- Research
- Open access
- Published:
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equa_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equb_HTML.gif)
has a unique solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equc_HTML.gif)
where the Green function G(t, s) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equd_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Eque_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equf_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equg_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equh_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equi_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equj_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equk_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equl_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equm_HTML.gif)
Thus, it follows from above that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equn_HTML.gif)
From the above, we conclude that TP ⊆ P. Also, one can verify that T is completely continuous by the Arzela-Ascoli theorem. □
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equo_HTML.gif)
Then, it is clear to see that 0 < l ≤ L < L.
Theorem 3.2. Assume (H1) to (H3) hold. In addition,
(H4)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equp_HTML.gif)
(H5) There exists a constant 2 ≤ p1 such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equq_HTML.gif)
(H6) There exists a constant p2 with such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equr_HTML.gif)
Then, problem (Equation 1) has one positive solution.
Proof. From (H4), there exists a 0 < η < ∞ such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ3_HTML.gif)
Choosing R1 ∈ (0, η), set Ω1 = {u ∈ E : ||u|| < R1}. We now prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ4_HTML.gif)
Let u ∈ P ∩ ∂Ω1. Since minθ≤t≤1-θu(t) ≥ θ ||u|| and ||u|| = R1, from Equation 3, (H1) and (H3), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equs_HTML.gif)
Then, Equation 4 holds.
On the other hand, from (H5), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ5_HTML.gif)
From (H6), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ6_HTML.gif)
Choosing , set Ω2 = {u ∈ E : ||u|| < R2}. We now prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ7_HTML.gif)
If u ∈ P ∩ ∂Ω2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equt_HTML.gif)
From Equations 5, 6, we can prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equu_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equv_HTML.gif)
It is easy to see that (H1) to (H3) hold. We also can have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equw_HTML.gif)
Take p1 = 2, then it is clear to see that (H4) and (H5) hold. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equx_HTML.gif)
then (H6) hold.
Theorem 3.3. Assume (H1) to (H3) hold. In addition,
(H7) There exists a constant 2 ≤ p1 such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equy_HTML.gif)
(H8) There exists a constant p2 with such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equz_HTML.gif)
(H9)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaa_HTML.gif)
Then, problem (Equation 1) has one positive solution.
Proof. From (H7), there exists η1 > 0 such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ8_HTML.gif)
From (H8), there exists η2 > 0 such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ9_HTML.gif)
Choosing , set Ω1 = {u ∈ E : ||u|| < R1}. We now prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ10_HTML.gif)
If u ∈ P ∩ ∂Ω1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equab_HTML.gif)
From Equations 8, 9, we can prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equac_HTML.gif)
Then, Equation 10 holds.
On the other hand, from (H7), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ11_HTML.gif)
Choosing , set Ω2 = {u ∈ E : ||u|| < R2}. We now prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ12_HTML.gif)
If u ∈ P ∩ ∂Ω2, Since minθ≤t≤1-θu(t) ≥ θ ||u|| and ||u|| = R2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equ13_HTML.gif)
By Equation 11, (H1) and (H3), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equad_HTML.gif)
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)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equae_HTML.gif)
(H11)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaf_HTML.gif)
(H12) There exists a constant a > 0 such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equag_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equah_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equai_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaj_HTML.gif)
Taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equak_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equal_HTML.gif)
by (H1) to (H3) and (H10).
Next, from (H11), there exists d' ∈ (0, a) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equam_HTML.gif)
Take . Then, for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equan_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equao_HTML.gif)
For u ∈ P (σ, a, b), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equap_HTML.gif)
for all t ∈ [θ, 1 -θ]. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equaq_HTML.gif)
by (H1) to (H3), (H12). In addition, for each u ∈ P (θ, a, c) with ||Tu|| > b, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equar_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equas_HTML.gif)
The proof is complete. □
Example. Let q = 2, h(t) = t, Φ(s) = 2 + s, φ(t) = 2t, and,
, namely,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equat_HTML.gif)
From a simple computation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-2770-2011-5/MediaObjects/13661_2011_Article_5_Equau_HTML.gif)
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.
References
Correa FJSA: On positive solutions of nonlocal and nonvariational elliptic problems. Nonliear Anal 2004, 59: 1147-1155.
Stanczy R: Nonlocal elliptic equations. Nonlinear Anal 2001, 47: 3579-3548. 10.1016/S0362-546X(01)00478-3
Kang P, Wei Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Anal 2009, 70: 444-451. 10.1016/j.na.2007.12.014
Kang P, Xub J, Wei Z: Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. Nonlinear Anal 2010, 72: 2767-2786. 10.1016/j.na.2009.11.022
Perera K, Zhang Z: Nontrivial solutions of Kirchhoff type problems via the Yang index. J Diff Equ 2006, 221(1):246-255. 10.1016/j.jde.2005.03.006
Pietramala P: A note on a beam equation with nonlinear boundary conditions. Boundary Value Problems 2011, 2011: 14. (Article ID 376782)
Wang F, An Y: Existence of nontrivial solution for a nonlocal elliptic equation with nonlinear boundary condition. Boundary Value Problems 2009, 2009: 8. (Article ID 540360)
Webb JRL, Infante G: Non-local boundary value problems of arbitrary order. J Lond Math Soc 2009, 79: 238-258.
Zhang Z, Perera K: Sign-changing solutions of Kirchhoff type problems via invariant sets of descent flow. J Math Anal Appl 2006, 317(2):456-463. 10.1016/j.jmaa.2005.06.102
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2011-5