In this thesis, we consider the following delay boundary value problem egin{eqnarray*}(BVP)left{begin{array}{l}y'(t)+q(t)f(t,y(sigma(t)))=0, tin(0,1)/{ au}, y(t)=xi(t), tin[- au_{0},0], y(1)=0,end{array} right. end{eqnarray*}, where the functions f and q satisfy certain conditions; $sigma(t)leq t$ is a nonlinear real valued continuous function. We use two different methods to establish some existence criteria for the solution of problem (BVP). We generalize the delay term to a nonlinear function and obtain more general and supplementary results for the known ones about linear delay term due to Agarwal and O¡¦Regan [1] and Jiang and Xu [5].