it　is　proven　that　if　the　leftmost　eigenvalue　is　weakly　linearly　degenerate,　then　the　Cauchy　problem　for　a　class　of　nonhomogeneous　quasilinear　hyperbolic　systems　with　small　and　decaying　initial　data　given　on　a　semi-bounded　axis　admits　a　unique　global　C-1　solution　on　the　domain　{(t,　x)　vertical　bar　t　＞=　0,　x　＞=　x(n)　(t)},　where　x　=　x(n)　(t)　is　the　fastest　forward　characteristic　emanating　from　the　origin.　As　an　application　of　our　result,　we　prove　the　existence　of　global　classical,　C-1　solutions　of　the　flow　equations　of　a　model　class　of　fluids　with　viscosity　induced　by　fading　memory　with　small　smooth　initial　data　given　on　a　semi-bounded　axis.　(c)　2006　Elsevier　Inc.　All　rights　reserved.