- Open Access
BVPs for higher-order integro-differential equations with ϕ-Laplacian and functional boundary conditions
© Guo et al.; licensee Springer. 2014
- Received: 24 June 2014
- Accepted: 22 October 2014
- Published: 4 November 2014
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.
- ϕ-Laplacian like operator
- integro-differential equation
- functional boundary condition
- Nagumo condition
- lower and upper solutions
for any , is measurable on ,
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.
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.
Then f satisfies the Nagumo condition with respect to v and w with .
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;
and is nondecreasing in the arguments , , , ;
(H5) for and , are nondecreasing in the arguments .
has a unique solution.
Then, from the fact that is continuous and increasing on R, a standard argument shows that there exists a unique solution of (2.22). □
where we take .
Proof This can be verified by direct computations, so we omit it. □
In this section, we will state and prove our existence results for BVP (1.1) and (1.2).
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).
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.
Thus as . This shows that is continuous.
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).
We obtain a contradiction again. Thus, and .
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.
This is a contradiction. Thus, for . By a similar argument, we see that for . Then (3.6) holds.
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).
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. □
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.
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).
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.
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.
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