Journal of Mathematics and System Science 4 (2014) 1-5 Existence Theorems for a Fifth-Order Boundary Value Problem Amir Elhaffaf 1 , Mostepha Naceri 2 1. Mathematics Faculty Of Science, Oran University, BP1524, Es-Senia Algeria 2. Economics, Commercial and Management Sciences, Preparatory School of Oran-Algeria Received: September 24, 2013 / Accepted: November 02, 2013 / Published: January 25, 2014. Abstract: In this paper,we study the existence of a solution for fifth-order boundary value problem ቊ ݑ ሺହሻ ሺݐሻ ൅ ሺݑ ,ݐሺݐሻ, ݑ ′′′ ሺݐሻሻ ൌ Ͳ , Ͳ൏ ݐ൏ͳ ݑሺͲሻ ൌ ݑ′ሺͲሻ ൌ ݑ′ሺͳሻ ൌ ݑ ′′′ ሺͳሻ ൌ ݑ ሺସሻ ሺͲሻ ൌ Ͳ Where  ܥאሺሾͲ,ͳሿ ൈ  ଶ , ሻ. By placing certain restrictions on the nonlinear term , we prove the existence of at least one solution to the boundary value problem with the use of the lower and upper solution method and Schauder fixed-point theorem. The construction of lower or upper solution is also presented. Boundary value problems of very similar type are also considered. Keywords: Fifth-order differential equations, boundary value problems, lower and upper solution method, Schauder fixed-point theorem. 1. Introduction Nonlinear fifth-order boundary value problems are one of the most important problems arising in the mathematical modeling of viscoelastic fluid flows [6, 10]. Some works have been done on nonlinear fifth-order boundary value problems. For instance, El-Shahed [8] and Odda [13-15], investigated the fifth-order boundary value problems using the Krasnoeslskii’s fixed point boundary value problem: ݑ ሺହሻ ሺݐሻ൅ ߣሺݐሻሺݑሺݐሻሻ ൌ Ͳ , Ͳ ൏ ݐ൏ͳ ݑሺͲሻ ൌ ݑ′ሺͲሻ ൌ ݑ′′ሺͲሻ ൌ ݑ′′′ሺͲሻ ൌ Ͳ, ݑ ሺ′′′ሻ ሺͳሻ ൌ Ͳ and ݑ ሺହሻ ሺݐሻ൅ ߣሺݐሻሺݑሺݐሻሻ ൌ Ͳ , Ͳ ൏ ݐ൏ͳ ݑሺͲሻ ൌ ݑ′′ሺͲሻ ൌ ݑ′′′ሺͲሻ ൌ ݑ ሺସሻ ሺͲሻ ൌ Ͳ, ݑߙ′ሺͳሻ ൅ ݑߚ′′ሺͳሻ ൌ Ͳ where ߣ൐Ͳ and ߚ ,ߙ൒Ͳ, ߙ൅ ߚ൐Ͳ. A large part of the literature on solution to higher-order boundary-value problems seems to be traced to Krasnoeslskii’s work on nonlinear operator Corresponding author: Amir Elhaffaf, Mathematics Faculty Of Science, Oran University, BP1524, Es-Senia Algeria. E-mail: elhaffaf1@yahoo.com. equations as well as other fixed-point theorem such as Legget-Williams fixed point theorem, see for example [2, 4, 12-14, 16]. The method of upper and lower solution is extensively developed for lower order equation with linear and nonlinear boundary condition. But there are only a few papers referred to lower and upper solutions of fifth-order equation consider the relationship between the property of nonlinear term and the construction of lower and upper solutions. The results presented in this paper seem to be new and original. They generalize several results obtained up to now in the study of nonlinear differential equations of several types. The purpose of this paper is to study the existence of solution for two class nonlinear fifth-order boundary value problems: ቊ ݑ ሺହሻ ሺݐሻ ൅ ሺݑ ,ݐሺݐሻ, ݑ ′′′ ሺݐሻሻ ൌ Ͳ , Ͳ൏ ݐ൏ͳ ݑሺͲሻ ൌ ݑ′ሺͲሻ ൌ ݑ′ሺͳሻ ൌ ݑ ′′′ ሺͳሻ ൌ ݑ ሺସሻ ሺͲሻ ൌ Ͳ (1) and ቊ ݑ ሺହሻ ሺݐሻ ൅ ሺݑ ,ݐሺݐሻ, ݑ ′′′ ሺݐሻ, ݑ ሺସሻ ሺݐሻሻ ൌ Ͳ, Ͳ൏ ݐ൏ͳ ݑሺͲሻ ൌ ݑ′ሺͲሻ ൌ ݑ′ሺͳሻ ൌ ݑ ′′′ ሺͳሻ ൌ ݑ ሺସሻ ሺͲሻ ൌ Ͳ (2) The method used here is not based on the D DAVID PUBLISHING