Abstract
In this paper, we study the existence of solutions of a class of higher-order integro-differential boundary value problems with ϕ-Laplacian like operator and functional boundary conditions. By giving the definition of a pair of coupled lower and upper solutions and some new hypotheses, we obtain some new existence results for boundary value problems with ϕ-Laplacian like operator by employing the Schauder fixed point theorem and an appropriate Nagumo condition. Finally, an example is given to illustrate the results.
MSC:39K10, 34B15.
Similar content being viewed by others
1 Introduction
Integro-differential equations have become more and more important in some mathematical models of real phenomena, especially in control, biological, medical, and informational models. Boundary value problems (BVPs) for nonlinear integro-differential equations are used to describe a great number of nonlinear phenomena in science (see [1, 2]), moreover, the theory of ϕ-Laplacian BVPs has emerged as an important area in recent years (see [3–6]). In this paper, we will consider the following BVPs of higher-order functional integro-differential ϕ-Laplacian like equations with functional boundary conditions:
where is an integer, ϕ is an increasing homeomorphism operator,
and (), , for , are continuous functions, and for ,
, , , , , and is a Carathéodory function, that is:
-
(i)
for any , is measurable on ,
-
(ii)
for a.e. , is continuous on ,
-
(iii)
for every compact set , there exists a nonnegative function such that
We say is a solution of BVP (1.1) and (1.2), that is, a function such that is absolutely continuous on , satisfies (1.1) a.e. on , and satisfies boundary condition (1.2).
As we know, higher-order boundary value problems for differential equations have received great attention in recent years (see [7–16]). We found that BVP (1.1) and (1.2) is more general in the literature, and the functional boundary condition (1.2) may not only cover many classical boundary conditions, such as various linear two-point, multi-point studied by many authors, but it may also include many new boundary conditions not studied so far in the literature. In recent years, BVPs with linear and nonlinear boundary conditions have been extensively investigated by numerous researchers. For a small sample of such work, we refer the reader to [17–26]. As is well known, a variety of methods and tools, such as lower and upper solution methods and various fixed point theorems, are very useful and have been successfully used to prove the existence of solutions of BVPs.
Motivated by the above mentioned works, we consider the BVPs of higher-order functional integro-differential equations (1.1) and (1.2) with ϕ-Laplacian like operator and functional boundary conditions in this paper. As we know, BVP (1.1) and (1.2) has not yet been considered. By introducing a definition for the coupled lower and upper solutions of BVP (1.1) and (1.2), we obtain the existence of solutions of the problem based on the assumption that there exists a pair of coupled lower and upper solutions.
This paper is organized as follows. In Section 2, we state some preliminaries and lemmas which will be used throughout this paper. In Section 3, some results concerning coupled lower and upper solutions are given. Finally, an example is given to illustrate our results in Section 4.
2 Preliminaries
Throughout this paper, let , for any , and
and
stand for the norms in E and , respectively, where denotes the Lebesgue measure of a set. In what follows, a functional is said to be nondecreasing if for any with on . A similar definition holds for y to be non-increasing.
Definition 2.1 Let be Carathéodory function, and satisfy
We say that f satisfies the Nagumo condition with respect to v and w if for
there exists a constant with
and functions , (), such that on ,
and
where for ,
and
Remark 2.1 Let satisfy (2.1). Assume that there exist , , and such that
Then f satisfies the Nagumo condition with respect to v and w with .
Definition 2.2 Let C be the constant introduced in Definition 2.1. Assume that there exist satisfying (2.1), and are absolutely continuous on . Then v and w are said to be a pair of coupled lower and upper solutions of BVP (1.1) and (1.2) if
and
For convenience, we first list the following hypotheses:
(H1) is increasing on R;
(H2) BVP (1.1) and (1.2) has a pair of coupled lower and upper solutions v and w satisfying (2.1);
(H3) the functional f satisfies the Nagumo condition with respect to v and w;
(H4) for with , , , , , and , , , we have
and is nondecreasing in the arguments , , , ;
(H5) for and , are nondecreasing in the arguments .
We assume that conditions (H1)-(H5) hold throughout this paper. For and , we define
then, for , is continuous on J, and
for and . Let be the constant introduced in Definition 2.1, and define
and a functional by
where
Then, in view of (H1) and (2.13), we find that is increasing and continuous (hence exists), and
What is more, for and , is continuous in u, and we can see that
Now, we consider the BVP consisting of the equation
and the boundary condition
Lemma 2.1 For any fixed , define by
where
Then the equation
has a unique solution.
Proof We first note that is continuous and increasing on R. From (2.16), we have
Then, from the fact that is continuous and increasing on R, a standard argument shows that there exists a unique solution of (2.22). □
Lemma 2.2 For , let
with being the unique solution of (2.22) and be defined by (2.15). Then is a solution of BVP (2.18) and (2.19) if and only if is a solution of the following equation:
where we take .
Proof This can be verified by direct computations, so we omit it. □
3 Main results
In this section, we will state and prove our existence results for BVP (1.1) and (1.2).
Theorem 3.1 Assume the hypotheses (H1)-(H5) hold. Then BVP (1.1) and (1.2) has at least one solution satisfying
and
where C is the constant introduced in Definition 2.1.
To prove Theorem 3.1, firstly, we want to show the following theorems.
Theorem 3.2 There exists at least one solution for BVP (2.18) and (2.19).
Proof By Lemma 2.2, for any , define an operator by
Then we can see that is a solution of BVP (2.18) and (2.19) if and only if is a fixed point of ℋ.
Let with as in E. We want to show that as in E. For f is a Carathéodory function, then it is easy to see . By the Lebesgue dominated convergence theorem, . Let be the unique solution of , where ℒ is given by (2.20). In view of (2.17), there exists such that
From (2.11), (2.14), (2.21) and the continuity of and , we see that is bounded and . Thus, is bounded. If is not convergent, then there exist two convergent subsequences and such that , . Then, by the continuity of and the Lebesgue dominated convergence theorem again, we have
and
which contradicts the fact that . Hence, is convergent, say . Thus, and . As a consequence, we also have
Thus as . This shows that is continuous.
From (3.3) and the fact that is bounded for , this means that ℋ is uniformly bounded on E, and is equicontinuous on J for . Now, we show that is equicontinuous on J. From the definition of ℋ and , we have
Thus, the equicontinuity of follows from the property of absolute of integrals. By the Arzela-Ascoli theorem, we see that is compact. From the Schauder fixed point theorem, ℋ has at least one fixed point , which is a solution of BVP (2.18) and (2.19). We complete the proof. □
Theorem 3.3 If u(t) is a solution of BVP (2.18) and (2.19), then satisfies (3.1).
Proof We first show that on J. To the contrary, suppose that there exists such that . If , then , from (2.19), (H5), (2.12), (2.14), and (2.8), we see that
which is a contradiction. Similarly, if , then , we have
We obtain a contradiction again. Thus, and .
Now, if such that . Then . Without loss of generality, we may assume that . Then and there exists a small right neighborhood Ω of such that and for all . We claim that there exists such that
If this is not true, then is strictly increasing in Ω. Hence, on Ω. This contradicts the assumption that is maximized at . Thus, (3.4) holds.
From (2.7), (2.13), and (2.14), we have , also, by (H4), (2.15), and (2.18), we have
which is a contradiction with (3.4). Thus, on J. By the same method as above, we can show that on J. Hence,
Next, we can see that the following inequality holds:
In fact, assume there exists such that , then, in view of (2.12), . Hence, from (2.19), (H5), (2.8) and (2.12),
This is a contradiction. Thus, for . By a similar argument, we see that for . Then (3.6) holds.
Finally, from (3.5) and integral inequality, we have
and using (3.6), we obtain . Similarly, we can show that satisfies (3.1). The proof is completed. □
Theorem 3.4 If is a solution of BVP (2.18) and (2.19), then satisfies (3.2).
Proof From Theorem 3.3, we know that satisfies (3.1). If (3.2) does not hold, then there exists such that or . By the mean value theorem, there exists such that . Then, from (2.2), (2.3), and (2.17), we see that
If there exist such that , and
where or . In the following, we only consider the case , since the other case can be treated similarly. From (2.14) and (3.7), on I, and in view of (2.11) and (3.1), we have for and . Thus, from (2.15),
Then, by a change of variables and from (2.4) and (2.18), we can obtain
Hence, the Hölder inequality implies
where ζ is defined by (2.6). But this contradicts with (2.5). Therefore, . If , by a similar argument as above, we can show that (3.2) holds. Hence the proof of the theorem is completed. □
Now we are in a position to prove Theorem 3.1.
Proof of Theorem 3.1 Note that any solution of BVP (2.18) and (2.19) satisfying (3.1), (3.2) is a solution of BVP (1.1) and (1.2). The conclusion readily follows from Theorem 3.2-3.4. □
4 Example
Example 4.1 Consider the boundary value problem consisting of the equation
and the boundary condition
BVP (4.1) and (4.2) has at least one solution satisfying
and
for .
In fact, if we let , ,
for , and , , , , , and
for , then it is easy to see that BVP (4.1) and (4.2) is of the form of BVP (1.1) and (1.2). Clearly, (H1), (H2) and (H5) hold.
Let and . Obviously, , satisfy (2.1). Define and . Then with , on , and
on , where is given by Definition 2.1. Thus (2.4) holds. For ξ defined by (2.2), we have with and , and it is easy to check that (2.3) holds. Through computations, we can obtain (2.5). Hence, f satisfies the Nagumo condition with respect to v, w i.e. (H3) holds. Moreover, a simple computation shows that and satisfy (2.7)-(2.10). Hence (H2) holds. Finally, obviously (H4) holds.
Therefore, by Theorem 3.1, BVP (4.1) and (4.2) has at least one solution satisfying (4.3)-(4.5).
5 Conclusions
In this paper, we obtain a new existence result for higher-order integro-differential BVPs with ϕ-Laplacian like operator and functional boundary conditions. Firstly, we state some preliminaries and lemmas such as the definitions if we have the Nagumo condition and a pair of coupled lower and upper solutions. Secondly, under conditions (H1)-(H5), we get the main result (Theorem 3.1), and we prove the result in three steps (Theorems 3.2-3.4) which mainly use lower and upper solutions and the Schauder fixed point theorem. Finally, an example is given to illustrate our main result.
References
Lakshmikantham V: Some problems in integro-differential equations of Volterra type. J. Integral Equ. 1985, 10: 137-146.
Guo D, Lakshmikantham V, Liu XZ (Eds): Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht; 1996.
Shi GL, Meng XR: Monotone iterative for fourth-order p -Laplacian boundary value problems with impulsive effects. Appl. Math. Comput. 2006, 181: 1243-1248. 10.1016/j.amc.2006.02.024
Cabada A, Otero-Espinar V: Existence and comparison results for difference ϕ -Laplacian boundary value problems with lower and upper solutions in reverse order. J. Math. Anal. Appl. 2002, 267: 501-521. 10.1006/jmaa.2001.7783
Misawa M: A Hölder estimate for nonlinear parabolic systems of p -Laplacian type. J. Differ. Equ. 2013, 254: 847-878. 10.1016/j.jde.2012.10.001
Agarwal RP: On fourth order boundary value problems arising in beam analysis. Differ. Integral Equ. 1989, 2: 91-110.
Wei Z: Existence of positive solutions for n th-order p -Laplacian singular sublinear boundary value problems. Appl. Math. Lett. 2014, 36: 25-30.
Davis JD, Henderson J, Wong PJY: General Lidstone problems: multiplicity and symmetry of solutions. J. Math. Anal. Appl. 2000, 251: 527-548. 10.1006/jmaa.2000.7028
Fialho JF, Minhós F: Higher order functional boundary value problems without monotone assumptions. Bound. Value Probl. 2013., 2013: Article ID 81
Agarwal RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore; 1986.
Lee EK, Lee YH: Multiple positive solutions of singular two points boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 2004, 158: 745-759. 10.1016/j.amc.2003.10.013
Eloe PW, Ahmad B: Positive solutions of a nonlinear n th order boundary value problem with nonlocal conditions. Appl. Math. Lett. 2005, 18: 521-527. 10.1016/j.aml.2004.05.009
Mawhin J: Homotopy and nonlinear boundary value problems involving singular ϕ -Laplacians. J. Fixed Point Theory Appl. 2013, 13: 25-35. 10.1007/s11784-013-0112-9
Graef JR, Yang B: Positive solutions to a multi-point higher order boundary value problems. J. Math. Anal. Appl. 2006, 316: 409-421. 10.1016/j.jmaa.2005.04.049
Graef JR, Kong L, Kong Q: Higher order multi-point boundary value problems. Math. Nachr. 2011, 284: 39-52. 10.1002/mana.200710179
Wang DB, Guan W: Multiple positive solutions for third-order p -Laplacian functional dynamic equations on time scales. Adv. Differ. Equ. 2014., 2014: Article ID 145
Graef JR, Kong L, Minhós FM: Higher order boundary value problems with ϕ -Laplacian and functional boundary conditions. Comput. Math. Appl. 2011, 61: 236-249. 10.1016/j.camwa.2010.10.044
Graef JR, Kong L: Existence of solutions for nonlinear boundary value problems. Commun. Appl. Nonlinear Anal. 2007, 14: 39-60.
Bai DY: A global result for discrete ϕ -Laplacian eigenvalue problems. Adv. Differ. Equ. 2013., 2013: Article ID 264
Wang W, Yang X, Shen J: Boundary value problems involving upper and lower solutions in reverse order. J. Comput. Appl. Math. 2009, 230: 1-7. 10.1016/j.cam.2008.10.040
Ehme J, Eloe PW, Henderson J: Upper and lower solution methods for fully nonlinear boundary value problems. J. Differ. Equ. 2002, 180: 51-64. 10.1006/jdeq.2001.4056
Franco D, Regan DO, Perán J: Fourth-order problems with nonlinear boundary conditions. J. Comput. Appl. Math. 2005, 174: 315-327. 10.1016/j.cam.2004.04.013
Cabada A, Pouso R, Minhós F: Extremal solutions to fourth-order functional boundary value problems including multipoint conditions. Nonlinear Anal. 2009, 10: 2157-2170. 10.1016/j.nonrwa.2008.03.026
Henderson J: Solutions of multipoint boundary value problems for second order equations. Dyn. Syst. Appl. 2006, 15: 111-117.
Cabada A, Pouso RL:Existence results for the problem with nonlinear boundary conditions. Nonlinear Anal. 1999, 35: 221-231. 10.1016/S0362-546X(98)00009-1
Kong L, Kong Q: Second-order boundary value problems with nonhomogeneous boundary conditions (I). Math. Nachr. 2005, 278: 173-193. 10.1002/mana.200410234
Acknowledgements
This work was supported by Natural Science Foundation of China Grant No. 11461021, Natural Science Foundation of Guangxi Grant No. 2014GXNSFDA118002, Scientific Research Foundation of Guangxi Education Department No. ZD2014131, No. 2013YB236, the open fund of Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis No. HCIC201305 and the Scientific Research Project of Hezhou University No. 2014ZC13. The authors wish to thank the anonymous reviewers for their helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guo, X., Lu, L. & Liu, Z. BVPs for higher-order integro-differential equations with ϕ-Laplacian and functional boundary conditions. Adv Differ Equ 2014, 285 (2014). https://doi.org/10.1186/1687-1847-2014-285
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2014-285