This paper is concerned with the existence and uniqueness of solutions for boundary value problems with p-Laplacian delay differential equations on the half-line. The existence of solutions is derived from the Schauder fixed point theorem, whereas the uniqueness of solution is established by the Banach contraction principle. As an application, an example is given to demonstrate the main results.
MSC: 34K10, 34B18, 34B40.
Keywords:delay differential equation; boundary value problem; infinite interval; Schauder fixed point theorem; Banach contraction principle
Boundary value problems on infinite intervals have many applications in physical problems. Such problems arise, for example, in the study of linear elasticity, fluid flows and foundation engineering (see [1,2] and the references cited therein). Boundary value problems on infinite intervals involving second-order delay differential equations are of specific interest in these applications. An interesting survey on infinite interval problems, including real world examples, history and various methods of proving solvability, can be found in the recent monograph by Agarwal et al. and Agarwal and O’Regan . Among the many articles dealing with boundary value problems of second-order delay differential equations, we refer the reader to  and the references cited therein.
Boundary value problems of second-order delay differential equations on infinite intervals are closely related to the problem of existence of global solutions with prescribed asymptotic behavior. Recently, there is a growing interest in the solutions of such boundary value problems; see, for example, [6-9]. For the basic theory of delay differential equations, the reader is referred to the books by Diekmann et al. as well as by Hale and Verduyn Lunel . For boundary value problems, we mention the monographs by Azbelev et al. and Azbelev and Rakhmatullina .
To the best of our knowledge, few authors have considered the existence of solutions on infinite intervals for delay differential equations. As far as we know, only in [6,8,9] the existence and uniqueness of solutions on infinite intervals for second-order delay boundary value problems are discussed. However, no work has been done on delay boundary value problems withp-Laplacian on infinite intervals. Motivated by the work mentioned above, this paper aims to fill in the gap, and we shall tackle the existence and uniqueness of solutions to a boundary value problem of delay differential equation withp-Laplacian on infinite interval, which has been rarely discussed until now. The results we obtain improve and generalize the results mentioned in the references.
Throughout this paper, for any intervals J and X of ℝ, we denote by the set of all continuous functions defined on J with values in X. Let r be a nonnegative real number. If t is a point in the interval and x is a continuous real-valued function defined at least on , the notation will be used for the function in defined by
In this paper we consider the delay differential equation with p-Laplacian
where , , with , and f is a real-valued function defined on the set , which satisfies the following continuity condition: is continuous with respect to t in for each given function which is continuously differentiable on the interval .
We are interested in global solutions of the p-Laplacian delay boundary value problem (1.1)-(1.3). By a solution on of (1.1)-(1.3), we mean a function which is continuously differentiable on the interval such that (1.1) is satisfied for all and the conditions (1.2) and (1.3) are also fulfilled.
The main results of this paper are stated in Section 2. In Theorem 2.1, sufficient conditions are established in order that (1.1)-(1.3) has at least one solution on , whereas Theorem 2.2 provides sufficient conditions for (1.1)-(1.3) to have a unique solution on . The proof of Theorem 2.1 and Theorem 2.2 is presented in Section 3, where we employ the Schauder fixed point theorem and the well known Banach contraction principle. In Section 4, we include an example to illustrate our main results.
2 Main results
and we have
Conversely, suppose that x is a solution on of the boundary value problem (1.1)-(1.3). In view of (1.2), we have for . Furthermore, from (1.1) it follows that x satisfies (2.4). Taking into account the fact that and , consequently we have
It follows that
We have thus proved that x has the expression (2.1). The proof of the lemma is complete. □
Theorem 2.1Suppose that
We also assume that
Theorem 2.2Let all the conditions of Theorem 2.1 be satisfied, i.e., (2.7), (C) and (A) hold. Moreover, suppose that
3 Proof of main results
To prove Theorem 2.1, we shall use the fixed point technique by applying the Schauder fixed point theorem , whereas Theorem 2.2 is established by the Banach contraction principle . We state our main tools below.
Schauder fixed point theorem
LetEbe a Banach space and Ω be any nonempty convex and closed subset ofE. IfMis a continuous mapping of Ω into itself andMΩ is relatively compact, then the mappingMhas at least one fixed point, i.e., there exists ansuch that.
We need the following compactness criterion for a subset of , which is a consequence of the well-known Arzela-Ascoli theorem. This compactness criterion is an adaptation of a lemma due to Avramescu . In order to formulate this criterion, we note that a set U of real-valued functions defined on is said to be equiconvergent at ∞ if all the functions in U are convergent in ℝ at the point ∞ and, in addition, for each , there exists such that, for any function , we have for .
Banach contraction principle
Throughout this section, let E denote the set of all functions x in , which are continuously differentiable on the interval and have bounded continuous derivatives on . The set E is a Banach space endowed with the norm given by
We shall first establish a lemma which will be needed to prove the main results.
Lemma 3.1Suppose that (2.7) holds, whereFis a nonnegative real-valued function defined on, which satisfies the continuity condition (C). Assume that (A) is satisfied. Let Ω be the subset of the Banach spaceEdefined by
Proof Since the condition (A) is satisfied, from (2.8) we have
First, we shall show that M is a mapping from Ω into E, i.e., . Let x be an arbitrary function in Ω. By the definition of Ω, the function x satisfies (1.2) and (2.11). Since , it follows from (1.2) that . By taking into account this fact and using (2.11), we can easily obtain
In view of (3.4), (2.11) and the condition (A), we get
On the other hand, (2.7) guarantees that
Thus, we have
From (3.2) and (3.5) it follows that
Furthermore, we can conclude that
Now, we shall prove that MΩ is relatively compact. We observe that, for any function , we have for . By taking into account this fact as well as the definition of the norm , we can easily conclude that it suffices to show that the set
In view of (3.2), this means that as , and we can easily verify that U is equicontinuous. Moreover, each function satisfies (3.9), where c is independent of x. This guarantees that U is uniformly bounded. Furthermore, for any , we have
and hence, noting (3.5), it follows that
Now (3.10) together with (3.2) implies that
Thus, by taking into account the fact that
This, together with the fact that
guarantees the pointwise convergence
For this purpose, we consider an arbitrary subsequence of . Since MΩ is relatively compact, there exists a subsequence of the sequence and a function u in E so that . As the convergence in the sense of implies the pointwise convergence to the same limit function, we must have , therefore (3.11) holds. Consequently, M is continuous. The proof is complete. □
Proof of Theorem 2.1 We shall apply the Schauder fixed point theorem. Let Ω be the subset of the Banach space E defined as in Lemma 3.1. Clearly, Ω is a nonempty convex and closed subset of E. By Lemma 3.1, the mapping is continuous and MΩ is relatively compact. We shall show that M maps Ω into itself, i.e., . Let us consider an arbitrary function . Following the argument in the proof of Lemma 3.1, we see that x satisfies (3.10), which together with (2.8) provides
By the Schauder fixed point theorem, there exists an such that . Hence, x has the expression (3.1), which coincides with (2.1). It follows from Lemma 2.1 that x is a solution on of the boundary value problem (1.1)-(1.3). Also, since , clearly x satisfies (2.11). Moreover, since , it follows from (2.11) that x also fulfills (2.10). This completes the proof of Theorem 2.1. □
Proof of Theorem 2.2 We shall employ the Banach contraction principle. Let Ω be the subset of the Banach space E defined in Lemma 3.1. Clearly, Ω is a nonempty closed subset of E. Following the argument in the proof of Theorem 2.1, we have .
Thus, using (3.17) in (3.16) yields
Combining (3.15) and (3.18), we get
where the function μ is defined by
We can rewrite (3.19) as
It follows that
So, from (3.20) we have
Now, using (3.17) and (3.21) in (3.14), we get
Finally, by the Banach contraction principle, the mapping has a unique fixed point having the expression (3.1), which coincides with (2.1). It follows from Lemma 2.1 that x is the unique solution on of the boundary value problem (1.1)-(1.3). Furthermore, as in the proof of Theorem 2.1, we conclude that this unique solution x of the boundary value problem (1.1)-(1.3) satisfies (2.10) and (2.11). The proof of Theorem 2.2 is now complete. □
Let us consider the delay boundary value problem with the p-Laplacian operator
If the boundary value problem (1.1)-(1.3) is to be equivalent to the boundary value problem (4.1), we must define for . Hence, by applying Theorem 2.1 to the boundary value problem (4.1), we can be led to the following result.
Corollary 4.1Assume that
Then the boundary value problem (4.1) has at least one solutionxsuch that (2.10) and (2.11) hold.
Example Consider the second-order nonlinear delay differential equation of Emden-Fowler type
where and are continuous real-valued functions on , and γ, β are positive real numbers. An application of Corollary 4.1 to the boundary value problem (4.2) leads us to conclude that if there exists a real number such that
then the boundary value problem (4.2) has at least one solution x satisfying (2.10) and (2.11).
In this case, (4.3) is reduced to
It is not difficult to verify that this inequality is equivalent to
has at least one solution x satisfying
The authors declare that they have no competing interests.
YW carried out the most part of the paper. PJYW gave the idea and revised it. All authors read and approved the final manuscript.
Project is supported by the National Natural Science Foundation of China (11061006) and the Bagui Scholars program of Guangxi.
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