Existence of multiple positive solutions for p-Laplacian multipoint boundary value problems on time scales

The theory of dynamic equation on time scales (or measure chains) was initiated by Stefan Hilger in his PhD thesis in 1988 [1] (supervised by Bernd Aulbach) as a means of unifying structure for the study of differential equations in the continuous case and study of finite difference equations in the discrete case. In recent years, it has gained a considerable amount of interest and attracted the attention of many researchers, see, for example, [2–13]. It is still a new area, and the research in this area is rapidly growing. The study of time scales has led to several important applications, e.g., in the study of insect population models, heat transfer, neural networks, phytoremediation of metals, wound healing, and epidemic models.

p-Laplacian equations for boundary value problems (BVPs) with nonlinearity depending on the first order derivative have been studied extensively, see [14–20] and references therein. However, there are few papers concerning p-Laplacian equations with nonlinearity depending on the first order derivative for BVPs on time scales, see [5, 7].

where ${\varphi }_p(u)$ is a p-Laplacian operator, i.e., ${\varphi }_p(u)=|u{|}^{p-2}u$, for $p>1$, with ${({\varphi }_p)}^{-1}={\varphi }_q$ and $\frac{1}{p}+\frac{1}{q}=1$. The usual notation and terminology for time scales as can be found in [21, 22], will be used here. The interesting point is that the nonlinear term f is involved with the first order derivative explicitly and the main tool is a fixed point theorem due to Bai and Ge. The results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting.

where ${\varphi }_p(u)$ is p-Laplacian operator, i.e., ${\varphi }_p(u)=|u{|}^{p-2}u$, for $p>1$, with ${({\varphi }_p)}^{-1}={\varphi }_q$ and $1/p+1/q=1$.

Recently, much attention has been focused on the study of multipoint positive solutions of BVPs on time scales. When the nonlinear term f does not depend on the first order derivative, many researchers study multipoint boundary conditions on time scales, see [11, 12, 23–26]. However, little work has been done on the existence of positive solutions for multipoint BVP on time scales when the nonlinear term is involved in the first order derivative explicitly, see [5].

where $\varphi :R\to R$ is an increasing homeomorphism and a positive homomorphism and $\varphi (0)=0$.

All the above-mentioned works about positive solutions were done under the assumption that f is allowed to depend just on u, while the first order derivative $u^{\mathrm{\Delta }}$ is not involved explicitly in the nonlinear term f.

where $\varphi :R\to R$ is an increasing homeomorphism and a homomorphism and $\varphi (0)=0$.

In recent years, much attention has been paid to the existence of positive solutions for nonlinear boundary value problems on time scales, see [11–13, 26–29] and the reference therein. On the other hand, multipoint nonlinear boundary value problems with p-Laplacian operators on time scales have also been studied extensively in the literature [26, 30–33]. However, to the best of our knowledge, there are few works on the increasing homeomorphism and homomorphism on time scales [34].

where ${\phi }_p(u)={|u|}^{p-2}u$, for $p>1$. ${\psi }_i:\mathbb{R}\to \mathbb{R}$ is continuous and nondecreasing $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{m-2}<\rho (T)$. By using the well-known Schauder fixed point theorem and upper and lower solution method, they obtained some new existence criteria for positive solutions of the boundary value problem.

where $\phi :\mathbb{R}\to \mathbb{R}$ is an increasing homeomorphism and a positive homomorphism and $\phi (0)=0$. They obtained the countably many positive solutions by using a fixed point index theory and fixed point theorem.

This work is motivated by recent papers [26, 34]. Existence of at least two positive solutions to BVP (1.5) and (1.6) are established by means of fixed point index theory. We also point out that when $\mathbb{T}=R$, $p=2$, (1.5) and (1.6) becomes a boundary value problem of differential equations and is just the problem considered in [35]. Our main results improve and extend the main results of [35, 36].

In this section, we provide some background materials from the theory of cones in Banach spaces. The following definitions can be found in the book by Deimling [37], as well as in the book by Guo and Lakshmikantham [38].

Every cone $P\subset E$ induces an ordering in E given by $x\le y$ if and only if $y-x\in P$.

for all $x,y\in P$ and $t\in [0,1]$.

for all $x,y\in P$ and $t\in [0,1]$.

Let ψ be a nonnegative continuous concave functional on P, and α and β be nonnegative continuous convex functionals on P.

To prove our main results, we need the following fixed point theorem, which comes from Bai and Ge in [14].

Lemma 2.1 [14]

Let P be a cone in a real Banach space E. Assume that constants $r_1$, b, d, $r_2$, $l_1$ and $l_2$ satisfy $0<r_1<b<d\le r_2$ and $0<l_1\le l_2$. If there exist two nonnegative continuous convex functionals α and β on P and a nonnegative continuous concave functional ψ on P such that

(A1) there exists $M>0$ such that $\parallel u\parallel \le Mmax\{\alpha (u),\beta (u)\}$ for all $u\in P$;

(A2) $P(\alpha ,r;\beta ,l)\ne \mathrm{\varnothing }$ for any $r>0$ and $l>0$;

(A3) $\psi (u)\le \alpha (u)$ for all $u\in \overline{P}(\alpha ,r_2;\beta ,l_2)$;

and if $F:\overline{P}(\alpha ,r_2;\beta ,l_2)\to (\alpha ,r_2;\beta ,l_2)$ is completely continuous operator, which satisfies

(B1) $\{u\in \overline{P}(\alpha ,d;\beta ,l_2;\psi ,b):\psi (u)>b\}\ne \mathrm{\varnothing }$, $\psi (Fu)>b$ for $u\in \overline{P}(\alpha ,d;\beta ,l_2;\psi ,b)$;

(B2) $\alpha (Fu)<r_1$, $\beta (Fu)<l_1$ for $u\in \overline{P}(\alpha ,r_1;\beta ,l_1)$;

(B3) $\psi (Fu)>b$ for $u\in \overline{P}(\alpha ,r_2;\beta ,l_2;\psi ,b)$ with $\alpha (Fu)>d$.

Clearly, P is a cone.

Throughout the section, we suppose that the following conditions are satisfied.

(H1) $0,T\in \mathbb{T}$, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{n-2}<\rho (T)$, ${\xi }_i\in \mathbb{T}$, ${\alpha }_i,{\beta }_i\in [0,\mathrm{\infty })$ satisfy $0\le {\sum }_{i=1}^{n-2}{\alpha }_i<1$ and $0\le {\sum }_{i=1}^{n-2}{\beta }_i<1$;

(H2) $\eta =min\{t\in \mathbb{T}:\frac{T}{2}\le t<T\}$ exists;

(H3) $q(t)\in C_{\mathrm{ld}}({[0,T]}_{\mathbb{T}},[0,\mathrm{\infty }))$ with $0<{\int }_{\eta }^{T}q(t)\mathrm{\Delta }t<\mathrm{\infty }$;

(H4) $f:{(0,T)}_{\mathbb{T}}\times [0,\mathrm{\infty })\times \mathbb{R}\to [0,\mathrm{\infty })$ is continuous;

(H5) $q(t)f(t,0,0)\not\equiv 0$, $f(t,0,0)\ge 0$ on ${[0,T]}_{\mathbb{T}}$.

Lemma 3.2 The solution of BVP (3.1) and (3.2) satisfies $u\ge 0$, for $t\in {[0,T]}_{\mathbb{T}}$.

This completes the proof. □

for $t\in {[0,T]}_{\mathbb{T}}$. By the definition of F, the monotonicity of ${\varphi }_q(u)$ and the assumptions (H1)-(H5), it is easy to see that for each $u\in P$, $Fu\in P$ and $Fu(T)$ is the maximum value of $Fu(t)$. Moreover, by direct calculation, we get that each fixed point of the operator F in P is a positive solution of the BVP (1.1) and (1.2). It is easy to see that $F:P\to P$ is completely continuous.

It is easy to see that $\alpha ,\beta :P\to [0,\mathrm{\infty })$ are nonnegative continuous convex functionals with $\parallel u\parallel =max\{\alpha (u),\beta (u)\}$; $\psi :P\to [0,\mathrm{\infty })$ is nonnegative concave functional. We have $\psi (u)\le \alpha (u)$ for $u\in P$ and the assumptions (A1), (A2) and (A3) in Lemma 2.1 hold.

Theorem 3.1 Assume that (H1)-(H5) hold, and there exists $0<r_1<b<2b\le r_2$, $0<l_1\le l_2$ such that $\frac{b}{\lambda }\le min\{r_2/L,l_2/Q\}$. If f satisfies the following conditions:

(D1) $f(t,w,v)\le min\{{\varphi }_p(r_2/L),{\varphi }_p(l_2/Q)\}$ for $(t,w,v)\in {[0,T]}_{\mathbb{T}}\times [0,r_2]\times [-l_2,l_2]$;

(D2) $f(t,w,v)>{\varphi }_p(b/\lambda )$ for $(t,w,v)\in {[\eta ,T]}_{\mathbb{T}}\times [b,2b]\times [-l_2,l_2]$;

(D3) $f(t,w,v)<min\{{\varphi }_p(r_1/L),{\varphi }_p(l_1/Q)\}$ for $(t,w,v)\in {[0,T]}_{\mathbb{T}}\times [0,r_1]\times [-l_1,l_1]$;

Proof In order to show that Lemma 2.1 holds, it is sufficient to show that the conditions in Lemma 2.1 are satisfied with respect to operator F.

Therefore, $F:\overline{P}(\alpha ,r_2;\beta ,l_2)\to \overline{P}(\alpha ,r_2;\beta ,l_2)$.

We can get that $F:\overline{P}(\alpha ,r_1;\beta ,l_1)\to P(\alpha ,r_1;\beta ,l_1)$.

So condition (B2) of Lemma 2.1 is satisfied.

So, for $u\in \overline{P}(\alpha ,2b;\beta ,l_2;\psi ,b)$, there are $b\le u(t)\le 2b$ and $|u^{\mathrm{\Delta }}(t)|\le l_2$ for $t\in {[\eta ,T]}_{\mathbb{T}}$.

So, we get $\psi (Fu)>b$ for $u\in \overline{P}(\alpha ,2b;\beta ,l_2;\psi ,b)$, and condition (B1) of Lemma 2.1 holds.

Finally, we prove that condition (B3) of Lemma 2.1 holds.

Otherwise, as $u_3$ satisfies $\alpha (u_3)\le 2\psi (u_3)$, we have ${max}_{t\in {[0,T]}_{\mathbb{T}}}{u}_3(t)<2b$. □

Clearly, the assumptions (H1)-(H5) hold, and $f(t,0,0)\not\equiv 0$ on ${[0,2]}_{\mathbb{T}}$.