- Research
- Open Access
- Published:
Positive solutions to PBVPs for nonlinear first-order impulsive dynamic equations on time scales
Advances in Difference Equations volume 2015, Article number: 83 (2015)
Abstract
By using the classical fixed point theorem for operators on a cone, in this paper, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained.
1 Introduction
The theory of dynamic equations on time scales has been a new important mathematical branch [1–3] since it was initiated by Hilger [4]. At the same time, the boundary value problems of impulsive dynamic equations on time scales have received considerable attention [5–21] since the theory of impulsive differential equations is a lot richer than the corresponding theory of differential equations without impulse effects [22–24].
In this paper, we concerned with the existence of positive solutions for the following PBVPs of impulsive dynamic equations on time scales
where \(\mathbb{T}\) is an arbitrary time scale, \(T>0\) is fixed, 0, \(T\in\mathbb {T}\), \(f\in C ( J\times[ 0,\infty ) , [ 0,\infty ) ) \), \(I_{k}\in C ( [ 0,\infty ) , [ 0,\infty ) ) \), \(p: [ 0,T ] _{\mathbb{T}}\rightarrow(0,\infty)\) is right-dense continuous, \(t_{k}\in ( 0,T ) _{\mathbb{T}}\), \(0< t_{1}<\cdots <t_{m}<T\), and, for each \(k=1,2,\ldots,m\), \(x(t_{k}^{+})=\lim_{h\rightarrow 0^{+}}x(t_{k}+h)\) and \(x(t_{k}^{-})=\lim_{h\rightarrow0^{-}}x(t_{k}+h)\) represent the right and left limits of \(x(t)\) at \(t=t_{k}\).
By using the Guo-Krasnoselskii fixed point theorem, Wang [18] considered the existence of one or two positive solutions to the problem (1.1).
In [20], by using the Schaefer fixed point theorem, Wang and Weng obtained the existence of at least one solution to the problem (1.1).
When \(I_{k}(x)\equiv0\), \(k=1,2,\ldots,m\), [25, 26] considered the existence of solutions to the problem (1.1) by means of the Schaefer fixed point theorem; when \(p(t)=0 \), the problem (1.1) reduces to the problem studied by [12, 19].
Motivated by the results mentioned above, in this paper, we shall obtain the existence of one and two solutions to the problem (1.1) by means of a fixed point theorem in cones. The results obtained in this paper improve the results in [18] intrinsically.
Throughout this work, we assume knowledge of time scales and the time-scale notation, first introduced by Hilger [4]. For more on time scales, please see the texts by Bohner and Peterson [2, 3].
In the remainder of this section, we state the following fixed point theorem [27].
Theorem 1.1
([27])
Let X be a Banach space and \(K\subset X\) be a cone in X. Assume \(\Omega_{1}\), \(\Omega_{2}\) are bounded open subsets of X with \(0\in\Omega_{1}\subset\overline{\Omega}_{1}\subset \Omega_{2}\) and \(\Phi:K\cap(\overline{\Omega}_{2}\backslash\Omega _{1})\to K\) is a completely continuous operator. If
-
(i)
there exists \(u_{0}\in K\backslash\{0\}\) such that \(u-\Phi u\neq \lambda u_{0}\), \(u\in K\cap\partial\Omega_{2}\), \(\lambda\geq0\); \(\Phi u\neq\tau u\), \(u\in K\cap\partial\Omega_{1}\), \(\tau\geq1\), or
-
(ii)
there exists \(u_{0}\in K\backslash\{0\}\) such that \(u-\Phi u\neq \lambda u_{0}\), \(u\in K\cap\partial\Omega_{1}\), \(\lambda\geq0\); \(\Phi u\neq\tau u\), \(u\in K\cap\partial\Omega_{2}\), \(\tau\geq1\),
then Φ has at least one fixed point in \(K\cap(\overline{\Omega}_{2}\backslash\Omega_{1})\).
2 Preliminaries
Throughout the rest of this paper, we always assume that the points of impulse \(t_{k}\) are right-dense for each \(k=1,2,\ldots,m\).
We define
where \(x_{k}\) is the restriction of x to \(J_{k}= \left.(t_{k},t_{k+1}]\right._{\mathbb{T}}\subset \left.(0,\sigma(T)]\right._{\mathbb{T}}\), \(k=1,2,\ldots,m\), and \(J_{0}=[0,t_{1}]_{\mathbb{T}}\), \(t_{m+1}=\sigma(T)\).
Let
with the norm \(\Vert x\Vert =\sup_{ t\in[0,\sigma (T)]_{\mathbb{T}}}\vert x(t)\vert \), then X is a Banach space.
Lemma 2.1
Suppose \(M>0\) and \(h:[0,T]_{\mathbb{T}}\rightarrow R\) is rd-continuous, then x is a solution of
where
if and only if x is a solution of the boundary value problem
Proof
Since the proof is similar to that of [18], Lemma 3.1, we omit it here. □
Lemma 2.2
Let \(G(t,s)\) be defined as in Lemma 2.1, then
Proof
It is obvious, so we omit it here. □
Remark 2.1
Let \(G(t,s)\) be defined as in Lemma 2.1, then \(\int_{0}^{\sigma(T)}G(t,s)\triangle s=\frac{1}{M}\).
Let \(m=\min_{t\in [ 0,T ] _{\mathbb{T}}}p(t)\), \(M=\max_{t\in [ 0,T ] _{\mathbb{T}}}p(t)\), then \(0< m\leq M<\infty\). For \(u\in X\), we consider the following problem:
It follows from Lemma 2.1 that the problem (2.1) has a unique solution:
where \(h_{u}(s)=Mu(\sigma(s))-p(t)u(\sigma(t))+f(s,u(\sigma(s)))\), \(s\in [0,T]_{\mathbb{T}}\).
We define the operator \(\Phi:X\rightarrow X\) by
It is obvious that fixed points of Φ are solutions of the problem (1.1).
Lemma 2.3
\(\Phi:X\rightarrow X\) is completely continuous.
Proof
Since the proof is similar to that of [18], Lemma 3.3, we omit it here. □
Let
where \(\delta=\frac{1}{e_{M}(\sigma(T), 0)}\in(0,1)\). It is not difficult to verify that K is a cone in X.
From Lemma 2.2, it is easy to obtain the following result.
Lemma 2.4
Φ maps K into K.
3 Main results
For convenience, we denote
and
Now we state our main results.
Theorem 3.1
Suppose that
- (H1):
-
\(f_{0}>M\), \(f^{\infty}< m\), \(I_{\infty}=0\) for any k; or
- (H2):
-
\(f_{\infty}>M\), \(f^{0}< m\), \(I_{0}=0\) for any k.
Then the problem (1.1) has at least one positive solution.
Proof
Firstly, we assume (H1) holds. Then there exist \(\varepsilon>0\) and \(\beta>\alpha>0\) such that
and
Let \(\Omega_{1}= \{ u\in X:\Vert u\Vert < r_{1} \} \), where \(r_{1}=\alpha\). Choose \(u_{0}=1\), then \(u_{0}\in K\backslash\{0\}\). We assert that
Suppose on the contrary that there exist \(\overline{u}\in K\cap\partial \Omega_{1}\) and \(\overline{\lambda}\geq0\) such that
Let \(\zeta=\min_{t\in[0,\sigma(T)]_{\mathbb{T}}}\overline{u}(t)\), then \(\zeta\geq\delta \Vert \overline{u}\Vert =\delta r_{2}=\beta\), and we have from (3.1)
Therefore,
which is a contradiction.
On the other hand, let \(\Omega_{2}= \{ u\in X:\Vert u\Vert < r_{2} \} \), where \(r_{2}=\frac{\beta}{\delta}\).
Then \(u\in K\cap\partial\Omega_{2}\), \(0<\delta\beta=\delta \Vert u\Vert \leq u(t)\leq\beta\), and in view of (3.2) and (3.3) we have
which yields \(\Vert \Phi(u)\Vert <\Vert u\Vert \).
Therefore
It follows from (3.4), (3.5), and Theorem 1.1 that Φ has a fixed point \(u^{*}\in K\cap(\overline{\Omega}_{2}\backslash\Omega_{1})\), and \(u^{*}\) is the desired positive solution of the problem (1.1).
Next, suppose that (H2) holds. Then we can choose \(\varepsilon ^{\prime }>0\) and \(\beta^{\prime}>\alpha^{\prime}>0\) such that
and
Let \(\Omega_{3}= \{ u\in X:\Vert u\Vert < r_{3} \} \), where \(r_{3}=\alpha^{\prime}\). Then for any \(u\in K\cap\partial\Omega_{3}\), \(0<\delta \Vert u\Vert \leq u(t)\leq \Vert u\Vert =\alpha ^{\prime }\).
It is similar to the proof of (3.5), by (3.7) and (3.8) we have
Let \(\Omega_{4}= \{ u\in X:\Vert u\Vert < r_{4} \} \), where \(r_{4}=\frac{\beta^{\prime}}{\delta}\). Then for any \(u\in K\cap\partial \Omega _{4}\), \(u(t)\geq\delta \Vert u\Vert =\delta r_{4}=\beta^{\prime}\), by (3.6) , it is easy to obtain
It follows from (3.9), (3.10), and Theorem 1.1 that Φ has a fixed point \(u^{*}\in K\cap(\overline{\Omega}_{4}\backslash\Omega_{3})\), and \(u^{*}\) is the desired positive solution of the problem (1.1). □
In particular, we have the following results, which are main results of [18].
Corollary 3.1
Suppose that
- (H1):
-
\(f_{0}=\infty\), \(f^{\infty}=0\), \(I_{\infty}=0\) for any k; or
- (H2):
-
\(f_{\infty}=\infty\), \(f^{0}=0\), \(I_{0}=0\) for any k.
Then the problem (1.1) has at least one positive solution.
Theorem 3.2
Suppose that
- (H3):
-
\(f^{0}< m\), \(f^{\infty}< m\), \(I_{0}=0\), \(I_{\infty}=0\);
- (H4):
-
there exists \(\rho >0\) such that
$$ \min\bigl\{ f(t,u)-p(t)u\mid t\in [ 0,T ] _{\mathbb{T}}, \delta \rho \leq u\leq\rho\bigr\} >0. $$(3.11)
Then the problem (1.1) has at least two positive solutions.
Proof
By (H3), from the proof of Theorem 3.1, we see that there exist \(\beta^{\prime\prime}>\rho>\alpha^{\prime\prime}>0\) such that
where \(\Omega_{5}= \{ u\in X:\Vert u\Vert < r_{5} \}\), \(\Omega _{6}= \{ u\in X:\Vert u\Vert < r_{6} \} \), \(r_{5}=\alpha^{\prime \prime}\), \(r_{6}=\frac{\beta^{\prime\prime}}{\delta}\).
By (3.11) of (H4), we can choose \(\varepsilon>0\) such that
Let \(\Omega_{7}= \{ u\in X:\Vert u\Vert <\rho \} \), for any \(u\in K\cap\partial\Omega_{7}\), \(\delta\rho=\delta \Vert u\Vert \leq u(t)\leq \Vert u\Vert =\rho\), from (3.14), it is similar to the proof of (3.4), and we have
By Theorem 1.1, from (3.12), (3.13), and (3.15) we conclude that Φ has two fixed points \(u^{**}\in K\cap(\overline{\Omega}_{6}\backslash \Omega _{7})\) and \(u^{***}\in K\cap(\overline{\Omega}_{7}\backslash\Omega_{5})\), and \(u^{**}\) and \(u^{***}\) are two positive solutions of the problem (1.1). □
Similar to Theorem 3.2, we have the following.
Theorem 3.3
Suppose that
- (H5):
-
\(f_{0}>M\), \(f_{\infty}>M\);
- (H6):
-
there exists \(\rho>0\) such that
$$\begin{aligned}& \max\bigl\{ f(t,u)-p(t)u\mid t\in [ 0,T ] _{\mathbb{T}}, \delta \rho \leq u\leq\rho\bigr\} < 0; \\& I_{k}(u)\leq\frac{[e_{M}(\sigma(T),0)-1]}{Mme_{M}(\sigma (T),0)}u, \quad\delta\rho\leq u\leq\rho\textit{ for any }k. \end{aligned}$$
Then the problem (1.1) has at least two positive solutions.
Remark 3.1
If (H3) in Theorem 3.2 is replaced by \(f^{0}=0\), \(f^{\infty}=0\), or if (H5) in Theorem 3.3 is replaced by \(f_{0}=\infty\), \(f_{\infty}=\infty\), then the results of Theorem 3.2 and Theorem 3.3 are also hold.
References
Agarwal, RP, Bohner, M: Basic calculus on time scales and some of its applications. Results Math. 35, 3-22 (1999)
Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Bohner, M, Peterson, A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
Hilger, S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 18, 18-56 (1990)
Belarbi, A, Benchohra, M, Ouahab, A: Existence results for impulsive dynamic inclusions on time scales. Electron. J. Qual. Theory Differ. Equ. 2005, 12 (2005)
Benchohra, M, Henderson, J, Ntouyas, SK, Ouahab, A: On first order impulsive dynamic equations on time scales. J. Differ. Equ. Appl. 6, 541-548 (2004)
Benchohra, M, Ntouyas, SK, Ouahab, A: Existence results for second-order boundary value problem of impulsive dynamic equations on time scales. J. Math. Anal. Appl. 296, 65-73 (2004)
Benchohra, M, Ntouyas, SK, Ouahab, A: Extremal solutions of second order impulsive dynamic equations on time scales. J. Math. Anal. Appl. 324, 425-434 (2006)
Chen, HB, Wang, HH: Triple positive solutions of boundary value problems for p-Laplacian impulsive dynamic equations on time scales. Math. Comput. Model. 47, 917-924 (2008)
Geng, F, Xu, Y, Zhu, D: Periodic boundary value problems for first-order impulsive dynamic equations on time scales. Nonlinear Anal. 69, 4074-4087 (2008)
Graef, JR, Ouahab, A: Extremal solutions for nonresonance impulsive functional dynamic equations on time scales. Appl. Math. Comput. 196, 333-339 (2008)
Guan, W, Li, DG, Ma, SH: Nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Electron. J. Differ. Equ. 2012, 198 (2012)
Henderson, J: Double solutions of impulsive dynamic boundary value problems on time scale. J. Differ. Equ. Appl. 8, 345-356 (2002)
Li, JL, Shen, JH: Existence results for second-order impulsive boundary value problems on time scales. Nonlinear Anal. 70, 1648-1655 (2009)
Li, YK, Shu, J: Multiple positive solutions for first-order impulsive integral boundary value problems on time scales. Bound. Value Probl. 2011, 12 (2011)
Liu, HB, Xiang, X: A class of the first order impulsive dynamic equations on time scales. Nonlinear Anal. 69, 2803-2811 (2008)
Wang, C, Li, YK, Fei, Y: Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales. Math. Comput. Model. 52, 1451-1462 (2010)
Wang, DB: Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales. Comput. Math. Appl. 56, 1496-1504 (2008)
Wang, DB: Periodic boundary value problems for nonlinear first-order impulsive dynamic equations on time scales. Adv. Differ. Equ. 2012, 12 (2012)
Wang, ZY, Weng, PX: Existence of solutions for first order PBVPs with impulses on time scales. Comput. Math. Appl. 56, 2010-2018 (2008)
Zhang, HT, Li, YK: Existence of positive periodic solutions for functional differential equations with impulse effects on time scales. Commun. Nonlinear Sci. Numer. Simul. 14, 19-26 (2009)
Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993)
Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)
Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995)
Dai, QY, Tisdell, CC: Existence of solutions to first-order dynamic boundary value problems. Int. J. Differ. Equ. 1, 1-17 (2006)
Sun, JP, Li, WT: Existence of solutions to nonlinear first-order PBVPs on time scales. Nonlinear Anal. 67, 883-888 (2007)
Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Acknowledgements
The author express her gratitude to the anonymous referee for his/her valuable suggestions. The research was supported by Natural Science Foundation of Gansu Province of China (Grant No. 1310RJYA080).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that she has no competing interests.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Guan, W. Positive solutions to PBVPs for nonlinear first-order impulsive dynamic equations on time scales. Adv Differ Equ 2015, 83 (2015). https://doi.org/10.1186/s13662-015-0428-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-015-0428-4
MSC
- 39A10
- 34B15
Keywords
- time scale
- periodic boundary value problem
- fixed point
- impulsive dynamic equation