The solution theory of the nonlinear qfractional differential equations Abstract   In this paper, we study the solution theory of the nonlinear    q   fractional differential equation:         c      D    q    xcexb1    x   (  t  )     f   (  t  ,  x   (  t  )   )   ,    0    xcexb1  ,    q    1    , with given initial value. We first give the existence theorem under the assumption that       t    xcexb2    f   (  t  ,  x  )      is continuous where     0  xe2x89xa4  xcexb2    xcexb1    . Then, by establishing a    q   analogue Gronwall inequality, we prove that the solution     x   (  t  )      is stable with respect to the initial value if     f   (  t  ,  x  )      satisfies the Lipschitz condition on variable    x   . This stability result also implies the uniqueness of solution.

Existence of solution for nonlinear integral equations of Hxc3xa9non type in critical case Abstract   In this paper, we will show that there exists a       xcexb4    xe2x88x97    u003e  0     such that when     xcexb1  u003e  0  ,  xcexb2  u003e  0     and     xcexb1    xcexb2      xcexb4    xe2x88x97      , the following problem    (0.1)       f      q    xcexb3    xe2x88x92  1     (  x  )       xe2x88xab    xcexa9           x       xcexb1         y       xcexb2    f   (  y  )          x  xe2x88x92  y       n  xe2x88x92  xcexb3        d  y    xcexbb    xe2x88xab    xcexa9      f   (  y  )          x  xe2x88x92  y       n  xe2x88x92  xcexb3  xe2x88x92  1        d  y  ,    x  xe2x88x88    xcexa9   xc2xaf       has at least one positive solution     f  xe2x88x88  C   (    xcexa9        )     , where    xcexa9    is the unit ball in       R    n     (  n  xe2x89xa5  3  )      centered at the origin,     1    xcexb3    n    ,     xcexbb  u003e  0     and       q    xcexb3    xe2x89x94    2  n    n    xcexb3      .