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
We notice that the set
In this paper we consider the delay differential equation with pLaplacian
subject to
and
where
We are interested in global solutions of the pLaplacian delay boundary value problem (1.1)(1.3). By a solution on
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
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
Lemma 2.1Letxbe a function in
Proof Let x be a function in
which implies that
and we have
which means that x satisfies (1.1). Thus, x is a solution on
Conversely, suppose that x is a solution on
It follows that
By integrating (2.6) and taking into account the fact that
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 on
(C)
We also assume that
(A) for each
where the function
Then the boundary value problem (1.1)(1.3) has at least one solutionxon
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 interval
Then the boundary value problem (1.1)(1.3) has a unique solutionxon
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 an
Let
We need the following compactness criterion for a subset of
Compactness criterion[16]
LetUbe an equicontinuous and uniformly bounded subset of the Banach space
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 unique
Throughout this section, let E denote the set of all functions x in
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
For
ThenMmaps Ω intoE. Moreover, MΩ is relatively compact and the mapping
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.,
By virtue of (1.2), (3.3) and (2.9), it follows 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
Next, using (3.5) and (2.8), from (3.1) we obtain for
Inequality (3.9) means that
Now, we shall prove that MΩ is relatively compact. We observe that, for any function
is relatively compact in the Banach space
In view of (3.2), this means that
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
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
For this purpose, we consider an arbitrary subsequence
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
Now, (3.12) and the fact that
By the Schauder fixed point theorem, there exists an
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
Furthermore, by using (2.12), from (3.1) we obtain for
This gives
By the definition of the norm
Next, from the definition of Ω, we have
Moreover, in view of the fact that
But, by the definition of the norm
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
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
Finally, by the Banach contraction principle, the mapping
4 Application
Let us consider the delay boundary value problem with the pLaplacian operator
where
If the boundary value problem (1.1)(1.3) is to be equivalent to the boundary value
problem (4.1), we must define
Corollary 4.1Assume that
whereHis a nonnegative continuous realvalued function on
Moreover, let there exist a real number
where the function
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
Example Consider the secondorder nonlinear delay differential equation of EmdenFowler type
where
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
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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