Volume 2011, Article ID 570493, 9 pagesdoi:10.1155/2011/570493 Research Article Nonlocal Four-Point Boundary Value Problem for the Singularly Perturbed Semilinear Differential Equations
Trang 1Volume 2011, Article ID 570493, 9 pages
doi:10.1155/2011/570493
Research Article
Nonlocal Four-Point Boundary Value Problem
for the Singularly Perturbed Semilinear
Differential Equations
Robert Vrabel
Institute of Applied Informatics, Automation and Mathematics, Faculty of Materials Science and
Technology, Hajdoczyho 1, 917 01 Trnava, Slovakia
Correspondence should be addressed to Robert Vrabel,robert.vrabel@stuba.sk
Received 21 April 2010; Revised 9 September 2010; Accepted 13 September 2010
Academic Editor: Daniel Franco
Copyrightq 2011 Robert Vrabel 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
This paper deals with the existence and asymptotic behavior of the solutions to the singularly perturbed second-order nonlinear differential equations For example, feedback control problems, such as the steady states of the thermostats, where the controllers add or remove heat, depending upon the temperature detected by the sensors in other places, can be interpreted with a second-order ordinary differential equation subject to a nonlocal four-point boundary condition Singular perturbation problems arise in the heat transfer problems with large Peclet numbers We show that the solutions of mathematical model, in general, start with fast transient which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to the solution of reduced problem with an arising new fast transient at the end of considered interval Our analysis relies on the method of lower and upper solutions
1 Motivation and Introduction
We will consider the nonlocal four-point boundary value problem
y ky ft, y
, t ∈ a, b, k < 0, 0 < 1, 1.1
y c − ya 0, yb − yd 0, a < c ≤ d < b. 1.2
We focus our attention on the existence and asymptotic behavior of the solutions y t
for singularly perturbed boundary value problem 1.1, 1.2 and on an estimate of the difference between y t and a solution ut of the reduced equation ku ft, u when a small parameter tends to zero.
Trang 2Singularly perturbed systems SPS normally occur due to the presence of small
“parasitic” parameters, armature inductance in a common model for most DC motors, small time constants, and so forth The literature on control of nonlinear SPS is extensive, at least
starting with the pioneering work of Kokotovi´c et al nearly 30 years ago1 and continuing
to the present including authors such as Artstein2,3, Gaitsgory et al 4 6
Such boundary value problems can also arise in the study of the steady-states of
a heated bar with the thermostats, where the controllers at t a and t b maintain a temperature according to the temperature registered by the sensors at t c and t d, respectively In this case, we consider a uniform bar of length b − a with nonuniform temperature lying on the t-axis from t a to t b The parameter represents the thermal
diffusivity Thus, the singular perturbation problems are of common occurrence in modeling the heat-transport problems with large Peclet number7
We show that the solutions of1.1, 1.2, in general, start with fast transient |y
a| →
∞ of y t from y a to ut, which is the so-called boundary layer phenomenon, and after decay of this transient they remain close to ut with an arising new fast transient of
y t from ut to y b |y
b| → ∞ Boundary thermal layers are formed due to the nonuniform convergence of the exact solution y to the solution u of a reduced problem in the neighborhood of the ends a and b of the bar.
The differential equations of the form 1.1 have also been discussed in 8 but with
the boundary conditions ya 0, yb − yc 0, that is, with free end y a Moreover, we show that the convergence rate of solutions y toward the solution u of a reduced problem is
at least O on every compact subset of a, b in 8, the rate of convergence is only of the
order O√
We will write s Or when 0 < lim → 0|s/r| < ∞.
The situation in the case of nonlocal boundary value problem is complicated by the fact that there are the inner points in the boundary conditions, in contrast to the “standard” boundary conditions as the Dirichlet problem, Neumann problem, Robin problem, periodic boundary value problem9 12, for example In the problem considered; there is not positive solution v of differential equation y− my 0, m > 0, 0 < i.e., v is convex such that
v c − v a uc − ua > 0 and v t → 0for t ∈ a, b and → 0, which could be used
to solve this problem by the method of lower and upper solutions The application of convex
functions is essential for composing the appropriate barrier functions α, β for two-endpoint
boundary conditions,see, e.g., 10 We will define the correction function v corrt which
will allow us to apply the method
In the past few years the multipoint boundary value problem has received a wide attentionsee, e.g., 13,14 and the references therein For example, Khan 14 have studied
a four-point boundary value problem of type yc − ν1ya 0, yb − ν2yd 0 where the
constants ν1, ν2are not simultaneously equal to 1 and 1.
As was said before, we apply the method of lower and upper solutions to prove the existence of a solution for problem1.1, 1.2 which converges uniformly to the solution u
of the reduced problemi.e., if we let → 0 in1.1 on every compact subset of interval
a, b As usual, we say that α ∈ C2a, b is a lower solution for problem 1.1, 1.2 if
α t kα t ≥ ft, α t and α c − α a 0, α b − α d ≤ 0 for every t ∈ a, b An upper solution β ∈ C2a, b satisfies β
t kβ t ≤ ft, β t and β c − β a 0,
β b − β d ≥ 0 for every t ∈ a, b.
α ≤ β , then there exists solution y of1.1, 1.2 with α ≤ y ≤ β
Proof of uniqueness of solution for1.1, 1.2 will be based on the following lemmas
Trang 3Lemma 1.2 cf 16, Theorem 1Peano’s phenomenon Assume that
i the function
h
t, y
ft, y
is nondecreasing with respect to the variable y for each t ∈ a, b,
ii h ∈ Ca, b × R.
If x , y are two solutions of 1.1, 1.2, then
a x t − y t C const in a, b,
b if C > 0 C < 0, then for each C1, 0 ≤ C1≤ C 0 ≥ C1 ≥ C the function y t C1is a solution of the problem1.1, 1.2.
Lemma 1.3 If h satisfies the strengthened condition (i)
i the function ht, y ft, y − ky is increasing with respect to the variable y for each
t ∈ a, b,
then there exists at most one solution of1.1, 1.2.
Proof Assume to the contrary that x , y are two solutions of the problem 1.1, 1.2
Lemma 1.2implies that y x c on a, b for some constant c/ 0 Thus
0 y
t − x
t ht, y t− ht, x t ht, x t c − ht, x t / 0. 1.4
This is a contradiction
The following assumptions will be made throughout the paper
A1 For a reduced problem ky ft, y, there exists C2 function u such that kut
ft, ut on a, b.
DenoteHu {t, y | a ≤ t ≤ b, |y − ut| < dt}, where dt is the positive
continuous function ona, b such that
d t
⎧
⎪
⎪
⎪
⎪
|uc − ua| δ, for a ≤ t ≤ a δ
2,
δ, for a δ ≤ t ≤ b − δ,
|ub − ud| δ, for b − δ
2 ≤ t ≤ b,
1.5
δ is a small positive constant.
A2 The function f ∈ C1Hu satisfies the condition
∂f
t, y
∂y
≤ w < −k for every
t, y
Trang 42 Main Result
the problem1.1, 1.2 has in Hu a unique solution, y , satisfying the inequality
−v corr t − C ≤ y t − ut v t C for uc − ua ≥ 0,
t − C ≤ y t − ut v t ≤ vcorr t C for uc − ua ≤ 0 2.1
on a, b where
v t u c − ua
m/b−t − e√m/t−b e√m/t−d − e√m/d−t
,
t |ub − ud|
m/t−a − e√m/a−t e√m/c−t − e√m/t−c
,
D
e√
m/b−a e√m/d−c e√m/c−b e√m/a−d
− e√
m/a−b e√m/c−d e√m/b−c e√m/d−a
,
2.2
m −k − w, C 1/m max{|ut|; t ∈ a, b} and the positive function
vcorr t w |uc − ua|√
m
· −O1 v t
uc − ua O
e
√
m/a−d t
|ub − ud| tO
e
√
m/χt
,
2.3
χt < 0 for t ∈ a, b and v corra vcorr c.
Remark 2.2 The function v t satisfies the following:
1 v
− mv 0;
2 v c − v a −uc − ua, v b − v d 0;
3 v t ≥ 0 ≤ 0 is decreasing increasing for a ≤ t ≤ b d/2 and increasing
decreasing for b d/2 ≤ t ≤ b if uc − ua ≥ 0≤ 0;
4 v t converges uniformly to 0 for → 0on every compact subset ofa, b;
5 v t uc − uaOe√m/χt where χt a − t for a ≤ t ≤ b d/2 and
χt t − b a − d for b d/2 < t ≤ b.
The function t satisfies the following:
0;
d |ub − ud|;
t ≥ 0 is decreasing for a ≤ t ≤ a c/2 and increasing for a c/2 ≤ t ≤ b;
Trang 5t converges uniformly to 0 for → 0on every compact subset ofa, b;
t |ub − ud|Oe√
The correction function v corrt will be determined precisely in the next section.
3 The Correction Function vcorrt
Consider the linear problem
y− my −2w|v t|, t ∈ a, b, > 0 3.1
with the boundary conditions1.2
We apply the method of lower and upper solutions We define
α t 0,
β t 2w
m max{|v t|, t ∈ a, b} 2w
m |v a|. 3.2
Obviously,|v a| |uc − ua|1 Oe√m/a−c and the constant functions α, β satisfy
the differential and boundary inequalities required on the lower and upper solutions for 3.1 and the boundary conditions1.2 Thus on the basis ofLemma 1.1for every > 0 the unique solution yLin
of linear problem3.1, 1.2 satisfies
0≤ yLin
t ≤ 2w
m |uc − ua| 1 O e
√
m/a−c
3.3
ona, b The solution we denote by v corrt, that is, the function
vcorr tdef yLin
and we compute vcorr t exactly as following:
vcorr t −
ψ a − ψ c
uc − ua v t
ψ d − ψ b
|ub − ud| t ψ t, 3.5 where
ψ t w |uc − ua|
D√
e√
m/b−t e√m/t−b − e√m/d−t − e√m/t−d
. 3.6
Trang 6ψ a − ψ c w |uc − ua|
D√
e√
m/b−a e√m/a−b − e√m/d−a − e√m/a−d
−w |uc − ua|
D√
e√
m/b−c e√m/c−b − e√m/d−c − e√m/c−d
w |uc − ua|√
ψ d − ψ b w |uc − ua|
D√
e√
m/b−d e√m/d−b− 2
−w |uc − ua|
D√
2− e√m/d−b − e√m/b−d
w |uc − ua|√
e√
m/a−d
,
ψ t w |uc − ua|√
e√
m/χt
.
3.7 Thus, we obtain
vcorr t w |uc − ua|√
m
· −O1 v t
uc − ua O
e√
m/a−d
t
|ub − ud| tO
e√
m/χt
.
3.8
Hence, taking into consideration3.8 and the fact that vcorr a vcorr c, the correction function vcorr converges uniformly to 0 ona, b for → 0.
First we will consider the caseuc − ua ≥ 0 We define the lower solutions by
α t ut v t − vcorr t − Γ 4.1 and the upper solutions by
β t ut v t Γ 4.2 HereΓ Δ/m where Δ is the constant which shall be defined below, α ≤ β on a, b and
satisfy the boundary conditions prescribed for the lower and upper solutions of1.1, 1.2
Trang 7Now we show that α t kα t ≥ ft, α t and β
t kβ t ≤ ft, β t Denote ht, y ft, y − ky By the Taylor theorem, we obtain
h t, α t ht, α t − ht, ut
∂h t, θ t
∂y
v t − v corr t − Γ
wheret, θ t is a point between t, α t and t, ut, and t, θ t ∈ Hu for sufficiently small Hence, from the inequalities m ≤ ∂ht, θ t/∂y ≤ m 2w in Hu we have
α t − ht, α t ≥ ut v
t − v corr
t − m 2wv t
mv corr t mΓ
4.4
Because v t |v t| we have −vcorr t − 2wv t mvcorr t 0; as follows from
differential equation 3.1, we get
α t − ht, α t ≥ ut mΓ ≥ − ut Δ. 4.5
For β t we have the inequality
h
t, β t− β
t ∂h
t, θ t
∂y v t Γ − β
t
mv t Γ − ut v
t
≥ Δ − ut ,
4.6
wheret, θ t is a point between t, ut and t, β t and t, θ t ∈ Hu for sufficiently small .
The Case: uc − ua ≤ 0
The lower solution
α t ut v t − Γ 4.7 and the upper solution
β t ut v t v corr t Γ 4.8
Trang 8α − ht, α u v
− ∂h
∂y v − Γ
u v
∂h
∂y −v Γ
≥ u v
m−v Γ
u Δ ≥ Δ − u ,
h
t, β
− β
∂h
∂y
v vcorr Γ
− u− v
− vcorr
≥ m 2wv m vcorr Γ
− u− v
− v corr
−2w|v | mv corr− vcorr Δ − u
Δ − u≥ Δ − u .
4.9
Now, if we choose a constantΔ such that Δ ≥ |ut|, t ∈ a, b, then α
t ≥ ht, α t and
β t ≤ ht, β t in a, b.
The existence of a solution for1.1, 1.2 satisfying the above inequality follows from
Lemma 1.1and the uniqueness of solution inHu follows fromLemma 1.3
Remark 4.1. Theorem 2.1implies that y t ut O on every compact subset of a, b
and lim → 0y a uc, lim → 0y b ud The boundary layer effect occurs at the point
a or/and b in the case when ua / uc or/and ub / ud.
Acknowledgments
This research was supported by Slovak Grant Agency, Ministry of Education of Slovak Republic under Grant no 1/0068/08 The author would like to thank the reviewers for helpful comments on an earlier draft of this article
References
1 P V Kokotovi´c, H K Khalil, and J O’Reilly, Singular Perturbation Methods in Control: Analysis and
Design, Academic Press, London, UK, 1986.
2 Z Artstein, “Stability in the presence of singular perturbations,” Nonlinear Analysis: Theory, Methods
& Applications, vol 34, no 6, pp 817–827, 1998.
3 Z Artstein, “Singularly perturbed ordinary differential equations with nonautonomous fast
dynamics,” Journal of Dynamics and Di fferential Equations, vol 11, no 2, pp 297–318, 1999.
4 Z Artstein and V Gaitsgory, “The value function of singularly perturbed control systems,” Applied
Mathematics and Optimization, vol 41, no 3, pp 425–445, 2000.
5 V Gaitsgory, “On a representation of the limit occupational measures set of a control system with
applications to singularly perturbed control systems,” SIAM Journal on Control and Optimization, vol.
43, no 1, pp 325–340, 2004
6 V Gaitsgory and M.-T Nguyen, “Multiscale singularly perturbed control systems: limit occupational
measures sets and averaging,” SIAM Journal on Control and Optimization, vol 41, no 3, pp 954–974,
2002
Trang 97 A Khan, I Khan, T Aziz, and M Stojanovic, “A variable-mesh approximation method for singularly
perturbed boundary-value problems using cubic spline in tension,” International Journal of Computer
Mathematics, vol 81, no 12, pp 1513–1518, 2004.
8 R Vr´abel, “Three point boundary value problem for singularly perturbed semilinear differential
equations,” Electronic Journal of Qualitative Theory of Di fferential Equations, vol 70, pp 1–4, 2009.
9 C De Coster and P Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, vol 205 of
Mathematics in Science and Engineering, Elsevier, Amsterdam, The Netherlands, 1st edition, 2006.
10 K W Chang and F A Howes, Nonlinear Singular Perturbation Phenomena: Theory and Applications, vol.
56 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1984.
11 R Vr´abel, “Asymptotic behavior of T-periodic solutions of singularly perturbed second-order
differential equation,” Mathematica Bohemica, vol 121, no 1, pp 73–76, 1996.
12 R Vr´abel, “Semilinear singular perturbation,” Nonlinear Analysis: Theory, Methods & Applications, vol.
25, no 1, pp 17–26, 1995
13 Y Guo and W Ge, “Positive solutions for three-point boundary value problems with dependence on
the first order derivative,” Journal of Mathematical Analysis and Applications, vol 290, no 1, pp 291–301,
2004
14 R A Khan, “Positive solutions of four-point singular boundary value problems,” Applied Mathematics
and Computation, vol 201, no 1-2, pp 762–773, 2008.
15 J Mawhin, Points Fixes, Points Critiques et Probl`emes aux Limites, S´eminaire de Math´ematiques
Sup´erieures, no 92, Presses de l’Universit´e de Montr´eal, Montreal, Canada, 1985
16 V ˇSeda, “On some nonlinear boundary value problems for ordinary differential equations,” Archivum
Mathematicum, vol 25, no 4, pp 207–222, 1989.
...satisfy the boundary conditions prescribed for the lower and upper solutions of1.1, 1.2
Trang 7Now... class="text_page_counter">Trang 9
7 A Khan, I Khan, T Aziz, and M Stojanovic, “A variable-mesh approximation method for singularly< /p>
perturbed. .. ≥ is decreasing for a ≤ t ≤ a c/2 and increasing for a c/2 ≤ t ≤ b;
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