subsets of E with , and be a completely continuous operator such that either

(i)
Au ≤ u, u ∈ P ∩ ∂Ω_{1} and Au ≥ u, u ∈ P ∩ ∂Ω_{2}; or

(ii)
Au ≥ u, u ∈ P ∩ ∂Ω_{1} and Au ≤ u, u ∈ P ∩ ∂Ω_{2}.
Then, A has a fixed point in .
Main result
Let E = C[0, 1] endowed norm u = max_{0≤t≤1}u, and define the cone P ⊆ E by
Then, it is easy to prove that E is a Banach space and P is a cone in E.
Define the operator T: E → E by
Lemma 3.1. T: E → E is completely continuous, and Te now prove thatP ⊆ P.
Proof. For any u ∈ P, then from properties of G(t, s), T (u)(t) ≥ 0, t ∈ [0, 1], and it follows from the definition of T that
Thus, it follows from above that
From the above, we conclude that TP ⊆ P. Also, one can verify that T is completely continuous by the ArzelaAscoli theorem. □
Let
Then, it is clear to see that 0 < l ≤ L < L.
Theorem 3.2. Assume (H1) to (H3) hold. In addition,
(H4)
(H5) There exists a constant 2 ≤ p_{1} such that
(H6) There exists a constant p_{2} with such that
Then, problem (Equation 1) has one positive solution.
Proof. From (H4), there exists a 0 < η < ∞ such that
Choosing R_{1} ∈ (0, η), set Ω_{1} = {u ∈ E : u < R_{1}}. We now prove that
Let u ∈ P ∩ ∂Ω_{1}. Since min_{θ≤t≤1θ}u(t) ≥ θ u and u = R_{1}, from Equation 3, (H1) and (H3), it follows that
Then, Equation 4 holds.
On the other hand, from (H5), there exists such that
From (H6), there exists such that
Choosing , set Ω_{2} = {u ∈ E : u < R_{2}}. We now prove that
If u ∈ P ∩ ∂Ω_{2}, we have
From Equations 5, 6, we can prove
Then, Equation 7 holds.
Therefore, by Equations 4 and 7 and the second part of Lemma 2.3, T has a fixed point in , which is a positive solution of Equation 1. □
Example. Let q = 2, h(t) = 1, Φ(s) = 2 + s, φ(t) = 2t, and , namely,
It is easy to see that (H1) to (H3) hold. We also can have
Take p_{1} = 2, then it is clear to see that (H4) and (H5) hold. Since
then (H6) hold.
Theorem 3.3. Assume (H1) to (H3) hold. In addition,
(H7) There exists a constant 2 ≤ p_{1} such that
(H8) There exists a constant p_{2} with such that
(H9)
Then, problem (Equation 1) has one positive solution.
Proof. From (H7), there exists η_{1} > 0 such that
From (H8), there exists η_{2} > 0 such that
Choosing , set Ω_{1} = {u ∈ E : u < R_{1}}. We now prove that
If u ∈ P ∩ ∂Ω_{1}, we have
From Equations 8, 9, we can prove
Then, Equation 10 holds.
On the other hand, from (H7), there exists such that
Choosing , set Ω_{2} = {u ∈ E : u < R_{2}}. We now prove that
If u ∈ P ∩ ∂Ω_{2}, Since min_{θ≤t≤1θ}u(t) ≥ θ u and u = R_{2}, we have
By Equation 11, (H1) and (H3), it follows that
Then, Equation 12 holds.
Therefore, by Equations 10 and 12 and the first part of Lemma 2.3, T has a fixed point in , which is a positive solution of Equation 1. □
Example. Let q = 2, h(t) = t, Φ(s) = 2 + s, φ(t) = 2t, and g(s) = s^{2}.
Theorem 3.4. Assume that (H1) to (H3) hold. In addition, φ(1) ≥ 1, and the functions f, g satisfy the following growth conditions:
(H10)
(H11)
(H12) There exists a constant a > 0 such that
Then, BVP (Equation 1) has at least three positive solutions.
Proof. For the sake of applying the LeggettWilliams fixed point theorem, define a functional σ(u) on cone P by
Evidently, σ: P → R^{+} is a nonnegative continuous and concave. Moreover, σ(u) ≤ u for each u ∈ P.
Now, we verify that the assumption of Lemma 2.2 is satisfied.
Firstly, it can verify that there exists a positive number c with such that .
By (H10), it is easy to see that there exists τ > 0 such that
Set
Taking
If , then
by (H1) to (H3) and (H10).
Next, from (H11), there exists d' ∈ (0, a) such that
Take . Then, for each , we have
Finally, we will show that {u ∈ P (σ, a, b): σ(u) > a} ≠ ∅ and σ(Tu) > a for all u ∈ P(σ, a, b).
In fact,
For u ∈ P (σ, a, b), we have
for all t ∈ [θ, 1 θ]. Then, we have
by (H1) to (H3), (H12). In addition, for each u ∈ P (θ, a, c) with Tu > b, we have
Above all, we know that the conditions of Lemma 2.2 are satisfied. By Lemma 2.2, the operator T has at least three fixed points u_{ i } (i = 1, 2, 3) such that
The proof is complete. □
Example. Let q = 2, h(t) = t, Φ(s) = 2 + s, φ(t) = 2t, and, , namely,
From a simple computation, we have
Then, it is easy to see that (H1) to (H3) and (H10) to (H11) hold. Especially, take a = 1, by and (H1), then (H12) holds.
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Authors' contributions
In this manuscript the authors studied the existence and multiplicity of positive solutions for an interesting nonlocal differential equation using the ConeCompression and ConeExpansion Theorem due to M. Krasnosel'skii for the existence result and LeggettWilliams fixed point Theorem for the multiplicity result. Moreover, in this work, the authors supplements the studies done in [1, 2], because here they consider the case nonlocal boundary value condition. All authors typed, read and approved the final manuscript.
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Wang, Y., Wang, F. & An, Y. Existence and multiplicity of positive solutions for a nonlocal differential equation. Bound Value Probl 2011, 5 (2011). https://doi.org/10.1186/1687277020115
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Keywords
 Nonlocal boundary value problems
 Cone
 Fixed point theorem