- Research
- Open Access
- Published:
Solvability for a fractional p-Laplacian multipoint boundary value problem at resonance on infinite interval
Advances in Difference Equations volume 2016, Article number: 183 (2016)
Abstract
In this paper, we study the multipoint boundary value problem for a fractional p-Laplacian equation at resonance on infinite interval and establish the existence result of solutions by using extension of Mawhin’s continuation theorem. Our paper enriches some known existing articles. In order to illustrate our main result, we give an example.
1 Introduction
Many boundary value problems (BVPs) on infinite interval arise naturally in the study of radially symmetric solutions of nonlinear elliptic equations (see [1]). There are numerous physical models giving us motivations to investigate BVPs on infinite interval, such as the following two important examples.
The first model is for determining the electric potential in an isolated atom derived independently by Thomas and Fermi:
Another famous model is the well-known Blasius boundary layer equation that shows flow past a semiinfinite flat plate in hydromechanics:
Although the Blasius equation is simple, it can clearly reveal the essence of the problems, which is convenient for people to carry out theoretic analysis and mathematical research for boundary layer problem. So, BVPs on infinite interval have important significance and have been received much attention (see [1]). There are a large number of papers discussing the existence of solutions for both integral-order and fractional-order differential BVPs on infinite interval by using the techniques of nonlinear analysis such as variational method (see [2]), fixed-point theorems (see [3–8]), upper and lower solutions method (see [9, 10]), fixed-point index theory (see [11, 12]), coincidence degree theory (see [13–15]), etc.
Jiang [13] proved the existence of solutions for the p-Laplacian boundary value problem
where \({\phi_{p}}(s) = \vert s\vert ^{p - 2}s\), \({\phi_{p}}(0) = 0\), \(p > 1\). The main method of the paper was the coincidence degree theory.
Su and Zhang [6] investigated the existence of unbounded solutions of boundary value problem
where \(D_{0 + }^{\alpha}\) and \(D_{0 +}^{\alpha-1}\) are the standard Riemann-Liouville fractional derivatives of order \(1 < \alpha \leq2\). The main result of this paper was obtained by using Schauder’s fixed point theorem.
Zhou, Kou, and Xie (see [15]) studied the existence of solutions for the following multipoint boundary value problem:
where \(D_{0 + }^{\alpha}\) is the standard Riemann-Liouville fractional derivative of order \(1 < \alpha \leqslant2\), \(\eta > 0\). The analysis of this paper relied on the coincidence degree of Mawhin.
Motivated by the results mentioned, in this paper, we use the extension of Mawhin’s continuation theorem (see [16]) to discuss the existence of solutions for the following multipoint boundary value problem of fractional p-Laplacian equation at resonance:
where \(1 < \alpha \le2\), \(D_{0 + }^{\alpha}\) is the standard Riemann-Liouville fractional derivative, \(0 < {\xi_{1}} < {\xi _{2}} < \cdots < {\xi_{n}} < + \infty\), \({\alpha_{i}} > 0\), \(\sum_{i = 1}^{n} {\alpha_{i}= 1} \), \({\phi_{p}}\) is reversible, and by \({\phi_{q}}\) we denote the inverse operator of \({\phi_{p}}\), where \(1 /p + 1/q = 1\).
As we know, fractional differential equations have been applied in various fields (see [17, 18]). So, it is meaningful to discuss the boundary value problems of fractional differential equations on infinite interval.
Throughout this paper, we suppose that the following hypothesis is satisfied:
- (\(A_{1}\)):
-
\(f \in C [ {0, + \infty} ) \times\mathbb{R}^{3} \to\mathbb {R}\) is an \(L^{1}\)-Caratheodory function, that is, f is a Caratheodory function, and for any \(r > 0\), there exists a nonnegative function \({g_{r}}(t) \in{L^{1}}[0, + \infty)\) such that
$$\bigl\vert {f(t,u,v,w)} \bigr\vert \le{g_{r}}(t),\quad \mbox{a.e. } t \in[ {0, + \infty} ),u,v,w \in\mathbb{R}, \Vert u \Vert \leq r,\Vert v \Vert \leq r,\Vert w \Vert \leq r. $$
2 Preliminaries
In this section, we introduce some definitions and lemmas.
Definition 2.1
The Rieman-Liouville fractional integral of order \(\alpha > 0\) for a function \(u:(0, + \infty) \to\mathbb{R} \) is defined as
where \(\Gamma(\alpha)\) is the gamma function, provided that the right-hand side is pointwise defined on \((0, + \infty)\).
Definition 2.2
The Riemann-Liouville functional derivative of order \(\alpha > 0\) for a function \(u:(0, + \infty) \to\mathbb{R}\) is defined as
where \(n = [ \alpha]+1\), provided that the right-hand side is pointwise defined on \((0, + \infty)\).
Lemma 2.1
Assume that \(u \in C[0,\infty) \cap L(0,\infty)\) with a fractional derivative of order \(\alpha > 0\) that belongs to \(C[0,\infty) \cap L(0,\infty)\). Then
for some \(c_{i} \in R\), \(i = 1,2, \ldots,n\), \(n = [ \alpha]+1\).
Definition 2.3
Let X and Y be two Banach spaces with norms \({\Vert \cdot \Vert _{X}}\) and \({\Vert \cdot \Vert _{Y}}\), respectively. A continuous operator \(M:X \cap\operatorname{dom}M \to Y\) is said to be quasi-linear if
-
(a)
\(\operatorname{Im}M: = M(X \cap\operatorname{dom}M)\) is a closed subset of Y, and
-
(b)
\(\operatorname{Ker}M: = \{ x \in X \cap\operatorname{dom}M: Mx = 0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\), \(n < \infty\).
Take \(X_{1}=\operatorname{Ker}M\) and let \(X_{2}\) be the complement space of \(X_{1}\) in X, so that, \(X =X_{1} \oplus X_{2}\). On the other hand, suppose that \(Y_{1}\) is a subspace of Y and \(Y_{2}\) is the complement space of \(Y_{1}\) in Y, so that \(Y =\mathrm{Y}_{1} \oplus Y_{2}\). Let \(P:X \to X_{1}\) and \(Q:Y \to Y_{1}\) be two projectors, and \(\Omega \subset X\) be an open bounded set with the origin \(\theta \in\Omega\). Throughout we use θ to denote the origin in a linear space (see [13, 16]).
Definition 2.4
Suppose that \(N_{\lambda}:\bar{\Omega}\to Y\), \(\lambda \in[0,1]\), is a continuous operator. Denote \(N_{1}\) by N. Let \(\sum_{\lambda}{ = \{ x \in\bar{\Omega}:Mx = N_{\lambda}x\} }\). \(N_{\lambda}\) is said to be M-compact in Ω̄ if there is a vector subspace \(Y_{1}\) of Y with \(\operatorname{dim}Y_{1} = \operatorname{dim}X_{1}\) and a continuous and compact operator \(R:\bar{\Omega}\times [0,1] \to X_{2}\) such that, for \(\lambda \in[0,1]\),
- (\(a_{1}\)):
-
\((I - Q)N_{\lambda}(\bar{\Omega}) \subset\operatorname{Im}M \subset(I - Q)Y\);
- (\(a_{2}\)):
-
\(QN_{\lambda}x = \theta\), \(\lambda \in(0,1) \Leftrightarrow QNx = \theta\);
- (\(a_{3}\)):
-
\(R( \cdot,0)\) is the zero operator, and \(R( \cdot,\lambda)\vert _{\sum{_{\lambda}} } = (I - P)\vert _{\sum{_{\lambda}} } \);
- (\(a_{4}\)):
-
\(M[P + R( \cdot,\lambda)] = (I - Q)N_{\lambda}\).
Lemma 2.2
(Extension of Mawhin’s continuation theorem)
Let X and Y be two Banach spaces with the norms \(\Vert \cdot \Vert _{X}\) and \(\Vert \cdot \Vert _{Y}\) respectively, and \(\Omega \subset X\) be an open bounded nonempty set. Suppose that
is a quasi-linear operator and
is M-compact on Ω̄. In addition, let the following conditions hold:
- (\(B_{1}\)):
-
\(Mx \ne N_{\lambda}x\), \(\forall(x,\lambda) \in\partial\Omega\times(0,1)\),
- (\(B_{2}\)):
-
\(QNx \ne0\), \(\forall x \in\operatorname{Ker}M \cap\partial\Omega\),
- (\(B_{3}\)):
-
\(\deg(JQN,\operatorname{Ker}M \cap\Omega,0) \ne0\).
Then the abstract equation \(Mx = Nx\) has at least one solution in \(\operatorname{dom}M \cap\bar{\Omega}\), where \(N = N_{1}\), \(Q:Y \to\operatorname{Im}Q\) is a projector, \(J:\operatorname{Im}Q \to\operatorname{Ker}M\) is a homeomorphism with \(J(\theta) = \theta\) (see [16]).
3 Main result
In this section, we give a theorem on the existence of solutions for BVP (1.1). Let
with norms
where \(\Vert y \Vert _{1} = \int_{0}^{ + \infty} {\vert {y(t)} \vert \,dt} \), \(\Vert {D_{0 + }^{\alpha}x} \Vert _{\infty}= \sup_{t \in[0, + \infty)} \vert {D_{0 + }^{\alpha}x(t)} \vert \), \(\Vert x \Vert _{0} = \Vert {\frac{{x(t)}}{ {1 + t^{\alpha}}}} \Vert _{\infty}\). Clearly, \((X,\Vert \cdot \Vert _{X} ) \) and \((Y,\Vert \cdot \Vert _{Y} )\) are Banach spaces.
Define the operators \(M:\operatorname{dom}M \subset X \to Y\) and \(N_{\lambda}:X \to Y\) as follows:
where
Then the BVP (1.1) is equivalent to \(Mx=Nx\), \(x \in\operatorname{dom}M\).
Lemma 3.1
For M defined as before, we have
and M is a quasi-linear operator.
Proof
For \(x \in\operatorname{Ker}M\), \(Mx= 0\), that is, \((\phi_{p} (D_{0 + }^{\alpha}x))' = 0\), by \(\phi_{p} (D_{0 +}^{\alpha}x( + \infty)) = \sum_{i = 1}^{n} \alpha_{i} \phi_{p}(D_{0 +}^{\alpha}x(\xi_{i}))\) we easily get
Based on Lemma 2.1, since \(x(0) = x'(0) = 0\), we have
Conversely, if \(x = ct^{\alpha}\), then \(Mx = 0\) by (3.1). If \(y \in\operatorname{Im} M\), then there exists a function \(x \in \operatorname{dom}M\) such that \(y(t) = (\phi_{p} (D_{0 + }^{\alpha}x(t)))'\). Then
that is,
On the other hand, if \(y \in Y\) satisfies (3.3), then take
Then \(x \in\operatorname{dom} M\) and \(Mx = y\). Hence, (3.2) holds. Clearly, \(\operatorname{dim}\operatorname{Ker}M = 1<+\infty\) and \(\operatorname{Im}M:=M(\operatorname{dom}M \cap X)\) is a closed subset of Y. Therefore, we get that M is a quasi-linear operator. □
Lemma 3.2
(See [1])
Let \(V \subset C_{\infty}= \{u \in C[0,\infty), \lim_{t \to + \infty} u(t)\textit{ exists} \}\). Then V is relatively compact if the following conditions hold:
- (\(b_{1}\)):
-
all functions from V are uniformly bounded,
- (\(b_{2}\)):
-
all functions from V are equicontinuous on any compact interval of \([0,+\infty)\),
- (\(b_{3}\)):
-
all functions from V are equiconvergent at infinity.
Remark 3.1
By Lemma 3.2, any set \(V \subset X\) (defined as before) is relatively compact, we only need to show that the sets
are uniformly bounded in X, equicontinuous on any compact intervals of \([0, + \infty)\), and equiconvergent at infinity.
Lemma 3.3
Let \(\Omega \subset X\) be nonempty, open, and bounded. Then \(N_{\lambda}\) is M-compact on Ω̄.
Proof
Define two projectors \(P:X \to X_{1}\) and \(Q:Y \to Y_{1}\) by
where \(X_{1} =\operatorname{Ker}M\) and \(Y_{1} =\operatorname {Im} Q\).
First, we show that (\(a_{1}\)) and (\(a_{2}\)) in Definition 2.4 hold. In fact, by the definition of P we get \(\operatorname{Im} P=\operatorname{Ker}M\) and \(P^{2} x(t) = Px(t)\). For \(x \in X\), since \(x = (x - Px) + Px\) and \(\operatorname{Im} P=\operatorname{Ker}M\), we have \((x - Px) \in\operatorname{Ker}P\), \(Px \in\operatorname{Ker}M\). We easily see that \(\operatorname{Ker}M \cap\operatorname{Ker}P= \{ 0 \}\). So, \(X=\operatorname{Ker}M \oplus\operatorname{Ker}P=X_{1} \oplus X_{2}\). Similarly, by the definition of Q, we can obtain
and \(\operatorname{Ker}Q = \operatorname{Im} M\). For \(y \in Y\), since \(y = (y - Qy) + Qy\) and \(\operatorname{Ker}Q = \operatorname{Im} M\), we have \((y - Qy) \in\operatorname{Ker}Q\), \(Qy\in\operatorname{Im} M\). Clearly, \(\operatorname{Im} Q \cap\operatorname{Im} M = \{ 0 \}\). So, we have \(Y = \operatorname{Im} Q \oplus\operatorname{Im} M = Y_{1} \oplus Y_{2}\) and \(\operatorname{dim} X_{1} = \operatorname{dim}\operatorname{Ker}M = \operatorname{dim} \operatorname{Im} Q = \operatorname{dim} Y_{1}\), where \(X_{2} =\operatorname{Ker}P\), \(Y_{2} = \operatorname{Im} M\). Let \(\Omega \subset X\) be open bounded, and let \(\theta \in\Omega\). On the one hand, for \(x \in\bar{\Omega}\), since \(Q(I - Q)\) is a zero operator, we have \(Q[(I - Q)N_{\lambda}x] = 0\); thus, \((I - Q)N_{\lambda}x \in\operatorname{Ker}Q=\operatorname{Im} M\), that is, \((I - Q)N_{\lambda}(\bar{\Omega}) \subset\operatorname{Im} M\). On the other hand, for \(y \in\operatorname{Im} M\), since \(y = (y - Qy) + Qy\) and \(\operatorname{Ker}Q=\operatorname{Im} M\), we have \(y \in(I - Q)Y\), that is, \(\operatorname{Im} M \subset(I - Q)Y\). Clearly, \(QN_{\lambda}x = 0\), \(\lambda \in(0,1) \Leftrightarrow QNx = 0\). So, conditions (\(a_{1}\)) and (\(a_{2}\)) of Definition 2.4 hold.
Second, we give the definition of operator R and aim to show that R is compact. For notational convenience, let
Define the operator \(R:\bar{\Omega}\times[0,1] \to X_{2}\) by
By (\(A_{1}\)) it is easy to know that \(R(x,\lambda)(t)\) is continuous on \(\bar{\Omega}\times[0,1]\).
Step 1. We prove that \(R(x,\lambda)(\bar{\Omega})\) is both uniformly bounded in X and equicontinuous on any compact interval of \([0, + \infty)\). In fact, since \(\Omega \subset X\) is nonempty, open, and bounded, by (\(A_{1}\)) there exist a constant \(r > 0\) and a nonnegative function \(g_{r} (t) \in L^{1} [0, + \infty)\) such that
Since
we have
Therefore, for any \(x \in\bar{\Omega}\), we have
and
That is, \(R(x,\lambda)(\bar{\Omega})\) is uniformly bounded in X. Next, we show that \(R(x,\lambda)(\bar{\Omega})\) is equicontinuous on any compact interval of \([0, + \infty)\). In fact, for any \(K > 0\), \(t_{1},t_{2} \in[0,K]\), \(x \in\bar{\Omega}\), \(\lambda \in [0,1]\), we have
So, \(\{ {\frac{{R(x,\lambda)(t)}}{ {1 +t^{\alpha}}},x \in\bar{\Omega}} \}\) is equicontinuous on \([0,K]\). Similarly, we obtain that \(\{ {\frac{{D_{0 + }^{\alpha - 1} R(x,\lambda)(t)}}{ {1 +t^{\alpha}}},x \in\bar{\Omega}} \}\) is equicontinuous on \([0,K]\). In addition, since
and
we have
By the absolute continuity of the integral, \(\{ {l(t,x,\lambda),x \in \bar{\Omega}} \}\) is equicontinuous on \([0,K]\), which combined with the uniform continuity of \(\phi_{q} (x)\) on \([ - \tilde{r} - \phi_{p} (r),\tilde{r} + \phi_{p} (r)]\), gives that \(\{ {D_{0 + }^{\alpha}R(x,\lambda)(t),x \in\bar{\Omega}} \}\) is equicontinuous on \([0,K]\).
Step 2. We establish the fact that \(R(x,\lambda)(\bar{\Omega})\) is equiconvergent at infinity. In fact, for any \(x \in\bar{\Omega}\), by (3.4) we have
Since \(\phi_{q} (x)\) is uniformly continuous on \([ - \tilde{r} - \phi_{p} (r),\tilde{r} + \phi_{p} (r)]\), for any \(\varepsilon > 0\), there exists a constant \(L_{1} > 0\) such that, for \(s \geq L_{1} \), we have
Therefore, \(\vert {h(s)} \vert < \varepsilon\) for \(s \geq L_{1}\) and \(\vert {h(s)} \vert \leq m\) for \(s < L_{1}\). On the other hand, since \(\lim_{t \to + \infty} \frac {{t^{\alpha - 1} }}{ {1 + t^{\alpha}}} = 0\) and \(\lim_{t \to + \infty} \frac{1}{ {1 + t^{\alpha}}} = 0\) for the above \(\varepsilon > 0\), there exists a constant \(L > L_{1} > 0\) such that, for any \(t_{1},t_{2} \geq L\) and \(0 \leq s\leq L_{1}\), we have
and
Then, for \(t_{1},t_{2} \geq L\), from the above we obtain
Similarly, we get
and, in addition,
So, \(R(x,\lambda)(\bar{\Omega})\) is equiconvergent at infinity. By Lemma 3.2, \(R:\bar{\Omega}\times[0,1] \to X_{2}\) is completely continuous.
Finally, we prove that the (\(a_{3}\)) and (\(a_{4}\)) in Definition 2.4 hold. Let \(x \in\sum_{\lambda}= \{ x \in\bar{\Omega}\vert Mx = N_{\lambda}x \}\). Then \((\phi_{p} (D_{0 +}^{\alpha}x(t)))'=N_{\lambda}x(t) \in\operatorname{Im} M = \operatorname{Ker}Q\) and
which, combined with boundary conditions, yields that
It is clear that \(R(x,0)(t)\) is a zero operator, and for any \(x \in\bar{\Omega}\), we have
By the above, \(N_{\lambda}\) is M-compact on Ω̄. □
Theorem 3.1
Suppose that (\(A_{1}\)) and the following conditions hold:
- (\(A_{2}\)):
-
there exist nonnegative functions \(a(t),b(t),c(t),d(t) \in Y\) such that
$$\begin{aligned} \bigl\vert {f(t,u,v,w)} \bigr\vert \leq& a(t) + b(t)\frac{\vert u \vert ^{p - 1}}{{(1+t^{\alpha})^{p - 1}}} + c(t) \frac{\vert v \vert ^{p - 1}}{{(1+t^{\alpha})^{p - 1}}} \\ &{} + d(t)\vert w \vert ^{p - 1} ,\quad \forall t \in[ {0, + \infty} ),(u,v,w) \in\mathbb{R}^{3}; \end{aligned}$$ - (\(A_{3}\)):
-
there exists a positive constant B such that one of the following inequalities hold:
$$\begin{aligned} &{ wf(t,u,v,w) > 0,\quad \forall t \in[ {0, + \infty} ),u,v \in \mathbb{R},\vert w \vert > {B},} \end{aligned}$$(3.5)$$\begin{aligned} &{ wf(t,u,v,w) < 0,\quad \forall t \in[ {0, + \infty} ),u,v \in \mathbb{R},\vert w \vert > {B}.} \end{aligned}$$(3.6)
Then BVP (1.1) has at least one solution in X, provided that \(\Vert {b} \Vert _{1}+\Vert {c} \Vert _{1}+ \Vert {d} \Vert _{1} < 1\).
Before we prove Theorem 3.1, we show two lemmas.
Lemma 3.4
Let \(\Omega_{1} = \{ {x \in\operatorname{dom}M\setminus\operatorname{Ker}M \vert Mx = N_{\lambda}x,\lambda \in(0,1)} \}\). Suppose that (\(A_{2}\)) and (\(A_{3}\)) hold. Then \(\Omega_{1}\) is bounded in X.
Proof
Let \(x \in\Omega_{1}\). Then \(Mx = N_{\lambda}x\) and thus \(QN_{\lambda}x = 0\), that is,
By Lemma 2.1 and the boundary conditions we have
Thus,
By (\(A_{3}\)) there exists a constant \(s_{0} \in[0, + \infty)\) such that \(\vert {D_{0 + }^{\alpha}x(s_{0} )} \vert \leqslant{B}\), which, combined with \(Mx = N_{\lambda}x\) and (\(A_{2}\)), gives
Then
Thus,
That is,
Therefore,
So, \(\Omega_{1}\) is bounded in X. □
Lemma 3.5
Let \(\Omega_{2} = \{ {x \in\operatorname{Ker}M\vert Nx \in\operatorname{Im} M} \}\). Suppose that (\(A_{3}\)) holds. Then \(\Omega_{2}\) is bounded in X.
Proof
Let \(x \in\Omega_{2}\), that is, \(x = ct^{\alpha}\), \(c \in\mathbb{R}\), \(QNx = 0\), so that
By (\(A_{3}\)) we have \(\vert c \Gamma(\alpha) \vert \leq{B}\), that is, \(\vert c \vert \leq \frac{B}{ {\Gamma(\alpha)}}\). Therefore,
So, \(\Omega_{2}\) is bounded in X. □
Proof of Theorem 3.1
Set \(\Omega = \{ {x \in X\vert {\Vert x \Vert } _{X} < \max \{ {B,\phi_{q} ( A ),C} \} + 1} \}\). By Lemma 3.1 and Lemma 3.3 we know that M is quasi-linear and \(N_{\lambda}\) is M-compact on Ω̄. From Lemma 3.4 and Lemma 3.5 we obtain:
- (\(B_{1}\)):
-
\(Mx \ne N_{\lambda}x\), \(\forall(x,\lambda) \in\partial\Omega \times(0,1)\),
- (\(B_{2}\)):
-
\(QNx \ne0\), \(\forall x \in\operatorname{Ker}M \cap\partial\Omega\).
Now we show (\(B_{3}\)) holds. Let \(J:\operatorname{Im} Q \to\operatorname{Ker}M\) be the homeomorphism defined by
Without loss of generality, we suppose that (3.6) holds. Define the homotopic mapping
Then \(H(x,\lambda) \ne0\), \(x \in\partial\Omega \cap\operatorname{Ker}M\), \(\lambda \in[0,1]\). Indeed, for \(x \in\partial\Omega \cap \operatorname{Ker}M\), we have \(x = ct^{\alpha}\) and thus
Clearly, \(H(x,1) \ne0\), \(x \in\partial\Omega \cap\operatorname{Ker}M\). For \(\lambda \in[0,1)\) and \(x = ct^{\alpha}\in\partial\Omega \cap \operatorname{Ker}M\), if \(H(x,\lambda) = 0\), then
which contradicts (3.6). If (3.5) holds, then defining the homotopic mapping
we also get contradiction in a similar way. Therefore, via the homotopy property of degree, we obtain
Applying Lemma 2.2, we conclude that (1.1) has at least one solution in Ω̄. □
4 Example
Example 4.1
Consider the BVP
where \(0 < \xi_{1} < \xi_{2} < \cdots < \xi_{n} < + \infty\), \(\alpha_{i} > 0\), \(\sum_{i = 1}^{n} {\alpha_{i} } = 1\). Let
We easily check (\(A_{1}\))-(\(A_{3}\)). By Theorem 3.1, problem (4.1) has at least one solution.
References
Agarwal, RP, O’Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001)
Gomes, JM, Sanchez, L: A variational approach to some boundary value problems in the half-line. Z. Angew. Math. Phys. 56(2), 192-209 (2005)
Arara, A, Benchohra, M, Hamidi, N, et al.: Fractional order differential equations on an unbounded domain. Nonlinear Anal., Theory Methods Appl. 72(2), 580-586 (2010)
Lian, H, Pang, H, Ge, W: Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Anal., Theory Methods Appl. 67(7), 2199-2207 (2007)
Liu, Y: Existence and unboundedness of positive solutions for singular boundary value problems on half-line. Appl. Math. Comput. 144(2), 543-556 (2003)
Su, X, Zhang, S: Unbounded solutions to a boundary value problem of fractional order on the half-line. Comput. Math. Appl. 61(4), 1079-1087 (2011)
Tian, Y, Ge, W, Shan, W: Positive solutions for three-point boundary value problem on the half-line. Comput. Math. Appl. 53(7), 1029-1039 (2007)
Zhao, X, Ge, W: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math. 109(2), 495-505 (2010)
Chen, SZ, Zhang, Y: Singular boundary value problems on a half-line. J. Math. Anal. Appl. 195(2), 449-468 (1995)
Liu, X, Jia, M: Multiple solutions of nonlocal boundary value problems for fractional differential equations on the half-line. Electron. J. Qual. Theory Differ. Equ. 2011, 56 (2011)
Yan, B: Multiple unbounded solutions of boundary value problems for second-order differential equations on the half-line. Nonlinear Anal., Theory Methods Appl. 51(6), 1031-1044 (2002)
Yan, B, O’Regan, D, Agarwal, RP: Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. J. Comput. Appl. Math. 197(2), 365-386 (2006)
Jiang, W: Solvability for p-Laplacian boundary value problem at resonance on the half-line. Bound. Value Probl. 2013(1), 207 (2013)
Lian, H, Pang, H, Ge, W: Solvability for second-order three-point boundary value problems at resonance on a half-line. J. Math. Anal. Appl. 337(2), 1171-1181 (2008)
Zhou, H, Kou, C, Xie, F: Existence of solutions for fractional differential equations with multi-point boundary conditions at resonance on a half-line. Electron. J. Qual. Theory Differ. Equ. 2011, 27 (2011)
Ge, W, Ren, J: An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p-Laplacian. Nonlinear Anal., Theory Methods Appl. 58(3), 477-488 (2004)
Wang, J, Ibrahim, AG, Feckan, M: Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Appl. Math. Comput. 257(2015), 103-118 (2015)
Wang, J, Zhou, Y, Feckan, M: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64(10), 3389-3405 (2012)
Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 11271364).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have made equal contributions to each part of this paper. All the authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, W., Liu, W. & Chen, T. Solvability for a fractional p-Laplacian multipoint boundary value problem at resonance on infinite interval. Adv Differ Equ 2016, 183 (2016). https://doi.org/10.1186/s13662-016-0878-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-016-0878-3
MSC
- 34A08
- 34B15
Keywords
- boundary value problem
- fractional p-Laplacian equation
- resonance
- infinite interval
- continuation theorem