Abstract
This paper is concerned with the existence and uniqueness of solutions for boundary value problems with pLaplacian delay differential equations on the halfline. 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 principle1 Introduction
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 secondorder 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.[3] and Agarwal and O’Regan [4]. Among the many articles dealing with boundary value problems of secondorder delay differential equations, we refer the reader to [5] and the references cited therein.
Boundary value problems of secondorder 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, [69]. For the basic theory of delay differential equations, the reader is referred to the books by Diekmann et al.[10] as well as by Hale and Verduyn Lunel [11]. For boundary value problems, we mention the monographs by Azbelev et al.[12] and Azbelev and Rakhmatullina [13].
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 secondorder delay boundary value problems are discussed. However, no work has been done on delay boundary value problems withpLaplacian 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 withpLaplacian 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 realvalued function defined at least on , the notation will be used for the function in defined by
We notice that the set is a Banach space equipped with the usual supnorm given by
In this paper we consider the delay differential equation with pLaplacian
subject to
and
where , , with , and f is a realvalued 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 pLaplacian 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
A useful integral representation of the boundary value problem (1.1)(1.3) is given by the following lemma. We note that has the inverse , where .
Lemma 2.1Letxbe a function inthat is continuously differentiable on. Thenxis a solution onof the boundary value problem (1.1)(1.3) if and only if
Proof Let x be a function in that is continuously differentiable on . Assume that x is given by (2.1). Then (1.2) is fulfilled. Moreover, we immediately obtain
which implies that , i.e., (1.3) holds. Furthermore, from (2.2) we get
and we have
which means that x satisfies (1.1). Thus, x is a solution on of the boundary value problem (1.1)(1.3).
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
By integrating (2.6) and taking into account the fact that , we obtain for every ,
We have thus proved that x has the expression (2.1). The proof of the lemma is complete. □
The first main result of this paper is the following theorem which provides sufficient conditions for (1.1)(1.3) to have at least one solution on .
Theorem 2.1Suppose that
whereFis a nonnegative realvalued function defined onwhich satisfies the continuity condition:
(C) is continuous with respect totinfor each given functionxinwhich is continuously differentiable on.
We also assume that
(A) for each, the functionis increasing onin the sense thatfor anywith (i.e., for) and anywith. Moreover, there exists a real numberso that
where the functiondepends onξ, cand is defined by
Then the boundary value problem (1.1)(1.3) has at least one solutionxonsuch that
and
The second main result is the following theorem that establishes conditions under which the boundary value problem (1.1)(1.3) has a unique solution on .
Theorem 2.2Let all the conditions of Theorem 2.1 be satisfied, i.e., (2.7), (C) and (A) hold. Moreover, suppose that
whereKis a nonnegative continuous realvalued function on the intervalsatisfying
Then the boundary value problem (1.1)(1.3) has a unique solutionxonsatisfying (2.10) and (2.11).
3 Proof of main results
To prove Theorem 2.1, we shall use the fixed point technique by applying the Schauder fixed point theorem [14], whereas Theorem 2.2 is established by the Banach contraction principle [15]. We state our main tools below.
Schauder fixed point theorem[14]
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.
Let be the Banach space of all bounded continuous realvalued functions on , endowed with the supnorm given by
We need the following compactness criterion for a subset of , which is a consequence of the wellknown ArzelaAscoli theorem. This compactness criterion is an adaptation of a lemma due to Avramescu [16]. In order to formulate this criterion, we note that a set U of realvalued 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 .
Compactness criterion[16]
LetUbe an equicontinuous and uniformly bounded subset of the Banach space. IfUis equiconvergent at ∞, it is also relatively compact.
Banach contraction principle[15]
LetEbe a Banach space and Ω be any nonempty closed subset ofE. IfMis a contraction of Ω into itself, then the mappingMhas a unique fixed point, i.e., there exists a uniquesuch that.
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 realvalued function defined on, which satisfies the continuity condition (C). Assume that (A) is satisfied. Let Ω be the subset of the Banach spaceEdefined by
ThenMmaps Ω intoE. Moreover, MΩ is relatively compact and the mappingis continuous.
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
By virtue of (1.2), (3.3) and (2.9), it follows that for all , which ensures that
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
and consequently,
Furthermore, we can conclude that
Since (3.8) holds for any function , we immediately see that the formula (3.1) makes sense for any , and this formula defines a mapping M from Ω into .
Next, using (3.5) and (2.8), from (3.1) we obtain for ,
Inequality (3.9) means that is bounded on the interval and so Mx belongs to E. We have thus proved 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
is relatively compact in the Banach space . Using (3.5), for any and any , with , we obtain
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
By using (3.2) and (3.10) again, we immediately see that U is equiconvergent at ∞. It now follows from the given compactness criterion that the set U is relatively compact in .
Finally, we shall prove that the mapping is continuous. Let with as . It is not difficult to verify that uniformly for and uniformly for . On the other hand, using (3.5) we have
Thus, by taking into account the fact that
we can apply the Lebesgue dominated convergence theorem to obtain, for every ,
This, together with the fact that
guarantees the pointwise convergence
It remains to show that this convergence is also convergence in the sense of , i.e.,
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
Now, (3.12) and the fact that for imply that . We have thus proved that .
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 .
Now, we shall prove that the mapping M is a contraction. For this purpose, let us consider two arbitrary functions x and in Ω. From (3.1), we have for , and consequently,
Furthermore, by using (2.12), from (3.1) we obtain for
This gives
By the definition of the norm , the last inequality and (3.13) imply
Next, from the definition of Ω, we have for , and so
Moreover, in view of the fact that , we get, for ,
But, by the definition of the norm , 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
i.e.,
It follows that
But, since μ is nondecreasing on , we have
So, from (3.20) we have
Now, using (3.17) and (3.21) in (3.14), we get
where the last inequality is due to (2.13). Hence, we have shown that the mapping is a contraction.
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. □
4 Application
Let us consider the delay boundary value problem with the pLaplacian operator
where , , and h is a continuous realvalued function on , and is a nonnegative continuous realvalue function on the interval with .
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
whereHis a nonnegative continuous realvalued function on. Suppose that for each, the functionis increasing onin the sense thatfor anywithand.
Moreover, let there exist a real numberso that
where the functionindepends onξ, cand is defined by
Then the boundary value problem (4.1) has at least one solutionxsuch that (2.10) and (2.11) hold.
Note that an interesting particular case is the one where the delay is a nonnegative real constant.
Example Consider the secondorder nonlinear delay differential equation of EmdenFowler type
where and are continuous realvalued 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).
For illustration purpose, suppose in (4.2) we have , , ,
In this case, (4.3) is reduced to
It is not difficult to verify that this inequality is equivalent to
which is satisfied if and only if . By taking , we can conclude that the boundary value problem
has at least one solution x satisfying
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YW carried out the most part of the paper. PJYW gave the idea and revised it. All authors read and approved the final manuscript.
Acknowledgements
Project is supported by the National Natural Science Foundation of China (11061006) and the Bagui Scholars program of Guangxi.
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