The solvability and explicit solutions of singular integraldifferential equations of nonnormal type via RiemannHilbert problem Abstract   In this article, we study nonnormal type singular integraldifferential equations involving convolution kernel and Cauchy kernel in class       0      , and we prove the existence of solutions and the Noether conditions. By using theory of Fourier analysis, the equations under consideration are transformed into boundary value problems for analytic functions (i.e., RiemannHilbert problems) with discontinuous coefficients. For such problems, we propose one method different from classical one, and we obtain the analytic solutions and the conditions of Noether solvability. At nodal points, all cases as regards the regularity and the asymptotic behaviors of the solution are discussed in detail. In particular, we generalize the case of normal type discussed in the literature (Li, 2017 7,8, 2020, 2019) to the case of nonnormal type.

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      .