Journal of Computational and Applied Mathematics 196 (26) 387 393 www.elsevier.com/locate/cam Upper and lower solution method for fourth-order four-point boundary value problems Qin Zhang a, Shihua Chen a,, Jinhu Lü b a School of Mathematics and Statistics, Wuhan University, Wuhan 4372, PR China b Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 18, PR China Received 18 November 24 Abstract This paper is concerned with the fourth-order ordinary differential equation u (4) (t) = f (t, u(t), u (t)), <t<1 with the four-point boundary conditions au ( ) bu ( ) =, cu (ξ 2 ) + du (ξ 2 ) =, where ξ i [, 1] (i = 1, 2) and a, b, c, d are nonnegative constants satisfying ad + bc + ac(ξ 2 )>. Some new existence results are obtained by developing the upper and lower solution method and the monotone iterative technique. 25 Elsevier B.V. All rights reserved. Keywords: Lower and upper solution method; Fourth-order ordinary differential equation; Four-point boundary conditions 1. Introduction It is well known that the upper and lower solution method is a powerful tool for proving existence results for boundary value problems. In many cases it is possible to find a minimal solution and a maximal solution between the lower solution and the upper solution by the monotone iterative technique. The upper and lower solution method has been used to deal with the multi-point boundary value problem for second-order ordinary differential equations and the two-point boundary value problem for higher-order ordinary differential equations [6,9,8,13]. There are fewer results on multi-point boundary value problems for higher-order equations in the literature of ordinary differential equations. For this reason, we consider the fourth-order nonlinear ordinary differential equation u (4) (t) = f (t, u(t), u (t)), <t<1 (1.1) Corresponding author. E-mail address: shcheng@whu.edu.cn (S. Chen). 377-427/$ - see front matter 25 Elsevier B.V. All rights reserved. doi:1.116/j.cam.25.9.7
388 Q. Zhang et al. / Journal of Computational and Applied Mathematics 196 (26) 387 393 together with the four-point boundary conditions au ( ) bu ( ) =, cu (ξ 2 ) + du (ξ 2 ) =, (1.2) where a, b, c, d are nonnegative constants satisfying ad + bc + ac(ξ 2 )>, < ξ 2 1 and f C([, 1] R R). Eq. (1.1) is often referred to as a generalized beam equation, which has been studied by several authors. A brief discussion, which is easily accessible to the nonexpert reader, of the physical interpretation of the beam equation associated with some boundary conditions can be found in Zill and Cullen [14, pp. 237 243]. Several authors have studied the equation when it takes a simple form such as f (t, u(t), u (t))=a(t)g(u(t)) or f (t, u(t), u (t))=f (t, u(t)) under a variety of boundary conditions [7,1,3,12,1]. The type of multi-point boundary conditions considered in (1.2) are also somewhat different from the conjugate [11], focal [2], and Lidstone [4] conditions that are commonly encountered in the literature. We will develop the upper and lower solution method for system (1.1) and (1.2) and establish some new existence results. 2. Preliminaries and lemmas In this section, we will give some preliminary considerations and some lemmas which are essential to our main results. Lemma 2.1. If α = ad + bc + ac(t 2 ) = and h(t) C[,t 2 ], then the boundary value problem u (t) = h(t), au( ) bu ( ) =, cu(t 2 ) + du (t 2 ) = (2.1) has a unique solution u(t) = G(t, s)h(s) ds, where 1 α (a(s ) + b)(d + c(t 2 t)), s<t t 2, G(t, s) = 1 α (a(t ) + b)(d + c(t 2 s)), t s t 2. Proof. Integrating the first equation of (2.1) over the interval [,t] for t [,t 2 ],wehave (2.2) t u(t) = u( ) + u ( )(t ) + (t s)h(s) ds, (2.3) which implies u(t 2 ) = u( ) + u ( )(t 2 ) + (t 2 s)h(s) ds, t 1 t2 u (t 2 ) = u ( ) + h(s) ds. (2.4)
Q. Zhang et al. / Journal of Computational and Applied Mathematics 196 (26) 387 393 389 Therefore, this equation together with the second equation of (2.1) leads to u( ) = 1 α (a(t 2 ) + b)(d + c(t 2 s))h(s) ds, u a ( ) = α (d + c(t 2 s))h(s) ds. Substituting this equation into Eq. (2.3), we have u(t) = 1 ( t (a(s ) + b)(d + c(t 2 t))h(s) ds α ) + (a(t ) + b)(d + c(t 2 s))h(s) ds, which implies the Lemma. t Lemma 2.2. Suppose a, b, c, d,, ξ 2 are nonnegative constants satisfying < ξ 2 1, b a, d c+cξ 2 and δ = ad + bc + ac(ξ 2 )>. If u(t) C 4 [, 1] satisfies u (4) (t), t (, 1), u(), u(1), au ( ) bu ( ), cu (ξ 2 ) + du (ξ 2 ) then u(t) and u (t) for t [, 1]. Proof. Let u (4) (t) = h(t), t (, 1), u() = x, u(1) = x 1, au ( ) bu ( ) = x 2, cu (ξ 2 ) + du (ξ 2 ) = x 3, where x, x 1, x 2, x 3, h(t) C[, 1] and h(t). In virtue of Lemma 2.1, we have 1 1 ξ2 u(t) = tx 1 + (1 t)x G 1 (t, ξ)r(ξ) dξ + G 1 (t, ξ) G 2 (ξ, s)h(s) ds dξ, ξ2 u (t) = R(t) G 2 (t, s)h(s) ds, (2.5) where R(t) = 1 δ ((a(t ) + b)x 3 + (c(ξ 2 t) + d)x 2 ), (2.6) { s(1 t), s<t 1, G 1 (t, s) = t(1 s), t s 1. 1 δ (a(s ) + b)(d + c(ξ 2 t)), s<t ξ 2, G 2 (t, s) = 1 δ (a(t ) + b)(d + c(ξ 2 s)), t s ξ 2. On the other hand, the assumptions of the lemma imply R(t) for t [, 1], and G 1 (t, s) and G 2 (t, s) for (t, s) [, 1] [, 1]. Thus, u(t) and u (t) for t [, 1], which complete the proof of the lemma. (2.7) (2.8)
39 Q. Zhang et al. / Journal of Computational and Applied Mathematics 196 (26) 387 393 Now, we introduce the following two definitions about the upper and lower solutions of the boundary value problem (1.1) (1.2). Definition 1. A function α(t) is said to be an upper solution of the boundary problem (1.1) (1.2), if it belongs to C 4 [, 1] and satisfies α (4) (t) f(t,α(t), α (t)), t (, 1), α(), α(1), aα ( ) bα ( ), cα (ξ 2 ) + dα (ξ 2 ). (2.9) Definition 2. A function β(t) is said to be a lower solution of the boundary problem (1.1) (1.2), if it belongs to C 4 [, 1] and satisfies β (4) (t) f(t,β(t), β (t)), t (, 1), β(), β(1), aβ ( ) bβ ( ), cβ (ξ 2 ) + dβ (ξ 2 ). (2.1) 3. Main results We are now in a position to present and prove our main results. In what follows, we will always assume that the nonnegative constants a, b, c, d,, ξ 2 satisfy < ξ 2 1, b a, d c+cξ 2 and ad+bc+ac(ξ 2 )>. Theorem 3.1. If there exist α(t) and β(t), upper and lower solutions, respectively, for the problem (1.1) (1.2) satisfying β(t) α(t) and β (t) α (t), (3.1) and if f :[, 1] R R R is continuous and satisfies f(t,u 1,v) f(t,u 2,v) for β(t) u 1 u 2 α(t), v R, t [, 1], f (t, u, v 1 ) f (t, u, v 2 ) for α (t) v 1 v 2 β (t), u R, t [, 1], (3.2) then, there exist two function sequences {α n (t)} and {β n (t)} that converge uniformly to the solutions of the boundary value problem (1.1) (1.2). Proof. We consider the operator T : C 2 [, 1] C 4 [, 1] defined by 1 ξ2 T u(t) = G 1 (t, ξ) G 2 (ξ, s)f (s, u(s), u (s)) ds dξ, (3.3) where G 1 (t, s) and G 2 (t, s) as in Eqs. (2.7) and (2.8), respectively, and define the set S by S ={u C 2 [, 1] β(t) u(t) α(t), α (t) u (t) β (t)}. (3.4) The operator T has the following two properties: (i) TS S. (ii) ω 1 (t) ω 2 (t), ω 1 (t) ω 2 (t), where ω 1(t) = Tu 1 (t) and ω 2 (t) = Tu 2 (t) for any u 1 (t) S,u 2 (t) S and u 1 (t) u 2 (t), u 1 (t) u 2 (t). In fact, for any u(t) S, let ω(t) = T u(t), then, ω (4) (t) = f (t, u(t), u (t)), <t<1, ω() = ω(1) =, aω ( ) bω ( ) =, cω (ξ 2 ) + dω (ξ 2 ) =.
Q. Zhang et al. / Journal of Computational and Applied Mathematics 196 (26) 387 393 391 Let ξ(t) = α(t) ω(t), from the definition of α(t) and conditions (3.2) of the theorem, we have ξ (4) (t) f(t,α(t), α (t)) f (t, u(t), u (t)), t (, 1), ξ(), ξ(1), aξ ( ) bξ ( ), cξ (ξ 2 ) + dξ (ξ 2 ). (3.5) In virtue of Lemma 2.2, we obtain that, ξ(t) and ξ (t), i.e., ω(t) α(t) and ω (t) α (t). Similarly, we can prove that, β(t) ω(t) and ω (t) β (t). Thus, TS S. Furthermore, if we let ω(t) = ω 2 (t) ω 1 (t), then, ω (4) (t) = f(t,u 2 (t), u 2 (t)) f(t,u 1(t), u 1 (t)), t (, 1), ω() =, ω(1) =, aω ( ) bω ( ) =, cω (ξ 2 ) + dω (ξ 2 ) =. So Lemma 2.2 implies that ω(t) and ω (t), i.e., ω 1 (t) ω 2 (t) and ω 1 (t) ω 2 (t). Thus, the second property of T is true. Now, we define the sequences {α n (t)} and {β n (t)} by α n (t) = T α n 1 (t), α (t) = α(t), β n (t) = T β n 1 (t), β (t) = β(t). From the properties of T,wehave α(t) α 1 (t) α 2 (t) α n (t) β(t), α (t) α 1 (t) α 2 (t) α n (t) β (t), β(t) β 1 (t) β 2 (t) β n (t) α(t), β (t) β 1 (t) β 2 (t) β n (t) α (t), (3.6) which together with the conditions (3.2) of the theorem imply that f(t,α(t), α (t)) f(t,α 1 (t), α 1 (t)) f(t,α n(t), α n (t)) f(t,β(t), β (t)), f(t,α(t), α (t)) f(t,β n (t), β n (t)) f(t,β 1(t), β 1 (t)) f(t,β(t), β (t)). So we have α (4) (t) α (4) 1 (t) α(4) 2 (t) α(4) n (t) β(4) (t), β (4) (t) β (4) 1 (t) β(4) 2 (t) β(4) n (t) α(4) (t). (3.7) Hence, from Eqs. (3.6) and (3.7) we know there exists a constant M such that α n (t) M, β n (t) M, t [, 1], n= 1, 2,...,n,..., α n (2) α (4) n (t) M, β(2) (t) M, t [, 1], n= 1, 2,...,n,..., n (t) M, β(4) (t) M, t [, 1], n= 1, 2,...,n,.... (3.8) n In addition, from the boundary condition we know that for each n N, there exist ξ n [, 1], η n [, 1] and a constant M 1, such that α n (ξ n) M 1, β n (η n) M 1. Thus, t α n (t) α (4) n (t) dt + α n (ξ n) M 1 + M, ξ n t β n (t) n (t) dt + β n (η n) M 1 + M, t [, 1], n= 1, 2,...,n,.... (3.9) η n β (4)
392 Q. Zhang et al. / Journal of Computational and Applied Mathematics 196 (26) 387 393 Using the boundary condition we get that for each n N, there exist γ n [, 1], τ n [, 1], such that α n (γ n) = and β n (τ n) =. Thus, t α n (t) = (t) dt M, β n (t) = t γ n α n τ n β n (t) dt M, t [, 1], n= 1, 2,...,n,.... (3.1) Hence, from Eqs. (3.8) (3.1), we know that {α n }, {β n } are uniformly bounded in C 4 [, 1]. On the other hand, for any x 1,x 2 [, 1], α n (x 1 ) α n (x 2 ) M x 1 x 2, β n (x 1 ) β n (x 2 ) M x 1 x 2, α n (x 1) α n (x 2) (M 1 + M) x 1 x 2, β n (x 1) β n (x 2) (M 1 + M) x 1 x 2. So {α n }, {β n } and {α n }, {β n } are equicontinuous. Thus, from the Arzela Ascoli theorem [5] and the Eqs. (3.6), (3.7) we know that {α n }, {β n } converge uniformly to the solutions of the problem (1.1) (1.2). The proof is completed as desired. 4. Applications To illustrate our results, we present the following examples. Example 1. Consider the boundary value problem u (4) (t) = u 2 1 π 1 (u ) 5 + 1 sin πt, 2 <t<1, au ( 1 3 ) bu ( 1 3 ) =, cu ( 2 1 ) + du ( 2 1 ) =, (4.1) where a, b, c, d are nonnegative constants satisfying 3b a> and 2d c>. It is easy to check that α(t) = sin πt, β(t) = are upper and lower solutions of (4.1), respectively, and that all assumptions of Theorem 3.1 are fulfilled. So the boundary problem (4.1) has at least one solution u(t) satisfying u(t) sin πt, π 2 sin πt u (t), t [, 1]. Example 2. Consider the fourth-order nonlinear ordinary differential equation u (4) (t) = f (t, u(t)), <t<1, (4.2) together with the four-point boundary conditions au ( ) bu ( ) =, cu (ξ 2 ) + du (ξ 2 ) =, (4.3) where a, b, c, d,, ξ 2 are nonnegative constants satisfying < ξ 2 1, b a, d c + cξ 2 and δ = ad + bc + ac(ξ 2 )>, f(t,u) C([, 1] [, + ), R + ) is nondecreasing in u and there exists a positive constant μ such that k μ f(t,u) f(t,ku) for any k 1.
Q. Zhang et al. / Journal of Computational and Applied Mathematics 196 (26) 387 393 393 It is easy to verify that the functions α(t)=k 1 g(t), β(t)=k 2 g(t)are lower and upper solutions of (4.2) (4.3), respectively, where k 1 min{1/a 2,(a 1 ) μ/(1 μ) }, k 2 max{1/a 1,(a 2 ) μ/(1 μ) }, and { a 1 = min 1, 1 } ξ2 2 min G 2 (t, s)f (s, s(1 s))ds, t [,1] { a 2 = max 1, 1 } ξ2 2 max G 2 (t, s)f (s, s(1 s))ds, t [,1] 1 ξ2 g(t) = G 1 (t, ξ) G 2 (ξ, s)f (s, s(1 s))ds dξ. (4.4) Thus, this boundary value problem has at least one solution. 5. Conclusion In this paper, we study a four-point boundary value problem for a fourth-order ordinary differential equation. Some new existence results are obtained by developing the upper and lower solution method and the monotone iterative technique. Furthermore, some applications are included in the paper. Acknowledgements This work was finished when the first two authors visited Academy of Mathematics and Systems Science, Chinese Academy of Sciences. The work is supported by the Natural Nature Science Foundation of P.R. China under Grant No. 757159 and No. 63417. The authors would like to thank the referees for their valuable comments. References [1] A.R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl. 116 (1986) 415 426. [2] R.P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic, Dordrecht, 1998. [3] M.A. Del Pino, R.F. Manasevich, Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc. 112 (1991) 81 86. [4] J.R. Graef, B. Yang, On a nonlinear boundary value problem for fourth order equations, Appl. Anal. 72 (1999) 439 448. [5] P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. [6] S. Leela, Monotone method for second order periodic boundary value problems, Nonlinear Anal. 7 (1983) 349 355. [7] T.F. Ma, Existence results and numerical solutions for a beam equation with nonlinear boundary conditions, Appl. Numer. Math. 47 (23) 189 196. [8] R.Y. Ma, J.H. Zhang, S.M. Fu, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl. 215 (1997) 415 422. [9] I. Rachunkova, Upper and lower solutions and topological degree, J. Math. Anal. Appl. 234 (1999) 311 327. [1] V. Shanthi, N. Ramanujam, A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations, Appl. Math. Comput. 129 (22) 269 294. [11] P.J.Y. Wong, Triple positive solutions of conjugate boundary value problems, Comput. Math. Appl. 36 (1998) 19 35. [12] Y. Yang, Fourth-order two-point boundary value problem, Proc. Amer. Math. Soc. 14 (1988) 175 18. [13] Z.X. Zhang, J. Wang, The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, J. Comput. Appl. Math. 147 (23) 41 52. [14] D.G. Zill, M.R. Cullen, Differential Equations with Boundary-value Problems, fifth ed., Brooks, Cole, 21.