An integral oscillation criterion for even order halflinear differential equations Abstract   Integral tests for detection of oscillatory behavior of solutions of even order halflinear differential equations of the form         L    n    x    q   (  t  )        x       xcexb2    sgn    x    0  ,    t  xe2x89xa5  a  ,  (A)      where         L    1    x   (  t  )       x    xe2x80xb2     (  t  )   ,      L    i    1    x   (  t  )             p    i     (  t  )          L    i    x   (  t  )          xcexb1    i      sgn      L    i    x   (  t  )         xe2x80xb2    ,    i    1  ,  xe2x80xa6  n  xe2x88x92  1  ,             xcexb1    i    ,    1  xe2x89xa4  i  xe2x89xa4  n  xe2x88x92  1    , and    xcexb2    being positive constants and       p    i    ,    1  xe2x89xa4  i  xe2x89xa4  n  xe2x88x92  1    , and    q    continuous functions from       a  ,  xe2x88x9e  )     to     (  0  ,  xe2x88x9e  )    , are established in the case where       xcexb1    1    xe2x8bxaf    xcexb1    n  xe2x88x92  1      xcexb2    .

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      .