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Nonlocal boundary value problems with resonant or nonresonant conditions
Boundary Value Problems volume 2013, Article number: 238 (2013)
Abstract
We study solvability of nonlocal boundary value problems for secondorder differential equations with resonance or nonresonance. The method of proof relies on Schauder’s fixed point theorem. Some examples are presented to illustrate the main results.
MSC:34B05.
1 Introduction
In this paper, we investigate the existence of solutions for the following boundary value problem:
where J=[0,1], 0<{a}_{i}\le 1 for 1\le i\le k, 0<{\eta}_{1}<{\eta}_{2}<\cdots <{\eta}_{k}<1, g:\mathbb{R}\to \mathbb{R}, f:J\times \mathbb{R}\to \mathbb{R}.
Nonlocal boundary value problems, studied by Il’in and Moiseev [1], have been addressed by many authors; see, for example, [2–9] and references therein. In the related literature, (1.1) is called resonance when {\sum}_{i=1}^{k}{a}_{i}=1, and nonresonance when {\sum}_{i=1}^{k}{a}_{i}\ne 1. For the boundary value problems at resonance, researchers usually use the continuity method or nonlinear alternative, which involves a complicated a priori estimate for the solution set; see [7, 10–12]. However, it is very difficult to obtain a related estimate for general differential equations. Here we list only a classical result about nonlocal boundary value problems at resonance of the form
where h:J\times \mathbb{R}\times \mathbb{R}\to \mathbb{R} is continuous, e:J\to \mathbb{R} is continuous and {a}_{i}>0 for 1\le i\le k, {\sum}_{i=1}^{k}{a}_{i}=1, 0<{\eta}_{1}<{\eta}_{2}<\cdots <{\eta}_{k}<1.
Theorem 1.1 [13]
Suppose that there are two constants M,\delta >0 such that
(A1) x[h(t,x,0)+e(t)]>\delta for any x>M, t\in J;
(A2) there exist constants {L}_{1},{L}_{2}:{L}_{1}>M, {L}_{2}<M such that
(A3) for (t,x,p)\in J\times [M,M]\times [{L}_{2},{L}_{1}],
Then (1.2) has at least one solution.
For (1.1), condition (A2) in Theorem 1.1 implies that f(t,x)\equiv g(x) for (t,x)\in J\times [M,M]. It follows that (1.1) has infinitely many solutions (x\equiv C\in [M,M] is the solution of (1.1)). At this point, Theorem 1.1 has little significance for (1.1). Moreover, there are few papers considering multiple results at resonance. For the case with nonresonance, there is an extensive literature; see [14–17] and the references therein.
The main purpose of this article is to discuss the existence of solutions of equation (1.1) by means of Schauder’s fixed point theorem. We only need to consider the behavior of g and f on some closed sets. Consequently, information on the location of the solution is obtained and multiple results are obtained if g and f satisfy the given conditions on distinct regions. Our approach is valid for the cases at resonance or nonresonance. In addition, some of our conditions are easily certified (see Corollaries 3.1 and 3.2).
The paper is organized as follows. Section 2 introduces an important lemma. Section 3 is devoted to the existence results of (1.1). In Section 4 we extend some results of Section 3 to the general boundary conditions.
2 Preliminaries
In this section, we consider the following boundary value problem for the linear differential equation:
where p>0, 0<{\alpha}_{i}\le 1, 0<{\eta}_{i}<1 for 1\le i\le k, 0<a:={\sum}_{i=1}^{k}{a}_{i}\le 1 and h\in C(J,\mathbb{R}).
Lemma 2.1

(1)
Boundary value problem (2.1) has a unique solution {x}_{h}\in {C}^{2}(J,\mathbb{R}).

(2)
If h\equiv C\in R on J and a=1, then {x}_{h}\equiv C/p on J.

(3)
If h(t)\ge 0 for all t\in J, then {x}_{h}\ge 0 on J; if h(t)\le 0 for all t\in J, then {x}_{h}\le 0 on J.

(4)
If h(t)\le C (C>0) on J, then {x}_{h}\le C/p on J.

(5)
Define an operator A:C(J,\mathbb{R})\to C(J,\mathbb{R}) by A(h)={x}_{h}, where \parallel h\parallel ={max}_{t\in J}h(t); then A is completely continuous.
Proof (1) Any solution of the differential equation {x}^{\u2033}(t)+px(t)=h(t) can be written as
where {c}_{1}, {c}_{2} are constants and \phi \in {C}^{2}(t,\mathbb{R}) is a particular solution of {x}^{\u2033}(t)+px(t)=h(t). From the boundary conditions, we obtain that
Since the above system has a unique solution ({c}_{1},{c}_{2}), (2.1) has a unique solution {x}_{h}\in {C}^{2}(t,\mathbb{R}).

(2)
The conclusion is obvious.

(3)
Here we only prove the case of h\ge 0. We consider two cases.
Case 3.1 Assume that {x}_{h}(t)\le 0 for all t\in J. We show that {x}_{h}(t)\equiv 0, t\in J. From (2.1), we obtain that
which implies that {x}_{h}^{\prime} is nonincreasing on J. Noting {x}_{h}^{\prime}(0)=0, we obtain that {x}_{h} is nonincreasing. Thus {x}_{h}(1)={min}_{t\in J}{x}_{h}(t)\le 0.
Since {x}_{h} is continuous, by the intermediate value theorem, there exists \theta \in (0,1) such that a{x}_{h}(\theta )={\sum}_{i=1}^{m}{a}_{i}{x}_{h}({\eta}_{i}).
If a=1, one can obtain from the monotonicity of {x}_{h} that {x}_{h}(t)\equiv {x}_{h}(1) on [\theta ,1]. Hence,
which implies that {x}_{h}(1)=0. Hence, {x}_{h}(t)\equiv 0 on J.
If 0<a<1, then {x}_{h}(1)={\sum}_{i=1}^{m}{a}_{i}{x}_{h}({\eta}_{i})=a{x}_{h}(\theta )\ge a{x}_{h}(1), which implies that {x}_{h}(1)=0. Hence, {x}_{h}(t)\equiv 0 on J.
Case 3.2 There exist {t}_{1},{t}_{2}\in J such that {x}_{h}({t}_{1})<0 and {x}_{h}({t}_{2})>0. We assume that {t}_{1}<{t}_{2}. Otherwise, {x}_{h}(t)\le 0 for all t\in [{t}_{1},1]. Similar to Case 3.1, one can show that {x}_{h}\equiv 0 for t\in [{t}_{1},1], which is impossible.
If there is \epsilon >0 such that {x}_{h}(t)\le 0 for t\in (0,\epsilon ), it is easy to check that {x}_{h}(t)\le 0 for all t\in J, a contradiction. Since {x}_{h}(1)\ge {\sum}_{\{i:{x}_{h}({\eta}_{i})<0\}}{a}_{i}{x}_{h}({\eta}_{i})\ge min\{{x}_{h}({\eta}_{i}):{x}_{h}({\eta}_{i})<0\}, there exists r\in (0,1) such that {x}_{h}(r)={min}_{t\in J}{x}_{h}(t). Noting that {x}_{h}^{\u2033}(r)\ge 0 and {x}_{h}(r)<0, we obtain that
which is a contradiction.
From Cases 3.1 and 3.2, one can easily obtain that {x}_{h}\ge 0 for all t\in J.

(4)
Since C\le h\le C, using the conclusion of (3), we have
{x}_{hC}\le 0,\phantom{\rule{2em}{0ex}}{x}_{h+C}\ge 0,\phantom{\rule{1em}{0ex}}t\in J.
Noting that {x}_{hC}={x}_{h}{x}_{C}, {x}_{h+C}={x}_{h}{x}_{C}, we obtain that
If a=1, then {x}_{C}=C/p and {x}_{C}=C/p. Thus {x}_{h}\le C/p on J.
Assume that a<1. There exist {t}_{1},{t}_{2}\in [0,1) such that
Noting that {x}_{C}^{\u2033}({t}_{1})\le 0 and {x}_{C}^{\u2033}({t}_{2})\ge 0, we have
and
From (2.3), (2.4) and (2.5), we obtain that {x}_{h}\le C/p on J.

(5)
Let {h}_{n}\to h in C(J,\mathbb{R}). For any \epsilon >0, there exists N(\epsilon )>0 such that
{h}_{n}(t)h(t)<\epsilon ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in J,\mathrm{\forall}n>N(\epsilon ).
Noting that A({h}_{n}h)={x}_{{h}_{n}}{x}_{h}, using the conclusion of (4), we obtain that
which implies that A is continuous. Let D be a bounded set in C(J,\mathbb{R}). Then there is {M}_{1}>0 such that \parallel h\parallel \le {M}_{1} for all h\in D. From the conclusion of (4), we obtain that A(h)\le {M}_{1}/p, which implies that A(D) is a uniformly bounded set. Since A(h), h are bounded for h\in D and
there exists {M}_{2}>0 such that
which implies that A(D) is equicontinuous. It follows that A(D) is relatively compact in C(J,\mathbb{R}) and A is a completely continuous operator. The proof is complete. □
Remark 2.1 Let h\equiv 1 and
By a direct computation, one can obtain that 0<{\beta}_{p}^{h=1}\le {\alpha}_{p}^{h=1}\le 1 and
The following wellknown Schauder fixed point theorem is crucial in our arguments.
Lemma 2.2 [18]
Let X be a Banach space and D\subset X be closed and convex. Assume that T:D\to D is a completely continuous map; then T has a fixed point in D.
3 Main results
The following theorem is the main result of the paper.
Theorem 3.1 Assume that there exist constants M>m, p>0 such that g\in C([m,M],\mathbb{R}), f\in C(J\times [m,M],\mathbb{R}), pu+g(u) is nondecreasing in u\in [m,M] and {\alpha}_{p}^{h=1}m\le {\beta}_{p}^{h=1}M. Further suppose that
Then (1.1) has at least one solution x with m\le x\le M.
Proof From Lemma 2.1, if x is a solution of (1.1), x satisfies
where A\circ H is composition of A and H defined as (A\circ H)x=A(Hx), and the operator H is defined in C(J,\mathbb{R}) as
Note that
Obviously, a fixed point of A\circ H is a solution of (1.1). Set \mathrm{\Omega}=\{x\in C(J,\mathbb{R}):m\le x(t)\le M,t\in J\}. Since the function pu+g(u) is nondecreasing in u\in [m,M], we obtain that for x\in \mathrm{\Omega},
Using (3.1), we have
for any x\in \mathrm{\Omega}. Since {\beta}_{p}^{h=1}\le pA(1)\le {\alpha}_{p}^{h=1}, and A is one nondecreasing operator, we obtain that for x\in \mathrm{\Omega},
Hence, (A\circ H)(\mathrm{\Omega})\subset \mathrm{\Omega}.
Also, the fact that A is completely continuous and H is continuous gives that A\circ H:\mathrm{\Omega}\to \mathrm{\Omega} is a continuous, compact map. By Lemma 2.2, A\circ H has at least one fixed point in Ω. The proof is complete. □
Remark 3.1 In Theorem 3.1, the condition that pu+g(u) is nondecreasing in u\in [m,M] can been replaced by the weaker condition
Corollary 3.1 Assume that a=1 and the following condition holds:
(H_{1}) There exist constants m<M such that g\in {C}^{1}([m,M],\mathbb{R}), f\in C(J\times [m,M],\mathbb{R}), and
Then (1.1) has at least one solution x with m\le x\le M.
Proof Since g\in {C}^{1}([m,M],\mathbb{R}), there exists p>0 such that the function pu+g(u) is nondecreasing in u\in [m,M]. When a=1, {\alpha}_{p}^{h=1}={\beta}_{p}^{h=1}=1. We directly apply Theorem 3.1 and this ends the proof. □
Corollary 3.2 Assume that 0<a<1 and the following condition holds:
(H_{2}) There exists constant M>0 such that g\in {C}^{1}([0,M],\mathbb{R}), f\in C(J\times [0,M],\mathbb{R}), and
Then (1.1) has at least one solution x with 0\le x\le M.
Proof Since g\in {C}^{1}([0,M],\mathbb{R}), there exists p>0 such that the function pu+g(u) is nondecreasing in u\in [0,M]. Condition (3.1) is satisfied if (3.4) holds. The proof is complete. □
Example 3.1 Consider the differential equation
Let {m}_{n}=(2n+0.5)\pi, {M}_{n}=(2n+1.5)\pi, g(u)=sinu, f(t,u)={t}^{2}{e}^{u} and n be a positive integer. For any t\in J and u\in [{m}_{n},{M}_{n}],
Hence, by Corollary 3.1, (3.5) has a solution {m}_{n}\le x\le {M}_{n}. Since n is an arbitrary positive integer, (3.5) has infinitely many solutions.
Example 3.2 Consider the differential equation
Using Corollary 3.2, we obtain that (3.6) has at least a nonnegative solution \overline{x}. Moreover, it is not difficult to show that \overline{x}\in (0,5) for any t\in J.
4 Generalization
In this section, we extend some results in the previous section to the following equation:
where {U}_{1} and {U}_{2} are linear operators defined as
where {a}_{i},{b}_{j},{c}_{s},{d}_{l}\in \mathbb{R} and 0<{\lambda}_{i},{\mu}_{j},{\nu}_{s},{\kappa}_{l}\le 1 for 1\le i\le {n}_{1}, 1\le j\le {n}_{2}, 1\le s\le {m}_{1}, 1\le l\le {m}_{2}.
Consider the boundary value problem for the linear differential equation:
where p>0 is sufficiently large, h:J\to \mathbb{R}. We introduce the following assumptions.
(P_{1}) The condition h\in C(J,\mathbb{R}) implies that boundary value problem (4.2) has a unique solution {x}_{h}\in {C}^{2}(J,\mathbb{R}).
(P_{2}) The condition h\equiv C\in \mathbb{R} implies that {x}_{h}\equiv C/p on J.
(P_{3}) The condition h\in C(J,\mathbb{R}):h\ge 0 (t\in J) implies that {x}_{h}\ge 0 on J.
(P_{4}) The condition h\in C(J,\mathbb{R}):h(t)\le C (C>0) implies that {x}_{h}\le C/p on J.
We say that the boundary condition {U}_{1}x={U}_{2}x=0 satisfies P_{123} if (4.2) satisfies conditions (P_{1}), (P_{2}), (P_{3}), and P_{134} if (4.2) satisfies conditions (P_{1}), (P_{3}), (P_{4}).
Theorem 4.1

(1)
Assume that the boundary condition {U}_{1}x={U}_{2}x=0 satisfies P_{123} and (H_{1}) holds. Then (4.1) has at least one solution x with m\le x\le M.

(2)
Assume that the boundary condition {U}_{1}x={U}_{2}x=0 satisfies P_{134} and (H_{2}) holds. Then (4.1) has at least one solution x with 0\le x\le M.
The proof of Theorem 4.1 is similar to that of Theorem 3.1 and we omit it.
Remark 4.1 The solution obtained in Corollary 3.2 or (2) of Theorem 4.1 may be trivial. Further suppose that
Then the solution obtained is nonnegative and nontrivial.
Remark 4.2 The boundary condition {U}_{1}x={U}_{2}x=0 satisfies P_{134} if it satisfies P_{123}.
Remark 4.3 Consider the twopoint boundary conditions:
One can easily check that boundary conditions (4.3), (4.4) satisfy P_{123}, and conditions (4.5), (4.6), (4.7) satisfy P_{134}.
Next, we consider the boundary conditions
where α, β, λ, μ, {b}_{i} (1\le i\le k), {c}_{j} (1\le j\le n) are constants and \alpha ,\lambda \in (0,+\mathrm{\infty}), \beta ,\mu ,{b}_{i},{c}_{j}\in [0,+\mathrm{\infty}), {\eta}_{i},{\xi}_{j}\in (0,1).
Theorem 4.2 Set b={\sum}_{i=1}^{k}{b}_{i}, c={\sum}_{j=1}^{n}{c}_{j}.

(1)
If b=\alpha, c=\lambda, then the boundary condition {U}_{1}x={U}_{2}x=0 satisfies P_{123}.

(2)
If b\in [0,\alpha ], c\in [0,\lambda ) or b\in [0,\alpha ), c\in [0,\lambda ], then the boundary condition {U}_{1}x={U}_{2}x=0 satisfies P_{134}.
Proof Without loss of generality, we assume that m=n=1. For any sufficiently large p>0 and h\in C(J,\mathbb{R}), the linear differential equation
has a unique solution {x}_{h}\in {C}^{2}(J,\mathbb{R}).
Suppose that h\ge 0 on J; if {x}_{h}\ge 0 is not true, by the maximum principle, we get that {x}_{h}(0)={min}_{t\in J}{x}_{h}(t)<0 or {x}_{h}(1)={min}_{t\in J}x(t)<0. If {x}_{h}(0)={min}_{t\in J}{x}_{h}(t)<0, then {x}_{h}^{\prime}(0)\ge 0. From the boundary conditions, we have
By the maximum principle, we obtain that {x}_{h}({\eta}_{1})\ge 0, which is a contradiction. If {x}_{h}(1)={min}_{t\in J}{x}_{h}(t)<0, then {x}_{h}^{\prime}(1)\le 0. From the boundary conditions, we have
By the maximum principle, we obtain that {x}_{h}({\xi}_{1})\ge 0, a contradiction.
If h\equiv C and b=\alpha, c=\lambda, then {x}_{h}\equiv C/p.
Now suppose that 0\le h\le C on J and h\not\equiv 0.
If there is \theta \in (0,1) such that {x}_{h}(\theta )={max}_{t\in J}x(t)>0, noting that {x}_{h}^{\u2033}(\theta )\le 0, we have
If {x}_{h}(0)={max}_{t\in J}x(t)>0, from the boundary conditions, we obtain that b=\alpha, \beta {x}_{h}^{\prime}(0)=0, which implies that {x}_{h}({\eta}_{1})={x}_{h}(0)={max}_{t\in J}x(t). The case has been discussed. If {x}_{h}(1)={max}_{t\in J}x(t)>0, from the boundary conditions, we obtain that c=\lambda, \mu {x}_{h}^{\prime}(1)=0, which implies that {x}_{h}({\xi}_{1})={x}_{h}(1)={max}_{t\in J}x(t). The case has also been discussed. The proof is complete. □
Example 4.1 Consider the differential equation
where \lambda >0, 0<\eta ,\xi <1 are constants.
Let g(u)={sin}^{1}u and f(t,u)={u}^{\lambda}tu. Set {m}_{n}=2n\pi +{(4n\pi )}^{\lambda 1}, {M}_{n}=2n\pi +0.5\pi. If n is a sufficiently large, positive integer, then for any t\in J, u\in [{m}_{n},{M}_{n}],
By Theorems 4.1 and 4.2, (4.10) has a solution {m}_{n}\le x\le {M}_{n}. Hence, (4.10) has infinitely many solutions.
Example 4.2 Consider the differential equation
where \mu >0, 0<{\xi}_{1},{\xi}_{2},{\xi}_{3}<1, 0\le \lambda \le 0.5 are constants.
In fact, g(u)={u}^{\mu}cosu and f(t)={t}^{2}. The boundary conditions in (4.11) satisfy P_{134} for \lambda \in [0,0.5] and P_{123} for \lambda =0.5.

(1)
Equation (4.11) has a solution 0\le \tilde{x}\le 1 and \tilde{x}(t)>0, t\in (0,1] for all 0\le \lambda \le 0.5. Set M=1, then g(M)=1\le f(t)\le g(0) for all t\in J. By Theorems 4.1 and 4.2, (4.11) has a solution 0\le \tilde{x}\le 1. Now we show that \tilde{x}(t)>0 for t\in (0,1]. Assume that there exists r\in (0,1) with \tilde{x}(r)=0. Since \tilde{x}(t)\ge 0, \tilde{x}(r) is minimum value and {\tilde{x}}^{\prime}(r)=0, {\tilde{x}}^{\u2033}(r)\ge 0. On the other hand, {\tilde{x}}^{\u2033}(r)={\tilde{x}}^{\mu}(r)cos\tilde{x}(r){r}^{2}={r}^{2}<0, a contradiction. If \tilde{x}(1)=0, then \tilde{x}({\xi}_{2})=0. This is impossible.

(2)
Equation (4.11) has infinitely many solutions for \lambda =0.5. Set {M}_{n}=2n\pi +2\pi, {m}_{n}=2n\pi +1.5\pi, where n>0 is an integer. Since g({M}_{n})={M}_{n}^{\mu}\le f(t)\le g({m}_{n})={m}_{n}^{\mu} for all t\in J, (4.11) has a solution {m}_{n}\le x\le {M}_{n}. Hence, (4.11) has infinitely many solutions.
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The work is supported by the NNSF of China (11171085) and Hunan Provincial Natural Science Foundation of China.
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Wang, W., Yang, X. Nonlocal boundary value problems with resonant or nonresonant conditions. Bound Value Probl 2013, 238 (2013). https://doi.org/10.1186/168727702013238
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DOI: https://doi.org/10.1186/168727702013238