Hindawi Publishing Corporation BoundaryValue Problems Volume 2011, Article ID 570493, 9 pages doi:10.1155/2011/570493 ResearchArticleNonlocalFour-PointBoundaryValueProblemfortheSingularlyPerturbedSemilinear 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 Copyright q 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 thesingularlyperturbed 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 nonlocalfour-pointboundary 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 thenonlocalfour-pointboundaryvalueproblem y ky f t, y ,t∈a, b,k<0, 0 < 1, 1.1 y c − y a 0,y b − y d 0,a<c≤ d<b. 1.2 We focus our attention on the existence and asymptotic behavior of the solutions y t forsingularlyperturbedboundaryvalueproblem 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. 2 BoundaryValue Problems Singularlyperturbed 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 ´ cetal. nearly 30 years ago 1 and continuing to the present including authors such as Artstein 2, 3, Gaitsgory et al. 4–6. Such boundaryvalue 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 number 7. We show that the solutions of 1.1, 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 theboundary conditions y a0, yb − yc0, 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, bin 8, the rate of convergence is only of the order O √ . We will write sOr when 0 < lim →0 |s/r| < ∞. The situation in the case of nonlocalboundaryvalueproblem is complicated by the fact that there are the inner points in theboundary conditions, in contrast to the “standard” boundary conditions as the Dirichlet problem, Neumann problem, Robin problem, periodic boundaryvalueproblem 9–12, for example. In theproblem 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 auc −ua > 0andv t → 0 for 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 corr t which will allow us to apply the method. In the past few years the multipoint boundaryvalueproblem has received a wide attention see, e.g., 13, 14 and the references therein. For example, Khan 14 have studied a four-pointboundaryvalueproblem of type yc − ν 1 ya0, yb − ν 2 yd0 where the constants ν 1 ,ν 2 are 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 forproblem 1.1, 1.2 which converges uniformly to the solution u of the reduced problem i.e., if we let → 0 in 1.1 on every compact subset of interval a, b. As usual, we say that α ∈ C 2 a, b is a lower solution forproblem 1.1, 1.2 if α tkα t ≥ ft, α t and α c − α a0, α b − α d ≤ 0 for every t ∈a, b. An upper solution β ∈ C 2 a, b satisfies β tkβ t ≤ ft, β t and β c − β a0, β b − β d ≥ 0 for every t ∈a, b. Lemma 1.1 see 15. If α , β are respectively lower and upper solutions for 1.1, 1.2 such that α ≤ β , then there exists solution y of 1.1, 1.2 with α ≤ y ≤ β . Proof of uniqueness of solution for 1.1, 1.2 will be based on the following lemmas. BoundaryValue Problems 3 Lemma 1.2 cf. 16, Theorem 1 Peano’s phenomenon. Assume that i the function h t, y f t, y − ky 1.3 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 C 1 , 0 ≤ C 1 ≤ C 0 ≥ C 1 ≥ C the function y tC 1 is a solution of theproblem 1.1, 1.2. Lemma 1.3. If h satisfies the strengthened condition (i) i the function ht, yft, y − ky is increasing with respect to the variable y for each t ∈a, b, then there exists at most one solution of 1.1, 1.2. Proof. Assume to the contrary that x ,y are two solutions of theproblem 1.1, 1.2. Lemma 1.2 implies t hat y x c on a, b for some constant c / 0. Thus 0 y t − x t h t, y t − h t, x t h t, x t c − h t, 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 C 2 function u such that kut ft, ut on a, b. Denote Hu{t, y | a ≤ t ≤ b, |y − ut| <dt}, where dt is the positive continuous function on a, b such that d t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ | u c − u a | δ, for a ≤ t ≤ a δ 2 , δ, for a δ ≤ t ≤ b − δ, | u b − u d | δ, for b − δ 2 ≤ t ≤ b, 1.5 δ is a small positive constant. A2 The function f ∈ C 1 Hu satisfies the condition ∂f t, y ∂y ≤ w<−k for every t, y ∈H u . 1.6 4 BoundaryValue Problems 2. Main Result Theorem 2.1. Under the assumptions (A1) and (A2) there exists 0 such that for every ∈ 0, 0 theproblem 1.1, 1.2 has in Hu a unique solution, y , satisfying the inequality −v corr t − v t − C ≤ y t − u t v t ≤ v t C for u c − u a ≥ 0, −v t − C ≤ y t − u t v t ≤ v corr t v t C for u c − u a ≤ 0 2.1 on a, b where v t u c − u a D · e √ m/b−t − e √ m/t−b e √ m/t−d − e √ m/d−t , v t | u b − u d | D · e √ 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{|u t|; t ∈a, b} and the positive function v corr t w | u c − u a | √ m · −O 1 v t u c − u a O e √ m/a−d v t | u b − u d | tO e √ m/χt , 2.3 χt < 0 for t ∈ a, b and v corr av corr 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 d0; 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 → 0 on every compact subset of a, b; 5 v tuc − uaOe √ m/χt where χta − t for a ≤ t ≤ b d/2and χtt − b a − d for b d/2 <t≤ b. The function v t satisfies the following: 1 v − mv 0; 2 v c − v a0, v b − v d|ub − ud|; 3 v t ≥ 0 is decreasing for a ≤ t ≤ a c/2 and increasing for a c/2 ≤ t ≤ b; BoundaryValue Problems 5 4 v t converges uniformly to 0 for → 0 on every compact subset of a, b; 5 v t|ub −ud|Oe √ m/ χt where χtc −b a −t for a ≤ t<a c/2and χtt − b for a c/2 ≤ t ≤ b. The correction function v corr t will be determined precisely in the next section. 3. The Correction Function v corr t Consider the linear problem y − my −2w | v t | ,t∈a, b,>0 3.1 with theboundary conditions 1.2. 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 theboundary conditions 1.2. Thus on the basis of Lemma 1.1 for every >0 the unique solution y Lin of linear problem 3.1, 1.2 satisfies 0 ≤ y Lin t ≤ 2w m | u c − u a | 1 O e √ m/a−c 3.3 on a, b. The solution we denote by v corr t, that is, the function v corr t def y Lin t 3.4 and we compute v corr t exactly as following: v corr t − ψ a − ψ c u c − u a v t ψ d − ψ b | u b − u d | v t ψ t , 3.5 where ψ t w | u c − u a | D √ m t e √ m/b−t e √ m/t−b − e √ m/d−t − e √ m/t−d . 3.6 6 BoundaryValue Problems Hence ψ a − ψ c w | u c − u a | D √ m a e √ m/b−a e √ m/a−b − e √ m/d−a − e √ m/a−d − w | u c − u a | D √ m c e √ m/b−c e √ m/c−b − e √ m/d−c − e √ m/c−d w | u c − u a | √ m O 1 , ψ d − ψ b w | u c − u a | D √ m d e √ m/b−d e √ m/d−b − 2 − w | u c − u a | D √ m b 2 − e √ m/d−b − e √ m/b−d w | u c − u a | √ m O e √ m/a−d , ψ t w | u c − u a | √ m tO e √ m/χt . 3.7 Thus, we obtain v corr t w | u c − u a | √ m · −O 1 v t u c − u a O e √ m/a−d v t | u b − u d | tO e √ m/χt . 3.8 Hence, taking into consideration 3.8 and the fact that v corr av corr c, the correction function v corr converges uniformly to 0 on a, b for → 0 . 4. Proof of Theorem 2.1 First we will consider the case uc − ua ≥ 0. We define the lower solutions by α t u t v t − v corr t − v t − Γ 4.1 and the upper solutions by β t u t v t v t Γ . 4.2 Here Γ Δ/m where Δ is the constant which shall be defined below, α ≤ β on a, b and satisfy theboundary conditions prescribed forthe lower and upper solutions of 1.1, 1.2. BoundaryValue Problems 7 Now we show that α tkα t ≥ ft, α t and β tkβ t ≤ ft, β t. Denote ht, yft, y − ky. By the Taylor theorem, we obtain h t, α t h t, α t − h t, u t ∂h t, θ t ∂y v t − v corr t − v t − Γ , 4.3 where t, 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 − h t, α t ≥ u t v t − v corr t − v t − m 2w v t mv corr t mv t mΓ . 4.4 Because v t|v t| we have −v corr t − 2wv tmv corr t0; as follows from differential equation 3.1,weget α t − h t, α t ≥ u t mΓ ≥− u t Δ. 4.5 For β t we have the inequality h t, β t − β t ∂h t, θ t ∂y v t v t Γ − β t m v t v t Γ − u t v t v t ≥ Δ − u t , 4.6 where t, 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 u t v t − v t − Γ 4.7 and the upper solution β t u t v t v corr t v t Γ 4.8 8 BoundaryValue Problems satisfy α − h t, α u v − v − ∂h ∂y v − v − Γ u v − v ∂h ∂y −v v Γ ≥ u v − v m −v v Γ u Δ ≥ Δ − u , h t, β − β ∂h ∂y v v corr v Γ − u − v − v corr − v ≥ m 2w v m v corr v Γ − u − v − v corr − v −2w | v | mv corr − v corr Δ − u Δ − u ≥ Δ − u . 4.9 Now, if we choose a constant Δ such that Δ ≥|u t|, t ∈a, b, then α t ≥ ht, α t and β t ≤ ht, β t in a, b. The existence of a solution for 1.1, 1.2 satisfying the above inequality follows from Lemma 1.1 and t he uniqueness of solution in Hu follows from Lemma 1.3. Remark 4.1. Theorem 2.1 implies that y tutO on every compact subset of a, b and lim →0 y auc, lim →0 y bud. Theboundary 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 t hank the reviewers for helpful comments on an earlier draft of this article. References 1 P. V. K o kotovi ´ 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 Differential Equations, vol. 11, no. 2, pp. 297–318, 1999. 4 Z. Artstein and V. Gaitsgory, “The value function of singularlyperturbed 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 singularlyperturbed control systems,” SIAM Journal on Control and Optimization, vol. 43, no. 1, pp. 325–340, 2004. 6 V. Gaitsgory and M T. Nguyen, “Multiscale singularlyperturbed control systems: limit occupational measures sets and averaging,” SIAM Journal on Control and Optimization, vol. 41, no. 3, pp. 954–974, 2002. BoundaryValue Problems 9 7 A. Khan, I. Khan, T. Aziz, and M. Stojanovic, “A variable-mesh approximation method forsingularlyperturbed 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 boundaryvalueproblemforsingularlyperturbedsemilinear differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 70, pp. 1–4, 2009. 9 C. De Coster and P. Habets, Two-Point BoundaryValue 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 singularlyperturbed 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 b oundary 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 boundaryvalue 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 ´ edeMontr ´ eal, Montreal, Canada, 1985. 16 V. ˇ Seda, “On some nonlinear boundaryvalue problems for ordinary diff erential equations,” Archivum Mathematicum, vol. 25, no. 4, pp. 207–222, 1989. . Corporation Boundary Value Problems Volume 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. < ∞. 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. 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