Volume 2008, Article ID 752827,14pages doi:10.1155/2008/752827
Research Article
Existence and Uniqueness of Solutions for
a Second-Order Delay Differential Equation
Boundary Value Problem on the Half-Line
Yuming Wei1, 2
1Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China 2School of Mathematical Science, Guangxi Normal University, Guilin 541004, China
Correspondence should be addressed to Yuming Wei,[email protected]
Received 5 May 2008; Accepted 8 September 2008
Recommended by Ivan T. Kiguradze
This paper is concerned with the existence and uniqueness of solutions for the second-order nonlinear delay differential equations. By the use of the Schauder fixed point theorem, the existence of the solutions on the half-line is derived. Via the Banach contraction principle, another result concerning the existence and uniqueness of solutions on the half-line is established. The main results in this paper extend some of the existing literatures.
Copyrightq2008 Yuming Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Boundary value problems on unbounded interval have many applications in physical problems. Such problems arise, for example, in the study of linear elasticity, fluid flows, and foundation engineeringsee1and the references therein. An interesting overview on
unbounded interval problems, including real-world examples, history, and various methods of proving solvability, can be found in the recent book by Agarwal and O’Regan 2.
Boundary value problems on unbounded interval concerning second-order delay differential equations are of specific interest in these applications.
For second-order delay differential equations, boundary value problems on the half-line are closely related to the problems of existence of global solutions on the half-half-line with prescribed asymptotic behavior. Recently, there is, in particular, a growing interest in solutions of such boundary value problems see, e.g., 3. For the basic theory of delay
differential equations, the reader is referred to the books by Diekmann et al. 4 and Hale and Verduyn Lunel 5. In particular, concerning initial value problems, we refer to the
monograph by Lakshmikantham and Leela6, while, regarding boundary value problems,
However, to the best of our knowledge, the literature on the existence and uniqueness of solutions on the half-line for delay differential equations seems to be rather limited. Motivated by the papers by Agarwal et al.9and Mavridis et al.10, this paper aims to
fill this gap by improving and generalizing results mentioned in the references.
Throughout the paper, for any intervalJ of the real lineRand any subsetX ofR, by
CJ, Xwe will denote the set of all continuous functions defined onJand having values in
X. Moreover,r will be a nonnegative real number. Furthermore, iftis a point in the interval 0,∞andxis a continuous real-valued function defined at least ont−r, t, the notationxt
will be used for the function inC−r,0,Rdefined by the formula
xtτ xtτ for −r≤τ ≤0. 1.1
We notice that the setC−r,0,Ris a Banach space endowed by the usual sup-norm| · |:
|ψ| max
−r≤τ≤0
ψτ forψ∈C
−r,0,R. 1.2
Consider the second-order nonlinear delay differential equation
ptxtft, xt, ptxt
0, 1.3
wherepis a positive continuous real-valued function on the interval0,∞such that
0 dt
pt <∞, 1.4
andfis a real-valued function defined on the set0,∞×C−r,0,R×R,which satisfies the continuity condition:ft, xt, ptxtis continuous with respect totin0,∞for each given
functionxinC−r,∞,Rthat is continuously differentiable on the interval0,∞.
Our interest will be concentrated on global solutions of the delay differential equation 1.3, for example, on solution of1.3on the whole interval0,∞.By a solution on0,∞
of1.3, we mean a functionxinC−r,∞,R,which is continuously differentiable on the
interval0,∞and such thatpxis continuously differentiable on0,∞and1.3is satisfied for allt >0.
With the delay differential equation1.3, one associates a condition of the form
x0φ, 1.5
or, equivalently,
xt φt for −r≤t≤0, 1.5’
whereφinC−r,0,Ris given with
Also, together with1.3, we specify another condition of the form
lim
t→∞ptx
t L, 1.7
whereLis a given real number.
Equations 1.3–1.7 constitute a boundary value problem BVP on the half-line. A solution on0,∞ of the delay differential equation 1.3 satisfying the boundary value conditions1.5and1.7is said to be a solution on0,∞of the boundary value problem 1.3–1.7or, more briefly, a solution on0,∞of the BVP1.3–1.7.
In the sequel, byPtwe will denote the positive continuous real-valued function on the interval0,∞defined by the formula
Pt t
0 ds
ps fort >0. 1.8
A useful integral representation of the BVP 1.3–1.7 is given by the following lemma, which will be used in proving the main result of the paper.
Lemma 1.1. Let x be a function inC−r,∞,Rthat is continuously differentiable on the interval 0,∞. Thenxis a solution on0,∞of the BVP1.3–1.7if and only if it satisfies
xt ⎧ ⎪ ⎨ ⎪ ⎩
φt for−r ≤t≤0,
LPt t
0
1
pσ ∞
σ
fs, xs, psxs
ds dσ fort >0. 1.9
Proof. Assume thatxsatisfies1.9. Then1.5’or1.5is fulfilled. Moreover, we immediately obtain
ptxt L ∞
t
fs, xs, psxs
ds, 1.10
which implies that limt→∞ptxt L, for example, 1.7 holds true. Furthermore, from
1.10we get
ptxt−ft, xt, ptxt
fort >0, 1.11
which means thatxis a solution on0,∞of1.3. Thus,xis a solution on0,∞of the BVP 1.3–1.7.
Conversely, let us suppose thatxis a solution on0,∞of the BVP1.3–1.7. In view
of1.5’we havext φtfor −r ≤t≤0.Furthermore, from1.3it follows thatxsatisfies 1.11and consequently
ptxt− lim
T→∞pTx
T −t ∞f
s, xs, psxs
So, because of 1.7, we conclude that 1.10 is satisfied. By using 1.10 and taking into account the fact thatx0 φ0 0,we obtain for everyt >0,
xt x0 L t
0 dσ pσ
t
0
1
pσ ∞
σ
fs, xs, psxs
ds dσ
LPt t
0
1
pσ ∞
σ
fs, xs, psxs
ds dσ.
1.13
We have thus proved thatxsatisfies1.9. The proof of the lemma is complete.
Our results in this paper are presented in the form of two theoremsTheorems2.1and
2.2. InTheorem 2.1, sufficient conditions are established in order that the BVP1.3–1.7has at least one solution in the interval0,∞.Theorem 2.2provides sufficient conditions for the
BVP1.3–1.7to have exactly one solution on the interval0,∞. The results of this paper
are stated inSection 2. The proofs of Theorems2.1and2.2are given inSection 3. The proof ofTheorem 2.1is based on the use of the classical Schauder fixed point theorem, while the well-known Banach contraction principle is used in the proof ofTheorem 2.2.
2. Main results
The first main result of this paper is the following theorem, which provides sufficient conditions for BVP1.3–1.7to have at least one solution on the interval0,∞.
Theorem 2.1. Suppose that
ft, ψ, z≤F
t,|ψ|,|z| ∀t, ψ, z∈0,∞×C−r,0,R×R, 2.1
where F is a nonnegative real-valued function defined on0,∞×C−r,0,0,∞×0,∞,which satisfies the following continuity condition:
(C)Ft,|xt|, pt|xt|is continuous with respect totin0,∞for each given functionxin
C−r,∞,Rwhich is continuously differentiable on the interval0,∞.
Assume that
(A) for eacht > 0, the function Ft,·,·is increasing onC−r,0,0,∞×0,∞in the sense that Ft, ψ, z ≤ Ft, ω, v for any ψ, ω in C−r,0,0,∞with ψ ≤ ωi.e., ψτ ≤ ωτfor −r ≤ τ ≤ 0and anyz, vin0,∞withz≤ v.Let a real numbercexist withc > |L|
such that
∞
0F
t, ηt, c
dt≤c− |L|, 2.2
where the functionηinC−r,∞,0,∞depends onφ, cand is defined by
ηt ⎧ ⎨ ⎩
φt for −r≤t≤0,
Then, the BVP1.3–1.7has at least one solutionxon the interval0,∞such that
−c|L|LPt≤xt≤c− |L|LPt for everyt >0, 2.4 −c|L|L≤ptxt≤c− |L|L for everyt >0. 2.5
In addition, for this solutionxof the BVP1.3–1.7,
lim
t→0ptx t ξ
x, 2.6
whereξxis some real number (depending on the solutionx); the numberξxis given by
ξxL
∞
0f
s, xs, psxs
ds. 2.7
It is remarkable that2.6implies
lim
t→0 xt
Pt ξx, 2.8
provided thatpis such that
∞
0
1
pt ∞. 2.9
Our second main result is Theorem 2.2 below. This theorem establishes conditions under which the BVP1.3–1.7has exactly one solution on the interval0,∞.
Theorem 2.2. Let the following generalized Lipschitiz condition be satisfied:
ft, ψ, z−ft, ω, v≤Ktmax
|ψ−ω|,|z−v|,
∀t, ψ, z,t, ω, vin0,∞×C−r,0,R×R, 2.10
whereKis a nonnegative continuous real-valued function on the interval0,∞such that
∞
0max
1, PtKtdt <1. 2.11
Moreover, suppose that2.1holds, whereFis a nonnegative real-valued function on the set0,∞× C−r,0,0,∞×0,∞, which satisfies the continuity condition (C). Assume that (A) is satisfied.
Let there exist a real number c with c > |L|so that 2.2 holds, where the function η in
C−r,∞,0,∞depends onφ, cand is defined by2.3. Then, the BVP1.3–1.7has exactly one solutionxon the interval0,∞with
this unique solutionxis such that2.4and2.5hold. In addition, for this unique solutionxof the BVP1.3–1.7, statement2.6is true, whereξxis some real number (depending on the solutionx);
the numberξxis given by2.7.
3. Proofs of Theorems2.1and2.2
To prove Theorem 2.1, we will use the fixed point technique, by applying the following Schauder theoremsee11.
The Schauder theorem
Let E be a Banach space and Ω any nonempty convex and closed subset of E. If M is a continuous mapping ofΩinto itself andMΩis relatively compact, then the mappingMhas at least one fixed pointi.e., there exists anx∈ΩwithxMx.
Let BC0,∞,Rbe the Banach space of all bounded continuous real-valued functions on the interval0,∞, endowed with the sup-norm · defined by
λsup
t>0
λt forλ∈BC
0,∞,R. 3.1
We need the following compactness criterion for subset of BC0,∞,R, which is
a consequence of the well-known Arzela-Ascoli theorem. This compactness criterion is an adaptation of a lemma due to Avramescu12. In order to formulate this criterion, we note
that a setUof real-valued functions defined on the interval0,∞is called equiconvergent at 0 resp.,∞if all the functions inUare convergent inRat the point 0 resp.,∞and, in addition, for each >0, there existsT ≡T>0 such that, for all the functionsλinU, it holds that|λt−lims→0λs|< for 0< t < Tresp.,|λt−lims→∞λs|< fort > T.
Compactness criterion
Let U be an equicontinuous and uniformly bounded subset of the Banach space BC0,∞,R.IfUis equiconvergent at 0and at∞, it is also relatively compact.
In order to proveTheorem 2.2, we will make use of the well-known Banach contraction principlesee13.
The Banach contraction principle
LetEbe a Banach space andΩany nonempty closed subset ofE. IfMis a contraction ofΩ into itself, then the mappingMhas exactly one fixed pointi.e., there exists anx∈ Ωwith
xMx.
Throughout the remainder of this section,E stands for the set of all functionsu in
C−r,∞,R,which are continuously differentiable on the interval0,∞and such thatpu
is bounded on0,∞.The setEis a Banach space endowed with the norm · Edefined as
follows:
uEmax
max
−r≤t≤0
ut,sup
t>0
ptut
In order to prove Theorems2.1and2.2, we will first establish the following proposi-tion.
Proposition 3.1. Suppose that2.1holds, whereFis a nonnegative real-valued function defined on the set0,∞×C−r,0,0,∞×0,∞, which satisfies the continuity condition (C). Assume that
(A) is satisfied. Letcbe a positive real number such that
∞
0
Ft, ηt, c<∞, 3.3
where the functionηinC−r,∞,0,∞depends onφ, cand is defined by2.3. Also, letΩbe the subset of the Banach spaceEconstituted of all functionsxinE, which satisfies1.5’and2.12. Then
the formula
Mxt ⎧ ⎪ ⎨ ⎪ ⎩
φt for −r≤t≤0,
LPt t
0
1
pσ ∞
σ
fs, xs, psxs
ds dσ fort >0 3.4
makes sense for any functionxinΩ, and this formula defines a mappingMofΩintoE. Moreover,
MΩis relatively compact and the mappingMis continuous.
Proof. Letxbe an arbitrary function inΩ. By the definition ofΩ, the functionxsatisfies1.5’
and2.12. Sinceφ0 0,it follows from1.5’thatx0 0. By taking into account this fact and using2.12, we can easily obtain
xt≤cPt for everyt≥0. 3.5
By virtue of1.5’,3.5, and2.3, it holds that|xt| ≤ηt∀t≥ −r, which ensures that
xt≤ηt for everyt≥0. 3.6
In view of3.6and2.12and the assumptionA, we get
Ft,xt, ptxt≤F
t, ηt, c
fort >0. 3.7
On the other hand, the hypothesis2.1guarantees that
f
t, xt, ptxt≤F
t,xt, ptxt fort >0. 3.8
Thus, we have
f
t, xt, ptxt≤F
t, ηt, c
From3.3and3.9, it follows that ∞
0 f
t, xt, ptxtdt <∞, 3.10
and consequently
∞
0f
t, xt, ptxt
dtexists inR. 3.11
Furthermore, by taking into account3.10and using the hypothesis that01/ptdt<∞, we can conclude that
0
1
pσ ∞
σ
f
t, xt, ptxtdt dσ <∞. 3.12
It follows from3.12that
0
1
pσ ∞
σ
ft, xt, ptxt
dt dσ exists inR. 3.13
As3.13holds true for all functionsxinΩ, we can immediately see that the formula 3.4makes sense for any functionxinΩ, and this formula defines a mappingMofΩinto
C−r,∞,R. We will show thatMis a mapping ofΩintoE, for example, thatMΩ⊆E.For this purpose, let us consider an arbitrary functionxinΩ. Then, by taking into account3.9,
from3.4we obtain fort >0,
ptMxtL ∞
t
fs, xs, psxs
ds
≤ |L| ∞
t
fs, xs, psxsds
≤ |L| ∞
t
Fs, ηs, c
ds
≤ |L| ∞
0F
s, ηs, c
ds.
3.14
Hence,
ptMxt≤Q for everyt >0, 3.15
here
Q|L| ∞
0
Fs, ηs, c
Note that, because of3.3,Q is a nonnegative real constant. Inequality 3.15means that
pMxis bounded on the interval0,∞and soMxbelongs toE. We have thus proved that, for anyx∈Ω, Mx∈E,for example, thatMΩ⊆E.
Now we will prove thatMΩis relatively compact. We observe that, for any function
xinΩ, we haveMxt φtfor−r ≤t≤0.By taking into account this fact as well as the definition of the norm · E, we can easily conclude that it is enough to show that the set
UpMx|0,∞
:x∈Ω 3.17
is relatively compact in the Banach space BC0,∞,R. By using3.9, for any functionxin Ωand everyt1, t2with 0< t1 ≤t2, we obtain
p t1
Mxt1
−pt2
Mxt2 t2
t1
fs, xs, psxs
ds
≤ t2
t1
f
s, xs, psxsds
≤ t2
t1
Fs, ηs, c
ds.
3.18
So, by taking into account3.3, we can easily verify thatUis equicontinuous. Moreover, each functionxinΩsatisfies3.15, where the nonnegative real numberQis defined by3.16 and
it is independent ofx. This guarantees thatUis uniformly bounded. Furthermore, ifxis an arbitrary function inΩ, then we have fort >0,
ptMxt−L∞
t
fs, xs, psxs
ds
≤ ∞
t
f
s, xs, psxsds,
3.19
and hence, because of3.9, it holds that
ptMxt−L≤∞
t
Fs, ηs, c
ds for everyt >0. 3.20
For any functionxinΩ,3.20together with3.3implies that
lim
t→∞ptMx
By again using3.3and3.20, we immediately see thatUis equiconvergent at∞.Now, for each functionxinΩ, we defineξxby2.7. For any functionxinΩ,3.11guarantees thatξx
is a real number. Ifxis an arbitrary function inΩ, then we obtain fort >0,
ptMxt−ξ
x
t
0
fs, xs, psxs
ds
≤ t
0 f
s, xs, psxsds,
3.22
and so, by the virtue of3.9, we get
ptMxt−ξ
x≤
t
0F
s, ηs, c
ds for everyt >0. 3.23
It follows from3.23and3.3that
lim
t→0ptMx t ξ
x, 3.24
for any functionxinΩ. Also, by again using3.3and3.23and taking into account3.24,
we conclude thatUis equiconvergent at 0.Furthermore, by the given compactness criterion, the setUis relatively compact in BC0,∞,R.
Next we will prove that the mappingMis continuous. Consider an arbitrary function
xinΩ, and let{xn}, n≥1,be any sequence of functions inΩsuch that limn→∞xn xin
the sense of · E. It is not difficult to verify that limn→∞xnt xtuniformly int∈−r,∞
and limn→∞xnt xtuniformly int∈0,∞.On the other hand, by3.9, it holds that f
t, xtn, ptxnt≤Ft, ηt, c
for everyt >0, ∀n≥1. 3.25
Thus by taking into account the fact that
0
1
pσ ∞
σ
Ft, ηt, cdt dσ <∞ 3.26
which is a consequence of3.3and the hypothesis that01/ptdt < ∞, we can apply
the Lebesgue dominated convergence theorem to obtain, for everyt >0,
lim
n→ ∞ t
0
1
pσ ∞
σ
fs, xsn, ps
xnsds dσt 0
1
pσ ∞
σ
fs, xs, psxs
ds dσ.
3.27
This, together with the fact that
guarantees the pointwise convergence
lim
n→∞
Mxnt Mxt fort≥ −r. 3.29
It remains to show that this convergence is also convergent in the sense of · E, for example,
that
lim
n→∞Mx
nMx. 3.30
For this purpose, we consider an arbitrary subsequenceMxk of Mxn. SinceMΩ is
relatively compact, there exist a subsequenceMxjof the sequenceMxkand a function uinEso that limj→∞Mxj uin the sense of · E. As the · Econvergence implies the
pointwise one to the same limit function, we must haveu Mx.That is,3.30holds true. Consequently,Mis continuous. The proof of the proposition has been finished.
Proof ofTheorem 2.1. First of all, we observe that the hypothesis2.2implies3.3. LetΩbe the subset of the Banach space ofEdefined as in the proposition. Clearly,Ωis a nonempty convex and closed subset of E. By our proposition, the formula3.4makes sense for any function xinΩ, and this formula defines a continuous mapping Mof ΩintoE such that
MΩis relatively compact. We will show thatMis a mapping ofΩinto itself, for example, thatMΩ⊆Ω.Let us consider an arbitrary functionxinΩ. The functionxsatisfies3.20. By
using the hypothesis2.2, from3.20we obtain
ptMxt−L≤c− |L| for everyt >0, 3.31
which obviously gives
ptMxt≤c for everyt >0. 3.32
By the definition of Ω,3.32 together with the fact that Mxt φt for −r ≤ t ≤ 0 guarantees thatMxbelongs toΩ. We have thus proved that, for eachxinΩ, Mx ∈ Ω, for
example, thatMΩ⊆Ω.
Now the Schauder theorem guarantees the existence of anxinΩ such that1.9is satisfied. By ourLemma 1.1,xis a solution on the interval0,∞of the BVP1.3–1.7. Also,
asx∈ΩandxMx,3.31ensures that the solutionxsatisfies
ptxt−L≤c− |L| for everyt >0. 3.33
That is,xsatisfies2.5. Moreover, sincex0 φ0 0,it follows from2.5thatxis also such that2.4holds. Furthermore, sincex ∈ Ω,3.11and3.24are satisfied, whereξx is
given by2.7. Because of3.11,ξxis a real number. Asx Mx,it follows from3.24that
the solution satisfies2.6. The proof of the theorem is complete.
proposition guarantees that the formula3.4makes sense for any functionxinΩ, and this formula defines a mappingMofΩintoE. As in the proof ofTheorem 2.1, we can use the hypothesis2.2to show thatMis a mapping ofΩinto itself.
Now, we will prove that the mapping M is a contraction. For this purpose, let us consider two arbitrary functionsxandxinΩ. In view of3.4, we haveMxt Mxt φtfor−r≤t≤0,and consequently
max
−r≤t≤0Mxt−
Mxt0. 3.34
Furthermore, by using2.10, from3.4we obtain fort >0,
ptMxt−ptMx t
∞
t
fs, xs, psxs
−fs,xs, psxs
ds
≤ ∞
t
f
s, xs, psxs
−fs,xs, psxsds
≤ ∞
t
Ksmaxxs−xs,psxs−psxsds.
3.35
This gives
sup
t>0
ptMxt−ptMx t≤
∞
0Ktmax
xt−xt,ptxt−ptxtdt.
3.36
By the definition of the norm · EinE, the last inequality together with3.34implies that
Mx−Mx
E≤
∞
0
Ktmaxxt−xt,ptxt−ptxtdt. 3.37
From the definition ofΩ, it follows thatxt xt φtfor−r≤t≤0,and so
xt−xt0 for −r≤t≤0. 3.38
Moreover, in view of the fact thatx0 x0 φ0 0,we get fort >0,
xt−xtt
0
xs−xsds
≤ t
0
xs−xsds
t
0
1
pspsx
s−psxsds.
But, by the definition of the norm · E, we have
ptxt−ptxt≤x−x
E for everyt >0. 3.40
Thus, we obtain
xt−xt≤Ptx−x
E for everyt >0. 3.41
This inequality and3.38can be written as
xt−xt≤μtx−x
E ∀t≥ −r, 3.42
where the functionμis defined by
μt 0 for −r≤t≤0, μt Pt fort >0. 3.43
Hence,
xtτ−xtτ≤μtτx−x
E fort >0, −r≤τ ≤0, 3.44
for example,
xtτ−xtτ≤μtτx−x
E fort >0, −r≤τ≤0. 3.45
Therefore,
max
−r≤τ≤0xtτ−xtτ≤
max
−r≤τ≤0μtτ
x−xE fort >0. 3.46
But sinceμis nondecreasing on−r,∞, we have
max
−r≤τ≤0μtτ μt Ptfort >0. 3.47
So it holds that
xt−xt≤Ptx−x
Next, by using3.48and3.40, from3.37we obtain
Mx−Mx
E≤
∞
0
KtmaxPtx−xE,x−xEdt
∞
0Ktmax
Pt,1dtx−xE.
3.49
By taking into account2.11, we see that the mappingM:Ω→Ωis a contraction.
Finally, by the Banach contraction principle, the mappingM : Ω → Ωhas a unique fixed point. Namely, there exists exactly onexin Ωsuch thatx Mx, for example, such that1.9is satisfied. So, by our lemma, the BVP 1.3–1.7 has exactly one solutionxon the interval0,∞such that2.12holds.Note that a solutionxon the interval0,∞of the BVP1.3–1.7belongs toΩif and only ifxsatisfies2.12.As in the proof ofTheorem 2.1, we conclude that this unique solutionx of the BVP1.3–1.7 satisfies2.4and 2.5. In
addition, for this solutionx,2.6holds true, where the real numberξxis given by2.7. The
proof of the theorem is now complete.
Acknowledgment
This work was supported by the NNSF of ChinaGrant no. 05046012.
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