2 Journal of Applied Mathematics, where the functions f p and g are continuous and monotonic h is a homeomorphic. We will develop the upper and lower solution method for the boundary value problem. y f x y y y,p y a y b y a y b 0,g y a y b y a y b y a y b 0. and establish some new existence results Furthermore some applications are also presented. 2 Preliminaries, In this section we will give some preliminary considerations and some lemmas which are. essential to our main results, Definition 2 1 Suppose the functions x and x C 3 a b satisfy. 3 x f x x x x,3 x f x x x x, Then x and x are respectively called the lower and upper solutions of the BVP 1 3. Because of Definition 2 1 it is clear that x x x a b Let D a b. Definition 2 2 Let C D R R denote the class of continuous functions from D R into R. and let f x y y y C D R R and x x C 3 a b be lower and upper solutions. of BVP 1 3 Suppose that there is a function W s C R 0 such that. f x y y y W y 2 2,for every x y y y D R where,ds max x min x 2 3. W s x a b x a b,Journal of Applied Mathematics 3,b a max a b b a 2 4. Then we say that f satisfies Nagumo s condition on the set D relative to x x. We assume throughout this paper the following, H1 There are lower and upper solutions x and x of BVP 1 3 as Definition 2 1. H2 Function f x y y y satisfies Nagumo s condition on the set D relative to. H3 Function f x y y y C a b R3 R is nonincreasing in y. H4 h a a b b is a homeomorphism with,h a b h a b 2 5. H5 Function p s t u v is continuous on R4 and nondecreasing in t u v and satisfies. p a b a b 0,p a b a b 0, H6 Function g x y z p q r is continuous on R6 and nondecreasing in x y q and. nonincreasing in r and it satisfies,g a b a b a b 0. g a b a b a b 0,It is not di cult to obtain the following lemma. Lemma 2 3 The boundary value problem,y f x y y y,y a 0 y a 0 y b 0. has a Green function,x s 2 b s a x 2,G x s 2 2 b a 2 9. 4 Journal of Applied Mathematics,Gx x s dt 2 10,Gxx x s dt b a. It is easy to prove the following lemma similarly to 12 page 25 Theorem 1 4 1. Lemma 2 4 Assume that H1 H2 hold Then for any solution y of y f x y y y with. x y x x x y x x on a b there exists a constant N 0 depending only. on W such that,y x N x a b 2 11,and one calls N is Nagumo s constant. Lemma 2 5 Assume that H1 H3 hold Then for any constant A a a B. a a C b b the boundary value problem,y f x y y y,at least has a solution y C 3 a b with. Proof By Lemma 2 3 it is clear that BVP 2 12 is equivalent to integral equation. y x G x s f s y s y s y s ds W x 2 14, where W x is a polynomial satisfying y 0 y a A y a B y b C. Journal of Applied Mathematics 5,m max max x max x N 1 2 15. x a b x a b,where N is Nagumo s constant,f x y y m y m. fm x y y y f x y y y m y m,f x y y m y m,fm x x y y y x. G x y y y fm x y y y x y x,f x x y y y x,F x y y y G x y y y x y x. Then F x y y y is bounded and continuous on a b R3 Suppose F x y y y. M W x K i 0 1 2 x a b, Now define an operator T on the set E C 2 a b R by. T y x G x s F s y s y s y s ds W x 2 17,If y E the norm is defined by. y max y x y x y x,It is clear that,T y x b a M K,6 Journal of Applied Mathematics. This shows that T maps the closed bounded and convex set. b a 3 M b a 2 M,B y y E y b a M 3K 2 20, into itself Also T is continuous and T y is bounded All of these considerations imply that. T is completely continuous by Ascoli s theorem Schauder fixed point theorem then yields. the fixed point y of T on B In other words the following boundary value problem. y F x y y y, has a solution y C 3 a b and satisfying x y x x and a y a a we. have x y x x x a b,In the following we prove that,x y x x x a b 2 22. In fact if it is invalid there is no harm in setting the right inequality to be not true the case. that the left inequality is not true can be proved in the same way By the assumption if. y x x for some x a b then there is a x0 a b such that. y x0 x0 max y x x 0 2 23,y x0 x0 y x0 x0 2 24,x0 y x0 x0. x0 y x0 x0 y x0 x0 2 25,They imply that x0 x0 x0 and. G x0 y x0 x0 x0 fm x0 x0 x0 x0 2 26,Journal of Applied Mathematics 7. F x0 y x0 y x0 y x0 f x0 x0 x0 x0,F x0 y x0 y x0 x0 f x0 x0 x0 x0. G x0 y x0 x0 x0 f x0 x0 x0 x0 2,1 y x0 2 27,G x0 y x0 x0 x0 f x0 x0 x0 x0. fm x0 x0 x0 x0 f x0 x0 x0 x0,f x0 x0 x0 x0 f x0 x0 x0 x0. which contradicts 2 24 hence 2 22 is true, Further by the definition of F y is a solution of the boundary value problem. y fm x y y y,Because there is a a b such that, so a b has a maximal subinterval c d with interior point for any x c d y x m. Hence for x c d y x is the solution of BVP 2 12 And by Lemma 2 4 we have y x. N m this contracts that c d is the maximal subinterval so we know c d a b. Consequently y is a solution of BVP 2 12,3 Main Results. Theorem 3 1 Assume H1 H4 H6 hold then BVP,y f x y y y. g y a y b y a y b y a y b 0,8 Journal of Applied Mathematics. has a solution y C 3 a b satisfying, Proof By Lemma 2 5 we know that the boundary value problem. y f x y y y,has a solution y C 3 a b with,on a b For any A a a B a a. For fixed A if B a then y a a y b b By H6 we know,g y a y b y a y b y a y b g a b a b a b 0 3 5. On the other hand if B a then y a a y b b By H6 we have. g y a y b y a y b y a y b g a b a b a b 0 3 6,Define the following sets. y y y f x y y y,y a A y a B y b h B A a a B a a,M1 B B a a y x y. g y a y b y a y b y a y b 0,M2 B B a a y x y,g y a y b y a y b y a y b 0. Journal of Applied Mathematics 9, Obviously y is nonempty If the theorem is not true we know that M1 and M1 are all. nonempty and M1 M2 a a we claim that M1 is closed To see this let Bn M1. with Bn B0 n Consider the following boundary value problem. y f x y y y, By Lemma 2 5 it is known that for every n N BVP 3 8 has a solution yn x C 3 a b. satisfying,x yn x x x yn x x x a b 3 9,and by Lemma 2 4 we know yn x N. Clearly sequences yn x yn x yn x are uniformly bounded and equicontinu. ous on a b Consequently there exists a subsequence of yn x which converges uniformly. on a b to a solution y0 x of the BVP,y f x y y y y. g y0 a y0 b y0 a y0 b y0 a y0 b 0 3 11,By assumption equality cannot occur so that. g y0 a y0 b y0 a y0 b y0 a y0 b 0 3 12, and thus B0 M1 Consequently M1 is closed Likewise we may show M2 is closed This is. a contradiction and proves the theorem, Similar to the proof of Theorem 3 1 we can obtain the following theorem. 10 Journal of Applied Mathematics,Theorem 3 2 Assume H1 H6 hold then BVP. y f x y y y,p y a y b y a y b 0,g y a y b y a y b y a y b 0. has a solution y C 3 a b satisfying,4 Applications. We all know it is di cult to find a solution of some nonlinear ordinary di erential equation. But according to Theorem 3 2 we can know whether a boundary value problem especially. a nonlinear boundary value problem has a solution and we also can know the existence. regions of the solution and its derivative, Example 4 1 Consider the following linear boundary value problem. y x y x 2y x x 0 1,y 1 2ey 0 0,3e 2 y 0 y 1 y 0 y 1 0. y 0 y 1 4y 0 5y 1 y 0 y 1 0, It is easy to know that x xex 1 x xex 1 are lower and upper solutions. of the linear boundary value problem respectively where. h s 2es p s t u v 3e 2 s t u v,g x y z p q r x y 4z 5p q r. Journal of Applied Mathematics 11, and all assumptions of Theorem 3 2 hold So the linear boundary value problem has a solution. y x satisfying,xex 1 y x xex 1,ex xex y x ex xex, Obviously the trivial solution of the linear boundary value problem is one. Example 4 2 Consider nonlinear boundary value problem. y x y x y x y x x,y sin 0 2 y,y2 y y y 0,4 4 2 8 4 2. y y y y y y 0,4 8 2 4 2 4 8 2, It is easy to verify that x sin x x 0 are lower and upper solutions of the. nonlinear boundary value problem respectively where. h s sin 2 s p s t u v s2 t u v,g x y z p q r x y z2 p2 q r. and all assumptions of Theorem 3 2 hold so the BVP has a solution y x satisfying. sin x y x 0 cos x y x 0 4 6,5 Conclusion, In this paper we study a nonlinear mixed two point boundary value problem for a third. order nonlinear ordinary di erential equation Some new existence results are obtained by. developing the upper and lower solution method Furthermore some applications are also. Acknowledgments, The work is supported by the Fundamental Research Funds for the Central Universities no. ZXH2012 K004 and Civil Aviation University of China Research Funds no 2012KYM05. The authors would like to thank the referees for their valuable comments. 12 Journal of Applied Mathematics,References, 1 L H Erbe Existence of solutions to boundary value problems for second order di erential equa. tions Nonlinear Analysis vol 6 no 11 pp 1155 1162 1982. 2 S Leela Monotone method for second order periodic boundary value problems Nonlinear Analysis. vol 7 no 4 pp 349 355 1983, 3 Z Zhang and J Wang The upper and lower solution method for a class of singular nonlinear second. order three point boundary value problems Journal of Computational and Applied Mathematics vol. 147 no 1 pp 41 52 2002, 4 I Rachunkova Upper and lower solutions and topological degree Journal of Mathematical Analysis. and Applications vol 234 no 1 pp 311 327 1999, 5 Q Zhang S Chen and J Lu Upper and lower solution method for fourth order four point. boundary value problems Journal of Computational and Applied Mathematics vol 196 no 2 pp 387. 6 J Ehme P W Eloe and J Henderson Upper and lower solution methods for fully nonlinear. boundary value problems Journal of Di erential Equations vol 180 no 1 pp 51 64 2002. 7 A Cabada The method of lower and upper solutions for second third fourth and higher order. boundary value problems Journal of Mathematical Analysis and Applications vol 185 no 2 pp 302. 8 G A Klaasen Di erential inequalities and existence theorems for second and third order boundary. value problems Journal of Di erential Equations vol 10 pp 529 537 1971. 9 Q Yao and Y Feng The existence of solution for a third order two point boundary value problem. Applied Mathematics Letters vol 15 no 2 pp 227 232 2002. 10 M Ruyun Z Jihui and F Shengmao The method of lower and upper solutions for fourth order. two point boundary value problems Journal of Mathematical Analysis and Applications vol 215 no 2. pp 415 422 1997, 11 Y X Gao Existence of solutions of three point boundary value problems for nonlinear fourth order. di erential equation Applied Mathematics and Mechanics vol 17 no 6 pp 543 550 1996. 12 S R Bernfeld and V Lakshmikantham An Introduction to Nonlinear Boundary Value Problems. Academic Press New York NY USA 1974,Advances in Advances in Journal of Journal of. Operations Research,Hindawi Publishing Corporation. Decision Sciences,Hindawi Publishing Corporation,Applied Mathematics. Hindawi Publishing Corporation,Hindawi Publishing Corporation. Probability and Statistics,Hindawi Publishing Corporation. http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014. The Scientific International Journal of,World Journal. Hindawi Publishing Corporation,Differential Equations. Hindawi Publishing Corporation, http www hindawi com Volume 2014 http www hindawi com Volume 2014. Submit your manuscripts at,http www hindawi com,International Journal of Advances in. Combinatorics,Hindawi Publishing Corporation,Mathematical Physics. Hindawi Publishing Corporation, http www hindawi com Volume 2014 http www hindawi com Volume 2014. Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in. Complex Analysis,Hindawi Publishing Corporation,Mathematics. Hindawi Publishing Corporation,in Engineering,Hindawi Publishing Corporation. Applied Analysis,Hindawi Publishing Corporation,Nature and Society. Hindawi Publishing Corporation, http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014. International,Journal of Journal of,Mathematics and. Mathematical,Discrete Mathematics,Journal of International Journal of Journal of. Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014. Function Spaces,Hindawi Publishing Corporation,Stochastic Analysis. Hindawi Publishing Corporation,Optimization,Hindawi Publishing Corporation. http www hindawi com Volume 2014 http www hindawi com http www hindawi com Volume 2014 http www hindawi com Volume 2014 http www hindawi com Volume 2014.

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