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Positive solutions of thirdorder nonlocal boundary value problems at resonance
Boundary Value Problems volume 2012, Article number: 102 (2012)
Abstract
In this paper, we investigate the existence of positive solutions for a class of thirdorder nonlocal boundary value problems at resonance. Our results are based on the LeggettWilliams normtype theorem, which is due to O’Regan and Zima. An example is also included to illustrate the main results.
MSC:34B10, 34B15.
1 Introduction
This paper is devoted to the existence of positive solutions for the following thirdorder nonlocal boundary value problem (BVP for short):
where 0<{\xi}_{1}<{\xi}_{2}<\cdots <{\xi}_{m2}<1, {\alpha}_{i}\in {R}^{+} (i=1,2,\dots ,m2) and {\sum}_{i=1}^{m2}{\alpha}_{i}=1. The problem (1.1) happens to be at resonance in the sense that the associated linear homogeneous BVP
has nontrivial solutions. Clearly, the resonant condition is {\sum}_{i=1}^{m2}{\alpha}_{i}=1. Thirdorder differential equations arise in a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a threelayer beam, electromagnetic waves or gravitydriven flows and so on [1].
Recently, the existence of positive solutions for thirdorder twopoint or multipoint BVPs has received considerable attention; we mention a few works: [2–11] and the references therein. However, all of the papers on thirdorder BVPs focused their attention on the positive solutions with nonresonance cases. It is well known that the problem of the existence of positive solutions to BVPs is very difficult when the resonant case is considered. Only few papers deal with the existence of positive solutions to BVPs at resonance, and just to secondorder BVPs [12–15]. It is worth mentioning that Infante and Zima [13] studied the existence of positive solutions for the secondorder mpoint BVP
by means of the LeggettWilliams normtype theorem due to O’Regan and Zima [16], where 0<{\eta}_{1}<{\eta}_{2}<\cdots <{\eta}_{m2}<1, {\alpha}_{i}\in {R}^{+} (i=1,2,\dots ,m2) and {\sum}_{i=1}^{m2}{\alpha}_{i}=1.
However, thirdorder or higherorder derivatives do not have the convexity; to the best of our knowledge, no results are available for the existence of positive solutions for thirdorder or higher order BVPs at resonance. The main purpose of this paper is to fill the gap in this area. Motivated greatly by the abovementioned excellent works, in this paper we will investigate the thirdorder nonlocal BVP (1.1) at resonance, where 0<{\xi}_{1}<{\xi}_{2}<\cdots <{\xi}_{m2}<1, {\alpha}_{i}\in {R}^{+} (i=1,2,\dots ,m2) and {\sum}_{i=1}^{m2}{\alpha}_{i}=1. Some new existence results of at least one positive solution are established by applying the LeggettWilliams normtype theorem due to O’Regan and Zima [16]. An example is also included to illustrate the main results.
2 Some definitions and a fixed point theorem
For the convenience of the reader, we present here the necessary definitions and a new fixed point theorem due to O’Regan and Zima.
Definition 2.1 Let X and Z be real Banach spaces. A linear operator L:domL\subset X\to Z is called a Fredholm operator if the following two conditions hold:

(i)
KerL has a finite dimension, and

(ii)
ImL is closed and has a finite codimension.
Throughout the paper, we will assume that
1^{∘}L is a Fredholm operator of index zero, that is, ImL is closed and dimKerL=codimImL<\mathrm{\infty}.
From Definition 2.1, it follows that there exist continuous projectors P:X\to X and Q:Z\to Z such that
and that the isomorphism
is invertible. We denote the inverse of L{}_{domL\cap KerP} by {K}_{P}:ImL\to domL\cap KerP. The generalized inverse of L denoted by {K}_{P,Q}:Z\to domL\cap KerP is defined by {K}_{P,Q}={K}_{P}(IQ). Moreover, since dimImQ=codimImL, there exists an isomorphism J:ImQ\to KerL. Consider a nonlinear operator N:X\to Z. It is known (see [17, 18]) that the coincidence equation Lx=Nx is equivalent to
Definition 2.2 Let X be a real Banach space. A nonempty closed convex set P is said to be a cone provided that

(1)
\lambda x\in P for all x\in P and \lambda \ge 0, and

(2)
x,x\in P implies x=\theta.
Note that every cone P\subset X induces a partial order ≤ in X by defining x\le y if and only if yx\in P. The following property is valid for every cone in a Banach space.
Lemma 2.3 ([19])
Let P be a cone in X. Then for everyu\in P\mathrm{\setminus}\{\theta \}, there exists a positive number\sigma (u)such that\parallel x+u\parallel \ge \sigma (u)\parallel x\parallelfor allx\in P.
Let \gamma :X\to P be a retraction, that is, a continuous mapping such that \gamma (x)=x for all x\in P. Set
and
Our main results are based on the following theorem due to O’Regan and Zima.
Theorem 2.4 ([16])
Let C be a cone in X and let{\mathrm{\Omega}}_{1}, {\mathrm{\Omega}}_{2}be open bounded subsets of X with{\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}andC\cap ({\overline{\mathrm{\Omega}}}_{2}\mathrm{\setminus}{\mathrm{\Omega}}_{1})\ne \theta. Assume that 1^{∘}is satisfied and if the following assumptions hold:
(H1) QN:X\to Zis continuous and bounded, and{K}_{P,Q}:X\to Xis compact on every bounded subset of X;
(H2) Lx\ne \lambda Nxfor allx\in C\cap \partial {\mathrm{\Omega}}_{2}\cap domLand\lambda \in (0,1);
(H3) γ maps subsets of{\overline{\mathrm{\Omega}}}_{2}into bounded subsets of C;
(H4) {d}_{B}([I(P+JQN)\gamma ]{}_{KerL},KerL\cap {\mathrm{\Omega}}_{2},\theta )\ne 0, where{d}_{B}stands for the Brouwer degree;
(H5) there exists{u}_{0}\in C\mathrm{\setminus}\{\theta \}such that\parallel x\parallel \le \sigma ({u}_{0})\parallel \mathrm{\Psi}(x)\parallelforx\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}, whereC({u}_{0})=\{x\in C:\mu {u}_{0}\le x\mathit{\text{for some}}\mu 0\}and\sigma ({u}_{0})is such that\parallel x+{u}_{0}\parallel \ge \sigma ({u}_{0})\parallel x\parallelfor allx\in C;
(H6) (P+JQN)\gamma (\partial {\mathrm{\Omega}}_{2})\subset C;
(H7) {\mathrm{\Psi}}_{\gamma}({\overline{\mathrm{\Omega}}}_{2}\mathrm{\setminus}{\mathrm{\Omega}}_{1})\subset C,
then the equationLx=Nxhas a solution in the setC\cap ({\overline{\mathrm{\Omega}}}_{2}\mathrm{\setminus}{\mathrm{\Omega}}_{1}).
3 Main results
For simplicity of notation, we set
where i=1,2,\dots ,m2, and
It is easy to check that G(t,s)\ge 0, t,s\in [0,1], and since 0\le {h}_{i}(s)\le \frac{1}{2}s(1s), we get
for every \delta \in (0,\frac{2{\xi}_{1}^{2}(32{\xi}_{1})}{3}]. We also let \delta :=min\{\frac{2{\xi}_{1}^{2}(32{\xi}_{1})}{3},\frac{1}{{max}_{t,s\in [0,1]}G(t,s)},1\}.
We can now state our result on the existence of a positive solution for the BVP (1.1).
Theorem 3.1 Assume that f:[0,1]\times [0,+\mathrm{\infty})\to R is continuous and

(1)
there exists a constant M\in (0,\mathrm{\infty}) such that f(t,x)>\delta x for all (t,x)\in [0,1]\times [0,M];

(2)
there exist r\in (0,M), {t}_{0}\in [0,1], a\in (0,1], b\in (0,1) and continuous functions g:[0,1]\to [0,\mathrm{\infty}), h:(0,r]\to [0,\mathrm{\infty}) such that f(t,x)\ge g(t)h(x) for (t,x)\in [0,1]\times (0,r], \frac{h(x)}{{x}^{a}} is nonincreasing on (0,r] with
\frac{h(r)}{{(r)}^{a}}{\int}_{0}^{1}G({t}_{0},s)g(s)\phantom{\rule{0.2em}{0ex}}ds\ge \frac{1b}{{b}^{a}}. 
(3)
{f}_{\mathrm{\infty}}={lim\hspace{0.17em}inf}_{x\to +\mathrm{\infty}}{min}_{t\in [0,1]}\frac{f(t,x)}{x}<0.
Then the resonant BVP (1.1) has at least one positive solution on[0,1].
Proof Consider the Banach spaces X=Z=C[0,1] with \parallel x\parallel ={max}_{t\in [0,1]}x(t).
Let L:domL\subset X\to Z and N:X\to Z with
be given by (Lx)(t)={x}^{\u2034}(t) and (Nx)(t)=f(t,x(t)) for t\in [0,1]. Then
and
Clearly, dimKerL=1 and ImL is closed. It follows from {Z}_{1}=Z\mathrm{\setminus}ImL that
In fact, for each y\in Z, we have
which shows that y{y}_{1}\in ImL, which together with {Z}_{1}\cap ImL=\{\theta \} implies that Z={Z}_{1}\oplus ImL. Note that dim{Z}_{1}=1 and thus codimImL=1. Therefore, L is a Fredholm operator of index zero.
Next, define the projections P:X\to X by
and Q:Z\to Z by
Clearly, ImP=KerL, KerQ=ImL and KerL=\{x\in X:{\int}_{0}^{1}x(s)\phantom{\rule{0.2em}{0ex}}ds=0\}. Note that for y\in ImL, the inverse {K}_{P} of {L}_{P} is given by
where
Considering that f can be extended continuously to [0,1]\times R, it is easy to check that QN:X\to Z is continuous and bounded, and {K}_{P,Q}N:X\to X is compact on every bounded subset of X, which ensures that (H1) of Theorem 2.4 is fulfilled.
Define the cone of nonnegative functions
Let
and
Clearly, {\mathrm{\Omega}}_{1} and {\mathrm{\Omega}}_{2} are bounded and open sets and
Moreover, C\cap ({\overline{\mathrm{\Omega}}}_{2}\mathrm{\setminus}{\mathrm{\Omega}}_{1})\ne \theta. Let J=I and (\gamma x)(t)=x(t) for x\in X. Then γ is a retraction and maps subsets of {\overline{\mathrm{\Omega}}}_{2} into bounded subsets of C, which means that (H3) of Theorem 2.4 holds.
Let 0<{r}^{\prime}=min\{{r}_{1},{r}_{2}\}, where {r}_{1} and {r}_{2} will be defined in the following proof.
Suppose that there exist {x}_{0}\in C\cap \partial {\mathrm{\Omega}}_{2}\cap domL and {\lambda}_{0}\in (0,1) such that L{x}_{0}={\lambda}_{0}N{x}_{0}. Then
Let {x}_{0}({t}_{1})=\parallel {x}_{0}\parallel ={max}_{t\in [0,1]}{x}_{0}(t)=M. Now, we verify that {t}_{1}\ne 0 and {t}_{1}\ne 1.
First, we show {t}_{1}\ne 0. Suppose, on the contrary, that {x}_{0}(t) achieves maximum value M only at t=0. Then {x}_{0}(0)={\sum}_{i=1}^{m2}{\alpha}_{i}{x}_{0}({\xi}_{i}) in combination with {\sum}_{i=1}^{m2}{\alpha}_{i}=1 yields that {max}_{1\le i\le m2}\{{x}_{0}({\xi}_{i})\}\ge M, which is a contradiction.
Next, we show {t}_{1}\ne 1. It follows from {x}_{0}^{\prime}(0)={x}_{0}^{\prime}(1)=0 that there is a constant \eta \in (0,1) such that {x}_{0}^{\u2033}(\eta )=0, and thus {x}_{0}^{\u2033}(t)={\lambda}_{0}{\int}_{\eta}^{t}f(s,{x}_{0}(s))\phantom{\rule{0.2em}{0ex}}ds. By the condition (3) we have, for x\ge {r}^{\prime} (M\ge {r}^{\prime}>r>0), there exists \sigma \in (0,\delta ) such that
Suppose, on the contrary, that {x}_{0}(1)=M. The step is divided into two cases:
Case 1. Assume that {x}_{0}(t)>0 on [\eta ,1]. Let {r}_{1}={min}_{t\in [\eta ,1]}{x}_{0}(t). Then (3.1) yields
which implies {x}_{0}^{\prime}(t) is increasing close to 1. This together with {x}_{0}^{\prime}(1)=0 induces {x}_{0}^{\prime}(t)<0 (t close to 1), that is, {x}_{0}(t) is decreasing close to 1, which contradicts {x}_{0}(1)=M.
Case 2. Assume that {x}_{0}(t) has zero points on [\eta ,1]; we may choose {\eta}^{\prime} nearest to 1 with {x}_{0}({\eta}^{\prime})=0. Then there is a constant \eta \in ({\eta}^{\prime},1) such that {x}_{0}^{\u2033}(\eta )=0. Similar to the above arguments, we easily get a contradiction too.
Hence, we can choose {t}_{1}\in (0,1) so that {x}_{0}({t}_{1})=M. This gives {x}_{0}^{\prime}({t}_{1})=0 and {x}_{0}^{\u2033}({t}_{1})\le 0. By {x}_{0}\in domL, we know {x}_{0}(t)\ge 0 and {x}_{0}^{\prime}(0)=0. Similarly, we also divide the part of the proof into two cases.
Case 1. If {x}_{0}(t)>0 on [0,{t}_{1}], then there is a constant \eta \in (0,{t}_{1}) such that {x}_{0}^{\u2033}(\eta )=0. Thus we have
Let {r}_{2}={min}_{t\in [\eta ,{t}_{1}]}{x}_{0}(t). Then it follows from (3.1) that
which is a contradiction.
Case 2. If {x}_{0}(t) has zero points on [0,{t}_{1}], we may choose {\eta}^{\prime} nearest to {t}_{1} with {x}_{0}({\eta}^{\prime})=0. Then there is a constant \eta \in ({\eta}^{\prime},{t}_{1}) such that {x}_{0}^{\u2033}(\eta )=0. Thus we have
Let {r}_{2}={min}_{t\in [\eta ,{t}_{1}]}{x}_{0}(t). Then it follows from (3.1) that
which is a contradiction. Therefore, (H2) of Theorem 2.4 holds.
Consider x\in KerL\cap {\mathrm{\Omega}}_{2}. Then x(t)\equiv c on [0,1]. Similar to [13], we define
Suppose H(x,\lambda )=0. Then in view of (1), we obtain
Hence, H(x,\lambda )=0 implies c\ge 0. Furthermore, if H(M,\lambda )=0, then we have
contradicting (3.1). Thus H(x,\lambda )=0 for x\in \partial {\mathrm{\Omega}}_{2} and \lambda \in [0,1]. Therefore,
However,
This gives
which shows that (H4) of Theorem 2.4 holds.
Let x\in {\overline{\mathrm{\Omega}}}_{2}\mathrm{\setminus}{\mathrm{\Omega}}_{1} and t\in [0,1]. Then
From (1), we know that
Hence {\mathrm{\Psi}}_{\gamma}(\overline{{\mathrm{\Omega}}_{2}}\mathrm{\setminus}{\mathrm{\Omega}}_{1})\subset C. Moreover, for x\in \partial {\mathrm{\Omega}}_{2}, we have
which shows that (P+JQN)\gamma (\partial {\mathrm{\Omega}}_{2})\subset P. These ensure that (H6), (H7) of Theorem 2.4 hold. It remains to show that (H5) is satisfied.
Taking {u}_{0}(t)\equiv 1 on [0,1] and \sigma ({u}_{0})=1, we confirm that
Let x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}. Then we have x(t)>0, t\in [0,1], 0<\parallel x\parallel <r and x(t)\ge b\parallel x\parallel, t\in [0,1]. Therefore, in view of (2), for all x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}, we obtain
That is, \parallel x\parallel \le \sigma ({u}_{0})\parallel \mathrm{\Psi}x\parallel for all x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}, which shows that (H5) of Theorem 2.4 holds.
Summing up, all the hypotheses of Theorem 2.4 are satisfied. Therefore, the equation Lx=Nx has a solution x\in C\cap ({\overline{\mathrm{\Omega}}}_{2}\mathrm{\setminus}{\mathrm{\Omega}}_{1}). And so, the resonant BVP (1.1) has at least one positive solution on [0,1]. □
4 An example
Consider the BVP
Here {\alpha}_{1}=1, {\xi}_{1}=\frac{1}{2}, g(t)=\frac{1}{4}({t}^{2}t1), h(x)=\sqrt{x+1} and
By a simple computation, we get
We may choose M=3, r=\frac{5}{2}, {t}_{0}=0, a=1, b=\frac{13}{14}. It is easy to check

(1)
f(t,x)>\frac{1}{3}x for all (t,x)\in [0,1]\times [0,3];

(2)
f(t,x)\ge g(t)h(x) for (t,x)\in [0,1]\times (0,\frac{5}{2}], \frac{\sqrt{x+1}}{x} is nonincreasing on (0,\frac{5}{2}] with
\frac{h(r)}{{(r)}^{a}}{\int}_{0}^{1}G(0,s)g(s)\phantom{\rule{0.2em}{0ex}}ds\ge \frac{7\sqrt{14}}{320}\ge \frac{1}{13}=\frac{1b}{{b}^{a}}; 
(3)
{f}_{\mathrm{\infty}}={lim\hspace{0.17em}inf}_{x\to +\mathrm{\infty}}{min}_{t\in [0,1]}\frac{f(t,x)}{x}=\frac{1}{8}.
Thus, all the conditions of Theorem 3.1 are satisfied. Then the resonant problem (4.1) has at least one positive solution on [0,1].
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (10801068) and the Education Scientific Research Foundation of Tangshan College (120186). The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
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Zhang, HE., Sun, JP. Positive solutions of thirdorder nonlocal boundary value problems at resonance. Bound Value Probl 2012, 102 (2012). https://doi.org/10.1186/168727702012102
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DOI: https://doi.org/10.1186/168727702012102