POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS

Electronic Journl of Differentil Equtions, Vol. 27(27), No. 156, pp. 1 8. ISSN: 172-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu (login: ftp) POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS ZENGQIN ZHAO Abstrct. In this pper, we present the Green s functions for second-order liner differentil eqution with three-point boundry conditions. We give exct expressions of the solutions for the liner three-point boundry problems by the Green s functions. As pplictions, we study uniqueness nd itertion of the solutions for nonliner singulr second-order three-point boundry vlue problem. 1. Introduction The Green s function plys n importnt role in solving boundry-vlue problems of differentil equtions. The exct expressions of the solutions for some liner ordinry differentil equtions boundry vlue problems cn be denoted by Green s functions of the problems (see [3, 12, 2]). The Green s function method might be used to obtin n initil estimte for shooting method. the Green s function method for solving the boundry vlue problem is n effective tool in numericl experiments [6]. Some boundry vlue problems for nonliner differentil equtions cn be trnsformed into the nonliner integrl equtions the kernel of which re the Green s functions of corresponding liner differentil equtions. The integrl equtions cn be solved by to investigte the property of the Green s functions (see [2, 4, 7, 8, 14]). The concept, the significnce nd the development of Green s functions cn be seen in [15]. The other study of second-order three-point boundry vlue problems cn be seen in [5, 9, 1, 16, 18, 19] nd its references. In bove litertures, the three-point boundry vlues re ll sme conditions u() =, u(1) = ku(η), the investigtion on the boundry condition u () =, u(1) = ku(η) cn be seen [1, 11, 13, 17], the investigtion for other three-point boundry conditions is few, since people my be not fmilir with their Green s functions. The solutions of the Green s functions diffuse in the literture, there is lck of uniform method. The undetermined prmetric method we use in this pper is universl method, the Green s functions of mny boundry vlue problems for ordinry differentil 2 Mthemtics Subject Clssifiction. 34A5, 34B27, 34B5, 34B15. Key words nd phrses. Green s function; second-order three-point boundry vlue problem; explicit solution; itertion. c 27 Texs Stte University - Sn Mrcos. Submitted Februry 5, 27. Published November 21, 27. Supported by grnts: 147175 from the Ntionl Nturl Science Foundtion of Chin, Y26A4 from the the Nturl Science Foundtion of Shndong Province of Chin, 264461 from the Doctorl Progrm Foundtion of Eduction Ministry of Chin. 1

2 Z. ZHAO EJDE-27/156 equtions cn obtined by similr the method. In ddition, our Green s functions hve orderly expressions. We consider the Green s function for the following second-order liner differentil eqution three-point boundry vlue problems subject to the boundry vlue conditions, respectively, u + f(t) =, t [, b], (1.1) u() =, u (b) = ku(η); (1.2) u() = ku(η), u (b) = ; (1.3) u() =, u (b) = ku (η); (1.4) u() = ku (η), u (b) = ; (1.5) u () =, u(b) = ku(η); (1.6) u () = ku(η), u(b) = ; (1.7) u () =, u(b) = ku (η); (1.8) u () = ku (η), u(b) = ; (1.9) where < η < b nd k is constnt. This pper is orgnized s follows. In 2, we study the Green s function for the eqution (1.1) stisfying the three-point boundry conditions (1.2) nd give the expression of the unique solution by the Green s function, tht incrnte the generl method of deriving the Green s functions for clss of boundry problems. In 3, for some interrelted boundry conditions, we give the Green s functions of the problems directly, omitting the prticulr of derivtion. The correctness of the Green s functions only need direct verifiction. As pplictions, in 4, we study the uniqueness of the solutions, the itertion nd the rte of convergence by the itertion for nonliner singulr second-order three-point boundry vlue problem. 2. The Green s Function of Equtions (1.1) with the Boundry Condition (1.2) About the boundry vlue problem (1.1)-(1.2), we hve the following conclusions. theorem 2.1. If k(η ) 1, then the second-order liner three-point boundry vlue problem (1.1)(1.2) hs unique solution u(t), which is given vi where u(t) = G 1 (t, s)f(s)ds, (2.1) k(t ) G 1 (t, s) = K(t, s) + K(η, s), (2.2) 1 k(η ) s, s t b K(t, s) = (2.3) t, t < s b. Remrk 2.2. We cll G 1 (t, s) the Green s function of the boundry vlue problem (1.1)-(1.2).

EJDE-27/156 POSITIVE SOLUTIONS 3 Proof. It is well known tht the second-order two-point liner boundry vlue problem hs unique solution u + f(t) =, t [, b], u() =, u (b) = w(t) = where K(t, s) is s described in (2.3). From (2.4) we obtin w() =, w (b) =, w(η) = K(t, s)f(s)ds. (2.4) K(η, s)f(s)ds. (2.5) Assume tht u(t) is solution of problem (1.1)-(1.2). Then u (t) = w (t) = f(t), thus, we cn be ssume tht u(t) = w(t) + c + dt, (2.6) where c, d re constnts tht will be determined. From (2.6) we know tht Equtions (2.5), (2.6) nd (2.7) imply Putting these into (1.2) yields u (t) = w (t) + d. (2.7) u() = c + d, u (b) = d, u(η) = c + dη + w(η). c + d =, d = k(c + dη + w(η)). Since k(η ) 1, solving the system of liner equtions on the unknown numbers c, d, we obtin hence, c + dt = c = kw(η) 1 k(η ), kw(η) d = 1 k(η ), k(t ) 1 k(η ) w(η). Putting this into (2.6), we obtin u(t) = w(t) + k(t ) 1 k(η ) w(η), which solution of (1.1)-(1.2). This together with (2.4) imply u(t) = K(t, s)f(s)ds + k(t ) 1 k(η ) K(η, s)f(s)ds. (2.8) Consequently, (2.1) holds. The uniqueness of solutions (1.1) (1.2) follows from the fct tht the corresponding homogeneous problem hs only the trivil solution. From Theorem 2.1 we obtin the following corollry.

4 Z. ZHAO EJDE-27/156 Corollry 2.3. Suppose the nonliner function g(t, u) is continuous on [, b] R, then if k(η ) 1, the nonliner three-point boundry-vlue problem u + g(t, u) =, u() =, is equivlent to the nonliner integrl eqution with G 1 (t, s) s in (2.2). u(t) = t [, b], u (b) = ku(η) G 1 (t, s)g(s, u(s))ds Exmple 2.4. The second-order three-point liner boundry vlue problem hs n unique solution u 1 (t) = 2 3 t sin (1) t cos ( 1 3 u (t) + cos(t) =, t [, 1], u() =, u (1) = 3 2 u(1 3 ) ) + t + cos (t) 1, t 1. (2.9) It cn be obtined by letting =, b = 1, η = 1 3, k = 3 2, f(t) = cos(t) in Theorem 2.1 tht u 1 (t) = B(t, s) cos(s)ds t B( 1, s) cos(s)ds, 3 where B(t, s) = mint, s}, t, s [, 1]. Therefore, (2.9) is obtined by direct computtion. Some properties of u 1 (t) re shown in the imge Figure 1..2.15.1.5.2.4 t.6.8 1 Figure 1. Grph of u 1 (t) 3. The Relte Results for Other Boundry Conditions In this section, we give the Green s functions for some boundry vlue problems directly vi the following tble, omitting the prticulr of derivtion. The proof is similr to tht of Theorem 2.1. Similrly, the unique solutions of the liner problems cn be denoted by its Green s functions. Some nonliner boundry vlue

EJDE-27/156 POSITIVE SOLUTIONS 5 problems cn be trnsformed into the nonliner integrl equtions the kernel of which re the Green s functions of the corresponding liner problems. Eqution: u (t) + f(t) =, t [, b], < η < b, k is constnt. no. ssume boundry Green s function 1 k(η ) 1 (1.2) G 1 (t, s) = K(t, s) + k(t ) 1 k(η ) K(η, s) 2 k 1 (1.3) G 2 (t, s) = K(t, s) + k 1 k K(η, s), 3 k 1 (1.4) G 3 (t, s) = K(t, s) + k(t ) 1 k K t(η, s), 4 (1.5) G 4 (t, s) = K(t, s) + kk t (η, s), 5 k 1 (1.6) G 5 (t, s) = H(t, s) + k 1 k H(η, s), 6 k(b η) 1 (1.7) G 6 (t, s) = H(t, s) k(b t) 1+k(b η) H(η, s), 7 (1.8) G 7 (t, s) = H(t, s) + kh t (η, s), 8 k 1 (1.9) G 8 (t, s) = H(t, s) k(b t) 1 k H t(η, s), where K(t, s) = H(t, s) = s, s t b t, t < s b; b t, s t b b s, t < s b; K t (η, s) = H t (η, s) =, s < η, 1, η < s b; 1, s < η,, η < s b. 4. Applictions in Nonliner Singulr Boundry Vlue Problems In this section,we study the itertion process for the following nonliner threepoint boundry vlue problem u + f(t, u) =, t (, 1), u() =, u (1) = ku(η), (4.1) with η (, 1), kη < 1, f(t, u) my be singulr t t = nd/or t = 1. Concerning the function f we impose the following hypotheses: f(t, u) is nonnegtive continuous on (, 1) [, + ), f(t, u) is monotone incresing on u, for fixed t (, 1), there exist q (, 1) such tht f(t, ru) r q f(t, u), < r < 1, (t, u) (, 1) [, + ). (4.2) Obviously, from (4.2) we obtin f(t, λu) λ q f(t, u), λ > 1, (t, u) (, 1) [, + ). (4.3) It is esy to see tht if < α i < 1, i (t) re nonnegtive continuous on (, 1), for i =, 1, 2,..., m, then f(t, u) = m i=1 i(t)u αi stisfy the condition (4.2). Concerning the boundry vlue problem (4.1), we hve following conclusions.

6 Z. ZHAO EJDE-27/156 theorem 4.1. Suppose the function f(t, u) stisfy the condition (4.2), nd < f(t, t)dt <. (4.4) Then the problem (4.1) hs n unique solution w(t) in D C 2 (, 1), here D = x C[, 1] M x m x >, such tht m x t x(t) M x t, t I}. Constructing successively the sequence of functions h n (t) = G(t, s)f(s, h n 1 (s))ds, n = 1, 2,..., (4.5) for ny initil function h (t) D, then h n (t)} must converge to w(t) uniformly on [, 1] nd the rte of convergence is mx h n(t) w(t) = O ( 1 N q n), (4.6) t [,1] where < N < 1, which depends on initil function h (t), G(t, s) = B (t, s) + Proof. Let J = (, 1), I = [, 1], R + = [, + ), F x(t) = ktb (η, s), B(t, s) = mint, s}, t, s [, 1]. (4.7) 1 kη P = x(t) x(t) C(I), x(t) }, G(t, s)f(s, x(s))ds, x(t) D. (4.8) It is esy to see tht the opertor F : D P is incresing. By direct verifictions we know tht if u D stisfies u(t) = F u(t), t I, (4.9) then u C 1 (I) C 2 (J) is solution of (4.1). For ny x D, there exist positive numbers < m x < 1 < M x such tht By (4.7) we hve m x s x(s) M x s, s I, (m x ) q f(s, s) f(s, x(s)) (M x ) q f(s, s), s J. (4.1) G(t, s) = B(t, s) + kt 1 kη B(η, s) t k B(η, s), (4.11) 1 kη G(t, s) t + kt ( B(η, s) t 1 + k ) B(η, s). (4.12) 1 kη 1 kη Using (4.8), (4.3), (4.1), (4.11), (4.12) nd the conditions (4.2), we obtin F x(t) = F x(t) t(m x ) q k 1 kη G(t, s)f(s, x(s))ds ( 1 + k t(m x ) q 1 kη Equtions (4.4), (4.13) nd (4.14) imply tht F : D D. B(η, s)f(s, s)ds, t I, (4.13) ) B(η, s) f(s, s)ds, t I. (4.14)

EJDE-27/156 POSITIVE SOLUTIONS 7 For ny h D, we let l h = sup l > : lh (t) (F h )(t), t I}, L h = inf L > : (F h )(t)) Lh (t), t I}, (4.15) m = min1, (l h ) 1 1 q }, M = mx1, (Lh ) 1 1 q }, u (t) = mh (t), v (t) = Mh (t), u n (t) = F u n 1 (t), v n (t) = F v n 1 (t), n =, 1, 2,.... (4.16) Since the opertor F is incresing, (4.2), (4.15) nd (4.16) imply u (t) u 1 (t) u n (t) v n (t) v 1 (t) v (t), t I. (4.17) For t = m/m, from (4.8), (4.2) nd (4.16), it cn obtined by induction tht From (4.17) nd (4.18) we know tht u n (t) (t ) qn v n (t), t I, n =, 1, 2,.... (4.18) u n+p (t) u n (t) v n (t) u n (t) ( 1 (t ) qn) Mh (t), n, p, (4.19) so tht there exists function w(t) D such tht u n (t) w(t), v n (t) w(t), (uniformly on I), (4.2) u n (t) w(t) v n (t), t I, n =, 1, 2,.... (4.21) From the opertor F being incresing nd (4.16) we hve u n+1 (t) = F u n (t) F w(t) F v n (t) = v n+1 (t), n =, 1, 2,.... This together with (4.2) nd uniqueness of the limit imply tht w(t) stisfy (4.9), hence w(t) C 1 (I) C 2 (J) is solution of (4.1). Form (4.5) (4.16) nd the opertor F being incresing, we obtin u n (t) h n (t) v n (t), t I, n =, 1, 2,..., (4.22) thus, it follows from (4.19), (4.21) nd (4.22) tht Therefore, h n (t) w(t) h n (t) u n (t) + u n (t) w(t) mx t I 2 v n (t) u n (t) ( 1 (t ) qn) M h (t). h n(t) w(t) ( 1 (t ) qn) M mx h (t). t I So tht (4.6) holds. From h (t) is rbitrry in D we know tht w(t) is the unique solution of the eqution (4.9) in D. Remrk 4.2. If f(t, u) is continuous on I R +, then it is quite evident tht the condition (4.4) holds. Hence the unique solution w(t) is in C 2 (I).

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