- Open Access
Existence and uniqueness of solutions for delay boundary value problems with p-Laplacian on infinite intervals
© Wei and Wong; licensee Springer. 2013
- Received: 2 March 2013
- Accepted: 3 May 2013
- Published: 31 May 2013
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.
- 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 with p-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 with p-Laplacian on infinite interval, which has been rarely discussed until now. The results we obtain improve and generalize the results mentioned in the references.
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.
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 .
which means that x satisfies (1.1). Thus, x is a solution on of the boundary value problem (1.1)-(1.3).
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 .
is continuous with respect to t in for each given function x in which is continuously differentiable on .
- (A)for each , the function is increasing on in the sense that for any with (i.e., for ) and any with . Moreover, there exists a real number so that(2.8)
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 .
Then the boundary value problem (1.1)-(1.3) has a unique solution x on satisfying (2.10) and (2.11).
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 
Let E be a Banach space and Ω be any nonempty convex and closed subset of E. If M is a continuous mapping of Ω into itself and M Ω is relatively compact, then the mapping M has at least one fixed point, i.e., there exists an such 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 .
Compactness criterion 
Let U be an equicontinuous and uniformly bounded subset of the Banach space . If U is equiconvergent at ∞, it is also relatively compact.
Banach contraction principle 
Let E be a Banach space and Ω be any nonempty closed subset of E. If M is a contraction of Ω into itself, then the mapping M has a unique fixed point, i.e., there exists a unique such that .
We shall first establish a lemma which will be needed to prove the main results.
Then M maps Ω into E. Moreover, M Ω is relatively compact and the mapping is continuous.
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 .
Inequality (3.9) means that is bounded on the interval and so Mx belongs to E. We have thus proved 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 .
For this purpose, we consider an arbitrary subsequence of . Since M Ω 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. □
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 .
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. □
where , , and h is a continuous real-valued function on , and is a nonnegative continuous real-value 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.
where H is a nonnegative continuous real-valued function on . Suppose that for each , the function is increasing on in the sense that for any with and .
Then the boundary value problem (4.1) has at least one solution x such that (2.10) and (2.11) hold.
Note that an interesting particular case is the one where the delay is a nonnegative real constant.
then the boundary value problem (4.2) has at least one solution x satisfying (2.10) and (2.11).
Project is supported by the National Natural Science Foundation of China (11061006) and the Bagui Scholars program of Guangxi.
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