New existence theory of positive solutions to fourth order p-Laplacian elasticity problems

Using a new technique for dealing with the bending term of the fourth order p-Laplacian elasticity problems, several new and more general results are obtained for the existence of at least single, twin or triple positive solutions. It is interesting to point out that estimates on the norms of these solutions will also be provided.

It is well known that many nonlinear problems can be reduced to the search for solutions of various differential equations. Fixed point methods are often used for proving the existence of such solutions. An overview of such results can be found in Guo et al. [1], in Guo and Lakshmikantham [2], and in Demling [3].

At the same time, we notice that some new techniques via appropriate transformation are proved to be very effective in studying the solvability of differential equations. Such techniques have attracted the attention of Zhang et al. [4] and Wong [5], etc. In [4], Zhang et al. considered the existence of positive solutions of the following problems:

where \(J=[0,1]\).

First, by means of the linear transformation

the authors converted (1.1) into

and

Then it follows from Lemma 2.3 in [4] that they converted the results obtained for problem (1.2) to the counterpart for problem (1.2).

In [5], Wong transformed the following problems:

into

and

by using

So the existence of a solution of the complementary Lidstone boundary value problem (1.3) follows from the existence of a solution of the Lidstone boundary value problem (1.4). From the proof of the main results of [5], we notice that the above paper requires F to satisfy some assumptions of monotonicity which are essential for the technique used.

Being directly inspired by [4, 5], in the present paper, by using a transformation technique and a fixed point method, we shall investigate the existence and multiplicity of positive solutions for the fourth order p-Laplacian elasticity problems

where \(a, b>0\), \(\phi_{p}(t)=|t|^{p-2}t\), \(p>1\), \(\phi_{q}=\phi _{p}^{-1}\), \(\frac{1}{p}+\frac{1}{q}=1\), \(F:[0,1]\times R\times R\rightarrow R\) is continuous.

Problem (1.5) occurs in beam theory; see [6, 7], for example for a beam with small deformation; a beam of a material which satisfies a nonlinear power-like stress and strain law; a beam with two-sided links which satisfies a nonlinear power-like elasticity law. If \(p=2\), \(g=0\) and \(h=0\), in mechanics, problem (1.5) is called the cantilever beam equation [8, 9]. The equation describes the deflection of an elastic beam fixed at the right and freed at the left.

Most research papers on fourth order elasticity problems consider nonlinear terms that as regards F in (1.5) involve y only, and derivative-dependent nonlinearities are seldom tackled; see [10–18] to name a few. Besides achieving new results, the presence of the derivative \(y^{\prime\prime}\) in the nonlinear term F requires a special technique to tackle the problem.

The main features of this paper are as follows. First, compared with [6–18], besides achieving new results, estimates on the norms of these solutions will also be provided. Second, we transform problem (1.5) into a differential systems without bending term, i.e., the technique to deal with bending term is completely different from that of [7, 19–21]. Finally, it is pointed out that we do not need any monotone assumption on F, which is weaker than the corresponding assumptions on F in [5].

The rest of the paper is organized as follows: In Section 2, we first present some definitions and lemmas which are needed throughout this paper and then state several fixed point results: Krasnosel’skii’s fixed point theorem in a cone, a new fixed point theorem due to Avery and Henderson and Leggett-Williams’ fixed point theorem. In Section 3, we use Krasnosel’skii’s fixed point theorem to obtain the existence of at least one or two positive solutions of problem (1.5). Section 4 will further discuss the existence of twin positive solutions of problem (1.5). Two new results will be presented by new fixed point theorem due to Avery and Henderson. Section 5 is due to develop existence criteria for (at least) three positive solutions of problem (1.5). In particular, our results in these sections are new when \(p=2\) (the semilinear case). Finally, in Section 6, an example is included to verify the theoretical results. To the best of the authors’ knowledge, it is the first paper where the transformation technique and fixed point method are applied to the fourth order p-Laplacian elasticity problems.

To establish the existence of positive solutions of problem (1.5), let us list the following assumptions, which will hold throughout this paper:

(H₁):

\(F\in C([0,1]\times[0,+\infty)\times(-\infty,0],[0,+\infty ))\);

(H₂):

\(g,h\in L^{1}[0,1]\) are nonnegative, \(\mu\in[0,a)\), \(\nu\in [0,1)\), where

$$\mu= \int_{0}^{1}g(s)\,ds, \qquad\nu= \int_{0}^{1}h(s)\,ds. $$

We shall reduce elasticity problem (1.5) to a system without the bending term. To this goal, first, by means of the transformation

It follows from (1.5) that Lemma 2.1 holds.

Lemma 2.1

(See [22])

Assume that (H₂) holds. Then for any \(x\in C[0,1]\), the boundary value problem

has a unique solution y given by

where

Taking into account (2.1) and (2.2), problem (1.5) reduces to the following problem:

Lemma 2.2

(See [22])

Assume that (H₁)-(H₂) hold. Then the boundary value problem (2.5) has a unique solution x given by

where

If problem (2.5) has a solution \(x^{*}\), then by (2.2), problem (1.5) has a solution given by

So the existence of a solution of problem (1.5) follows from the existence of a solution of problem (2.5).

Lemma 2.3

(See [22])

Let (H₂) hold. Then we have the following results:

where

Lemma 2.4

(See [22])

Let (H₂) hold. Then we obtain the following results:

where

It is clear from (2.2) that \(\|y^{*}\|\leq\rho_{2}\phi_{q}(\|x^{*}\|)\), moreover, if \(x^{*}\) is positive, so is \(y^{*}\).

Let \(J=[0,1]\), \(E=C[0,1]\). It is well known that E is a real Banach space with the norm \(\|\cdot\|\) defined by \(\|x\|=\max_{t\in J}|x(t)|\).

Define an operator \(T:E\rightarrow E\) as follows:

Let K be a cone in E which is defined as

where

Lemma 2.5

Let (H₁)-(H₂) hold. Then we have \(T(K)\subset K\), and \(T:K\rightarrow K\) is completely continuous.

Proof

In view of condition (2.15), we see that

Moreover, it follows from (2.15) that

This proves that \(T(K)\subset K\).

Next by standard methods and the Ascoli-Arzela theorem one can prove that \(T: K\rightarrow K\) is completely continuous. So this is omitted. □

In the rest of this section, we state several fixed point theorems which we needed later.

Lemma 2.6

(See [1–3])

Let P be a cone of real Banach space E, and \(\Omega_{1}\) and \(\Omega_{2}\) be two bounded open sets in E such that \(0\in\Omega_{1}\) and \(\bar{\Omega}_{1}\subset\Omega_{2}\). Let operator \(A:P\cap(\bar{\Omega}_{2}\setminus\Omega_{1})\rightarrow P\) be completely continuous. Suppose that one of the two conditions

1. (i)

   \(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\) and \(\|Ax\| \geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\),

2. (ii)

   \(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\) and \(\|Ax\| \leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\)

is satisfied. Then A has at least one fixed point in \(P\cap(\bar{\Omega}_{2}\setminus\Omega_{1})\).

Remark 2.1

To make clear what \(\bar{\Omega}_{2}\), \(\partial\Omega _{2}\), \(\partial\Omega_{1}\), and \(\Omega_{2}\backslash\bar{\Omega}_{1}\) mean, we give typical examples of \(\Omega_{1}\) and \(\Omega_{2}\), e.g.,

with \(0< r< R\), where \(\|x\|_{\infty}=\sup_{t\in J}|x(t)|\).

Given a nonnegative continuous functional ψ on a cone P of a real Banach space, we define, for each \(r>0\), the set

Lemma 2.7

(See [23])

Let P be a cone in a real Banach space. Let α and γ be increasing nonnegative continuous functional on P, and let θ be a nonnegative continuous functional on P with \(\theta(0)=0\) such that, for some \(c>0\) and \(H>0\),

for all \(x\in\overline{P(\gamma,c)}\). Suppose there exists a completely continuous operator \(A:\overline{P(\gamma,c)}\rightarrow P\) and \(0< r_{1}< r_{2}< c\) such that

and

1. (i)

   \(\gamma(Ax)>c\) for all \(x\in\partial P(\gamma,c)\);

2. (ii)

   \(\theta(Ax)< r_{2}\) for all \(x\in\partial P(\theta,r_{2})\);

3. (iii)

   \(P(\alpha,r_{1})\neq{\emptyset}\) and \(\alpha(Ax)>r_{1}\) for \(x\in\partial P(\alpha,r_{1})\).

Then A has at least two positive solutions \(x_{1}\) and \(x_{2}\) belonging to \(\overline{P(\gamma,c)}\) satisfying

Let \(0< r_{1}< r_{2}\) be given and let β be a nonnegative continuous concave functional on the cone K. Define the convex sets \(K_{r_{1}}\), \(K(\beta,r_{1},r_{2})\) by

Finally we state the Leggett-Williams fixed point theorem [24].

Lemma 2.8

(See [24])

Let K be a cone in a real Banach space E, \(A:\bar{K}_{c}\rightarrow\bar{K}_{c}\) be completely continuous and β be a nonnegative continuous concave functional on K with \(\beta(x)\leq\|x\|\) for all \(x\in K_{r_{1}}\). Suppose there exist \(0< d < r_{1} < r_{2} \leq c\) such that

1. (i)

   \(\{x\in K(\beta,r_{1},r_{2}): \beta(x)>r_{1}\} \neq{\emptyset}\) and \(\beta(Ax)>r_{1}\) for \(x\in K(\beta,r_{1},r_{2})\);

2. (ii)

   \(\|Ax\|< d\) for \(\|x\|\leq d\);

3. (iii)

   \(\beta(Ax)>r_{1}\) for \(x\in K(\beta,r_{1},c)\) with \(\|Ax\| >r_{2}\).

Then A has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\) satisfying

In this section, we apply Lemma 2.6 to establish the existence of positive solutions for problem (1.5). We begin by introducing the following conditions on \(F(t,u,v)\):

(H₃):

There exist two positive constants r, R with \((\rho _{2}+1)\phi_{q}(r)<(\rho_{1}+1)\phi_{q}(\delta R)\) such that

$$\begin{aligned}& F(t,u,v)\leq\frac{1}{\eta}r,\quad \forall t\in J, |u|+|v|\leq( \rho_{2}+1)\phi_{q}(r), \end{aligned}$$

(3.1)

$$\begin{aligned}& F(t,u,v)\geq\frac{1}{\rho\beta\delta} R,\quad \forall t\in J, |u|+|v|\geq( \rho_{1}+1)\phi_{q}(\delta R), \end{aligned}$$

(3.2)

where

$$\eta=\frac{1}{4}\gamma_{2}, \qquad \sigma=\rho_{1} \delta^{q-1} , \qquad \beta = \int_{0}^{1}s(1-s)\,ds=\frac{1}{6}. $$

Theorem 3.1

Assume that (H₁)-(H₃) hold. Then we have the following conclusions:

1. (i)

   Problem (2.5) has (at least) one positive solution \(x\in K\) such that

   $$ \delta r\leq x(t)\leq\frac{1}{\delta} R, \quad t\in J. $$

   (3.3)
2. (ii)

   Problem (1.5) has (at least) one positive solution y such that

   $$ \left \{ \textstyle\begin{array}{@{}l} y(t)=\int_{0}^{1}H_{1}(t,s)\phi_{q}(x(s))\,ds,\quad t\in J;\\ \|y\|\leq\rho_{2}\phi_{q}(\|x\|);\\ y(t)\geq\sigma\phi_{q}(\|x\|), \quad t\in J. \end{array}\displaystyle \right . $$

   (3.4)

We further have

Proof

Let T be the cone preserving, completely continuous operator that was defined by (2.16).

Let \(x\in K\) with \(\|x\|=r\). Then \(0\leq\phi_{q}(x(t))\leq\phi _{q}(r)\), \(t\in J\), and

And hence for \(x\in K\) with \(\|x\|=r\), we have

Then it follows from (3.1) that

Now if we let \(\Omega_{1}=\{x\in K:\|x\|< r\}\), then (3.6) shows that

Further, let

and

Then \(x\in K\) and \(\|x\|=R_{1}\) implies

that is,

Hence, \(\phi_{q}(x(t))\geq\phi_{q}(R)\) for all \(t\in J\), and

So

Using condition (3.2), it follows from \(x\in K\) and \(\|x\|=R_{1}\) that

that is, \(x\in\partial\Omega_{2}\) implies

It follows from Lemma 2.6 that problem (2.5) has (at least) one positive solution \(x\in\bar{\Omega}_{2}\setminus\Omega_{1}\) satisfying (3.3).

It is observed from (2.2) that problem (1.5) has (at least) one positive solution y such that

Moreover, since \(x\in K\), we get for \(t\in J\),

Then we get (3.4).

Further, it follows from (3.3) and (3.4) that (3.5) holds. □

In Theorem 3.2 we assume the following condition on \(F(t,u,v)\):

(H₄):

There exist two positive constants r, R with \((\rho _{2}+1)\phi_{q}(r)< R\) such that:

$$\begin{aligned}& F(t,u,v)\geq\frac{1}{\rho_{1}\phi_{p}(\rho_{1}+1)\beta\delta}\phi_{p} \bigl(|u|+|v|\bigr),\quad \forall t\in J, |u|+|v|\leq(\rho_{2}+1)\phi_{q}(r), \end{aligned}$$

(3.10)

$$\begin{aligned}& F(t,u,v)\leq\frac{1}{2\eta\phi_{p}(\rho_{2}+1)} \phi_{p}\bigl(|u|+|v|\bigr),\quad \forall t\in J, |u|+|v|\geq R, \end{aligned}$$

(3.11)

and we write

$$ M=\max_{\forall t\in J, |u|+|v|\leq R}F(t,u,v). $$

(3.12)

Theorem 3.2

Assume that (H₁)-(H₂) and (H₄) hold. Then we have the following conclusions:

1. (i)

   Problem (2.5) has (at least) one positive solution \(x\in K\) such that

   $$ \delta r\leq x(t)\leq\max \biggl\{ 2\eta M, \phi_{p} \biggl( \frac{R}{\phi _{q}(\delta)(\rho_{1}+1)} \biggr) \biggr\} , \quad t\in J. $$

   (3.13)
2. (ii)

   Problem (1.5) has (at least) one positive solution y such that (3.4) holds. We further have

   $$ \sigma\delta\phi_{q}(r)\leq y(t) \leq\max \biggl\{ \rho_{2} \phi _{q}(2\eta M), \frac{\rho_{2}R}{\phi_{q}(\delta)(\rho_{1}+1)} \biggr\} ,\quad t\in J. $$

   (3.14)

Proof

Let \(x\in K\) with \(\|x\|=r\). Then \(0\leq\phi _{q}(x(t))\leq\phi_{q}(r)\), \(t\in J\), and

And hence for \(x\in K\) with \(\|x\|=r\), we have

and it follows from condition (3.10) that

that is, \(x\in\partial\Omega_{1}\) implies that

From (3.11) and (3.12), we have

Further, let

and

Noticing that for \(x\in\partial\Omega_{3}\) we have

Thus, for \(x\in\partial\Omega_{3}\), it follows from (3.17) that

that is, \(x\in\partial\Omega_{3}\) implies

It now follows from Lemma 2.6 that problem (2.5) has (at least) one positive solution \(x\in\bar{\Omega}_{3}\setminus\Omega_{1}\) satisfying (3.13).

It follows from (2.2) that problem (1.5) has (at least) one positive solution y. Similar to the proof of (3.5), one can show that y satisfies (3.14). □

Theorem 3.3

Assume that (H₁)-(H₂), (3.2) of (H₃) and (3.10) of (H₄) hold. In addition, let f satisfies the following condition:

(H₅):

Let l, \(\zeta_{i} \) (\(i=1,2\)) and L satisfy

$$0< l< \delta\zeta_{1}< \zeta_{1}< \delta\zeta_{2}< \zeta_{2}< L< \infty. $$

then we have the following conclusions:

1. (i)

   Problem (2.5) has (at least) two positive solutions \(x_{1},x_{2}\in K\) such that

   $$ \delta l\leq x_{1}(t)\leq\zeta_{1}< \delta \zeta_{2}\leq x_{2}(t)\leq L,\quad t\in J. $$

   (3.20)
2. (ii)

   Problem (1.5) has (at least) two positive solutions \(y_{1}\), \(y_{2}\) such that, for \(i=1,2\),

   $$ \left \{ \textstyle\begin{array}{@{}l} y_{i}(t)=\int_{0}^{1}H_{1}(t,s)\phi_{q}(x_{i}(s))\,ds, \quad t\in J;\\ \|y_{i}\|\leq\rho_{2}\phi_{q}(\|x_{i}\|);\\ y_{i}(t)\geq\sigma\phi_{q}(\|x_{i}\|), \quad t\in J. \end{array}\displaystyle \right . $$

   (3.21)

We further have

Proof

If (3.10) of (H₄) holds, similar to the proof of (3.16), we can prove that

If (3.2) of (H₃) holds, similar to the proof of (3.9), we have

Finally, we show that

In fact, for \(x\in K\) with \(\|x\|=\zeta_{i}\) (\(i=1,2\)) then we have

and

Therefore,

and hence it follows from (H₅) that

which shows that (3.25) holds.

Applying Lemma 2.6 to (3.23), (3.24) and (3.25) shows that problem (2.5) has (at least) two positive solutions \(x_{1}\), \(x_{2}\) with \(x_{1}\in K_{l,\zeta_{1}}=\{x \in K, l \leq\|x\| \leq\zeta_{1} \}\), \(x_{2}\in K_{\zeta_{2},L}=\{x \in K, \zeta_{2}\leq\|x\| \leq L \}\). Hence, since for \(x_{i}\in K\) we have \(x_{i}(t)\geq \delta\|x_{i}\|\), \(t \in J\), \(i=1,2\), it follows that (3.20) holds.

Similar to the proof of (3.4) and (3.5), one can show that (3.21) and (3.22) hold. □

In Theorem 3.4 we assume the following condition on \(F(t,u,v)\):

(H₆):

There exist two constants r, R with \(0< r< \delta R\) such that

$$\begin{aligned}& F(t,u,v)\leq\frac{1}{\eta}r, \quad \forall(t,u,v)\in J\times \bigl[\rho _{1}\phi_{q}(\delta r),\rho_{2} \phi_{q}(r) \bigr]\times \bigl[-\phi_{q}(r),-\phi _{q}(\delta r) \bigr]; \\& F(t,u,v)\geq\frac{1}{\rho\beta}R, \quad \forall(t,u,v)\in J\times \bigl[ \rho_{1}\phi_{q}(\delta R),\rho_{2} \phi_{q}(R) \bigr]\times \bigl[-\phi _{q}(R),- \phi_{q}(\delta R) \bigr]. \end{aligned}$$

Theorem 3.4

Assume that (H₁)-(H₂) and (H₆) hold. Then we have the following conclusions:

1. (i)

   Problem (2.5) has (at least) one positive solution \(x\in K\) such that

   $$ \delta r\leq x(t)\leq R,\quad t\in J. $$

   (3.26)
2. (ii)

   Problem (1.5) has (at least) one positive solution y such that (3.4) holds. We further have

   $$ \sigma\phi_{q}(r)\leq y(t) \leq\rho_{2} \phi_{q}(R), \quad t\in J. $$

   (3.27)

Proof

For \(x\in K\) with \(\|x\|=r\), we have \(x(s)\in[\delta r,r]\), and

Then, for \(t\in J\), we have

i.e. \(x\in\partial\Omega_{1}\) implies

On the other hand, for \(x\in K\) with \(\|x\|=R\), we have \(x\in[\delta R,R]\), and \(\phi_{q}(x)\in[-\phi_{q}(R), -\phi_{q}(\delta R)]\), \(\int _{0}^{1}H_{1}(s,\tau)\phi_{q}(x(\tau))\,d\tau\in[\rho_{1}\phi_{q}(\delta R),\rho_{2}\phi_{q}(R)]\), \(\forall s\in J\). Then, for \(t\in J\), we have

i.e. \(x\in\partial\Omega_{2}\) implies

Applying Lemma 2.6 to (3.28) and (3.29) shows that T has a fixed point \(x\in K\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\) with (3.26). It is observed from (2.2) that problem (1.5) has one solution y such that

 □

We remark that the condition (H₆) in Theorem 3.4 can be replaced by the following condition:

(H₇):

There exist two constants r, R with \(0< r<\delta R\) such that

$$\begin{aligned}& F(t,u,v)\geq\frac{1}{\rho\beta}r, \quad \forall(t,u,v)\in J\times \bigl[ \rho_{1}\phi_{q}(\delta r),\rho_{2} \phi_{q}(r) \bigr]\times \bigl[-\phi_{q}(r),\phi _{q}(\delta r) \bigr]; \\& F(t,u,v)\leq\frac{1}{\eta}R, \quad \forall(t,u,v)\in J\times \bigl[\rho _{1}\phi_{q}(\delta R),\rho_{2} \phi_{q}(R) \bigr]\times \bigl[-\phi_{q}(R),\phi _{q}(\delta R) \bigr]. \end{aligned}$$

Corollary 3.1

If the condition (H₆) in Theorem  3.4 is replaced by (H₇), respectively, then the conclusions of Theorem  3.4 also hold.

In the previous section, we have obtained some results on the existence of at least two positive solutions for problem (1.5). In this section, we will further discuss the existence of two positive solutions for problem (1.5) in the more general case.

For \(x\in K\), we define the nonnegative increasing continuous functionals θ, \(\theta_{1}\) and \(\theta_{2}\) by

and

It is easy to see that, for each \(x\in K\),

In addition, for each \(x\in K\), \(\theta(x)=x(1)\geq\delta\|x\|\), which implies that

Finally, we also note that

where \(r>0\).

For notational convenience, we denote

We now present the results in this section.

Theorem 4.1

Suppose that there exist constants \(0< a< b< c\) such that

Assume F satisfies the following conditions:

(C₁):

\(F(t,u,v)>\frac{c}{N}\) for \((t,u,v)\in J\times[\rho _{1}\phi_{q}(c),\rho_{2}\phi_{q}(\frac{c}{\delta})]\times[-\phi _{q}(\frac{c}{\delta}),-\phi_{q}(c)]\);

(C₂):

\(F(t,u,v)<\frac{b}{N_{1}}\) for \((t,u,v)\in J\times[0,\rho _{2}\phi_{q}(\frac{b}{\delta})]\times[-\phi_{q}(\frac{b}{\delta}),0]\);

(C₃):

\(F(t,u,v)>\frac{a}{N} for (t,u,v)\in J\times[\rho _{1}\phi_{q}(\delta a),\rho_{2}\phi_{q}(a)]\times[-\phi_{q}(a),-\phi _{q}(\delta a)]\).

1. (i)

   Problem (2.5) has at least two positive solutions \(x_{1}\) and \(x_{2}\) such that

   $$ a< \theta_{2}(x_{1}), \quad \textit{with } \theta_{1}(x_{1})< b, \quad\textit{and}\quad b< \theta_{1}(x_{2}) \quad \textit{with } \theta(x_{2})< c. $$

   (4.3)
2. (ii)

   Problem (1.5) has (at least) two positive solutions \(y_{1}\) and \(y_{2}\) such that for \(i=1,2\)

   $$ \left \{ \textstyle\begin{array}{@{}l} y_{i}(t)=\int_{0}^{1}H_{1}(t,s)\phi_{q}(x_{i}(s))\,ds, \quad t\in J;\\ \|y_{i}\|\leq\rho_{2}\phi_{q}(\|x_{i}\|);\\ y_{i}(t)\geq\sigma\phi_{q}(\|x_{i}\|), \quad t\in J. \end{array}\displaystyle \right . $$

   (4.4)

We further have

Proof

By the definition of operator T and its properties, it suffices to show that the conditions of Lemma 2.7 hold with respect to T.

We first show that if \(x\in\partial K(\theta,c)\), then \(\theta(Tx)>c\).

In fact, if \(x\in\partial K(\theta,c)\), then \(\theta(x)=\min_{t\in J}x(t)=x(1)=c\). Since \(x\in K\), one can get \(x(t)\geq c\) for \(t\in J\). Noticing (4.2), we have

which shows that

As a consequence of (C₁)

Also, \(Tx\in K\), and hence we get

Next, we verify that \(\theta_{1}(Tx)< b\) for \(x\in\partial K(\theta_{1},b)\).

So, let us choose \(x\in\partial K(\theta_{1},b)\), then

This implies that \(0\leq x(t)\leq b\) for \(t\in[0,\frac{1}{2}]\), and, since \(x\in K\), we also have

which implies that

and then

It follows from (C₂) that

Noticing \(Tx\in K\), we obtain

Finally, we prove that \(K(\theta_{2},a)\neq{\emptyset}\) and \(\theta_{2}(Tx)>a\) for all \(x\in\partial K(\theta_{2},a)\).

In fact, the constant \(\frac{a}{3}\in K(\theta_{2},a)\). Moreover, for \(x\in\partial K(\theta_{2},a)\), we have

which implies that

and then

Using assumption (C₃), we have

As before \(Tx\in K\), and so

Thus, by Lemma 2.7, there exist at least two fixed points of T which are positive solutions \(x_{1}\) and \(x_{2}\), belonging to \(\overline{K(\theta,c)}\), of problem (2.5) such that

Noticing the definition of θ and \(\theta_{2}\), one can show that (4.4) and (4.5) hold. □

In this section, we shall study the existence of three positive solutions of problem (1.5). Let σ, N and \(N_{1}\) be defined in Theorem 3.1 and Theorem 4.1, respectively.

Theorem 5.1

Assume that (H₁)-(H₂) hold. Suppose that there exist positive constants \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\) with

such that the following hold:

(A₁):

\(F(t,u,v)\leq\frac{1}{N_{1}}\omega_{3}\) for \((t,u,v)\in J\times[0,\rho_{2}\phi_{q}(\omega_{3})]\times[-\phi _{q}(\omega_{3}),0]\);

(A₂):

\(F(t,u,v)> \frac{1}{N}\omega_{2}\) for \((t,u,v)\in J\times[\rho_{1}\phi_{q}(\omega_{2}),\rho_{2}\phi_{q}(\frac{1}{\delta }\omega_{2})]\times[-\phi_{q}(\frac{1}{\delta}\omega_{2}),-\phi _{q}(\omega_{2})]\);

(A₃):

\(F(t,u,v)< \frac{1}{N_{1}}\omega_{1}\) for \((t,u,v)\in J\times[0,\rho_{2}\phi_{q}(\omega_{1})]\times[-\phi _{q}(\omega_{1}),0]\).

1. (i)

   Problem (2.5) has (at least) three positive solutions \(x_{1},x_{2},x_{3}\in K\) such that

   $$ \left \{ \textstyle\begin{array}{@{}l} \|x_{1}\|< \omega_{1}; \\ x_{2}(t)>\omega_{2}, \quad t\in J;\\ \|x_{3}\|>\omega_{1} \textit{ and }\min_{t\in J}x_{3}(t)< \omega_{2}. \end{array}\displaystyle \right . $$

   (5.1)
2. (ii)

   Problem (1.5) has (at least) three positive solutions \(y_{1}\), \(y_{2}\), \(y_{3}\) such that, for \(i=1,2,3\),

   $$ \left \{ \textstyle\begin{array}{@{}l} y_{i}(t)=\int_{0}^{1}H_{1}(t,s)\phi_{q}(x_{i}(s))\,ds, \quad t\in J;\\ \|y_{i}\|\leq\rho_{2}\phi_{q}(\|x_{i}\|);\\ y_{i}(t)\geq\rho_{1}\phi_{q}(\delta\|x_{i}\|), \quad t\in J. \end{array}\displaystyle \right . $$

   (5.2)

We further have

Proof

Let \(x\in\bar{K}_{\omega_{3}}\), so \(0\leq\phi _{q}(x(s))\leq\phi_{q}(\omega_{3})\), \(s\in J\), and \(0\leq\int _{0}^{1}H_{1}(s,\tau)\phi_{q}(x(\tau))\,d\tau\leq\rho_{2}\phi_{q}(\omega _{3})\). By (A₁), we see that

This proves that \(T:\bar{K}(\omega_{3})\rightarrow\bar{K}(\omega_{3})\).

For convenience, we denote \(\omega_{4}=\frac{\omega_{2}}{\delta}\). Let \(\psi: K\rightarrow[0,+\infty)\) be defined by

Clearly, ψ is a nonnegative continuous concave functional on K and \(\psi(x)\leq\|x\|\) for all \(x\in K\).

Now we need to show that condition (i) of Lemma 2.8 is satisfied. It is obvious that

Then

Next, let \(x\in K(\psi,\omega_{2},\omega_{4})\). Then for \(s\in J\), we have \(\omega_{2}\leq x(s)\leq\omega_{4}\), and

Using condition (A₂), it follows that

Therefore, we have shown that \(\psi(Tx)>\omega_{2}\) for all \(x\in K(\psi ,\omega_{2},\omega_{4})\).

Next, we shall show that condition (ii) of Lemma 2.8 holds. Take \(x\in\bar{K}(\omega_{1})\) with \(\|x\|<\omega_{1}\). Then for \(s\in J\), we have \(x(s)\in[0,\omega_{1}]\) and

By condition (A₃), we have

Hence, we have proved that \(\|Tx\|<\omega_{1}\) for all \(x\in\bar{K}(\omega_{1})\).

Finally, we need to prove that condition (iii) of Lemma 2.8 is satisfied. Let \(x\in K(\psi,\omega_{2},\omega_{3})\) with \(\|Tx\|>\omega _{4}\). Then it follows from (2.16) that

So, we have proved that \(\psi(Tx)>\omega_{2}\) for all \(x\in K(\psi ,\omega_{2},\omega_{3})\) with \(\|Tx\|>\omega_{4}\).

It now follows from Lemma 2.8 that problem (2.5) has (at least) three positive solutions \(x_{1}, x_{2}, x_{3}\in\bar{K}(\omega_{3})\) satisfying (5.1).

It is observed from (2.2) that problem (1.5) has (at least) three positive solutions \(y_{1}\), \(y_{2}\), \(y_{3}\) such that, for \(i=1,2,3\),

Moreover, since \(x_{i}\in K\), we get for \(t\in J\),

Then we get (5.2).

Further, since \(x_{2}(t)>\omega_{2}\) for \(t\in J\), we have, for \(t\in J\),

Hence, we get (5.3). □

Remark 5.1

Leggett-Williams’ fixed point theorem is a valid tool to show the existence of multiple positive solutions for various boundary value problems where the nonlinear terms F is of the form \(F(t,y)\); see for example [25–27]. If F is of the form \(F(t,y,y'')\), then it is very difficult to obtain the existence result of three positive solutions; see [20, 21] to name a few. In [28], using the five functionals fixed point theorem (which is a generalization of the Leggett-Williams’ fixed point theorem), Avery et al. showed the existence of at least three positive solutions in the case that F is the form of \(F(y,-y'')\).

To illustrate how our main results can be used in practice, we present an example.

Let \(a=1\), \(b=\frac{1}{2}\) and \(g(t)=h(t)=\frac{1}{2}\). Consider the following fourth order p-Laplacian elasticity problem:

and the nonlinear term F given by

where

and the sets \(D_{i} \) (\(i=1,2,\ldots,7\)) are defined by

and \(N=\rho\beta\), \(N_{1}=\eta\). For details, see Figure 1.

The relationship of \(\pmb{D_{i}}\) , \(\pmb{i=1,2,3,4,5,6,7}\) .

Full size image

By calculation we obtain

Therefore, it follows from the definitions F, g and, h that (H₁)-(H₂) hold.

Choosing \(\omega_{1}=\phi_{p}(\frac{\delta\rho_{1}}{2\rho_{2}})\omega_{2}\), \(\omega_{3}=\frac{5N_{1}}{N}\omega_{2}\), then \(\omega_{i}\) (\(i=1,2,3\)) satisfy

Then, for \(t\in J\), \((u,v)\in D_{1}\), we have

For \(t\in J\), \((u,v)\in D_{2}\cup D_{3}\cup D_{4}\), we have

For \(t\in J\), \((u,v)\in D_{5}\), we have

For \(t\in J\), \((u,v)\in D_{6}\), we have

For \(t\in J\), \((u,v)\in D_{7}\), we have

We shall check the conditions of Theorem 5.1. First, condition (A₃) is obviously satisfied by (6.2). Next, from (6.3)-(6.6), it follows that

Hence, condition (A₁) holds. Finally, by (6.3), we see that

Condition (A₂) is satisfied.

It follows from Theorem 5.1 that problem (6.1) has (at least) three positive solutions \(y_{1}\), \(y_{2}\), \(y_{3}\) such that

 □

This work is sponsored by the project NSFC (11301178) and the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58). The authors are grateful to the anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.

Affiliations

1. Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, People’s Republic of China
   • Yaling Zhou
   •  & XueMei Zhang

Corresponding author

Correspondence to XueMei Zhang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All results belong to YZ and XZ. All authors read and approved the final manuscript.

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