Abstract
In this paper, the existence and multiplicity results of positive solutions for a nonlocal differential equation are mainly considered.
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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 Riemann-Stieltjes integrals.
Many authors consider the problem
because of the importance in numerous physical models: system of particles in thermodynamical equilibrium interacting via gravitational potential, 2-D fully turbulent behavior of a real flow, one-dimensional 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 Kras-noselskii 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 Leggett-Williams 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 ∈ C2(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 x1, x2, x3 satisfying
Lemma 2.3[10]. Let E be a Banach space, and let P ⊂ E be a closed, convex cone in E, assume Ω1, Ω2 are bounded open subsets of E with , and be a completely continuous operator such that either
-
(i)
||Au|| ≤ ||u||, u ∈ P ∩ ∂Ω1 and ||Au|| ≥ ||u||, u ∈ P ∩ ∂Ω2; or
-
(ii)
||Au|| ≥ ||u||, u ∈ P ∩ ∂Ω1 and ||Au|| ≤ ||u||, u ∈ P ∩ ∂Ω2.
Then, A has a fixed point in .
Main result
Let E = C[0, 1] endowed norm ||u|| = max0≤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 Arzela-Ascoli 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 ≤ p1 such that
(H6) There exists a constant p2 with such that
Then, problem (Equation 1) has one positive solution.
Proof. From (H4), there exists a 0 < η < ∞ such that
Choosing R1 ∈ (0, η), set Ω1 = {u ∈ E : ||u|| < R1}. We now prove that
Let u ∈ P ∩ ∂Ω1. Since minθ≤t≤1-θu(t) ≥ θ ||u|| and ||u|| = R1, 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 Ω2 = {u ∈ E : ||u|| < R2}. We now prove that
If u ∈ P ∩ ∂Ω2, 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 p1 = 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 ≤ p1 such that
(H8) There exists a constant p2 with such that
(H9)
Then, problem (Equation 1) has one positive solution.
Proof. From (H7), there exists η1 > 0 such that
From (H8), there exists η2 > 0 such that
Choosing , set Ω1 = {u ∈ E : ||u|| < R1}. We now prove that
If u ∈ P ∩ ∂Ω1, 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 Ω2 = {u ∈ E : ||u|| < R2}. We now prove that
If u ∈ P ∩ ∂Ω2, Since minθ≤t≤1-θu(t) ≥ θ ||u|| and ||u|| = R2, 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) = s2.
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 Leggett-Williams 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 Cone-Compression and Cone-Expansion Theorem due to M. Krasnosel'skii for the existence result and Leggett-Williams 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/10.1186/1687-2770-2011-5
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DOI: https://doi.org/10.1186/1687-2770-2011-5