BIFURCATION SOLUTIONS OF BOUNDARY VALUE PROBLEM M. A. Abdul Hussain University of Basrah, Basrah, Iraq This paper studies the bifurcation solutions of boundary value problem . <...> In some domains of parameters, the existence and stability solutions of a certain boundary value problem was shown in which the number of solutions is fixed in every domain. <...> INTRODUCTION It is known that many of the nonlinear problems in mathematics and physics can be written in the form of operator equation, fx b x O X b Y Rn =Œ Г Œ , , Œ (, ) ,ll . (1) in which f is a smooth Fredholm map of index zero, X, Y Banach spaces and O open subset of X. For these problems, the method of reduction to finite dimensional equation [1], Q(, ) , MN (2) xl b x =Œ b Œ , , can be used, where M and N are smooth finite dimensional manifolds. <...> Passage from equation (1) into equation (2) (variant local scheme of Lyapunov—Schmidt) with the conditions, that equation (2) has all the topological and analytical properties of equation (1) ( multiplicity, bifurcation diagram, etc.) dealing with [4, 8, 10, 11]. <...> In this work it was assume that fF:WЖ is a nonlinear Fredholm map of index zero. <...> A smooth map fF:WЖ has variational property, if there exist functional VR:WЖ such that fV = gradH or equivalently, ∂ where ( <·,·>H H ). ∂ ="ŒW, Œ . <...> Sup: Ж is a smooth Fredholm map of index zero, E, F are Banach spaces and ∂ where V is a smooth functional on E. Also it was assume thatEF HГГ , H is a Hilbert space, then by using method of finite dimensional reduction (Local scheme of Lyapunov—Schmidt ) the problem, © Abdul Hussain M. A., 2007 162 Vx extr x E Rn (, ) , , (, )xlЖŒ . <...> ЖŒ Œ can be reduce into equivalent problem, Wextr the function W(, )xl is called Key function. <...> The study of bifurcation solutions of functional V is equivalent to the study of bifurcation solutions of Key function. <...> If f has variational property, then it is <...>