Integral Equations and Operator Theory

, Volume 13, Issue 5, pp 660–670 | Cite as

Regularity of the solution of Hammerstein equations with weakly singular kernel

  • Hideaki Kaneko
  • Richard Noren
  • Yuesheng Xu
Article

Abstract

Some regularity properties of the solution to a class of weakly singular Hammerstein equations are derived. The results obtained in this paper extend the results of C. Schneider [4], where he obtains similar properties for the solution to weakly singular Fredholm equations of the second kind.

Keywords

Regularity Property Singular Kernel Fredholm Equation Weakly Singular Kernel Hammerstein Equation 

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Bibliography

  1. [1]
    Giraud, G. — Sur certains problemes non-linearires de Neumann et sur certains problemes non-linearires mixtes. Ann. Sci. Ecole Norm. Sup. 49(1932), 3–17.Google Scholar
  2. [2]
    Mikhailov, L.G. — A new class of singular integral equations. Groningen, Wolters-Noordhoff Publishing, 1970.Google Scholar
  3. [3]
    Rice, J.R. — On the degree of convergence of nonlinear spline approximation, in Approximations with Special Emphasis on Spline Functions (I.J. Schoenberg, ed), Academic Press, N.Y. 1969, 349–365.Google Scholar
  4. [4]
    Schneider, C. — Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Eqs. and Op. Thy. 2(1979), 62–68.Google Scholar
  5. [5]
    Schneider, C. — Product integration for weakly singular integral equations, Math. of Comp. 36(1981) 207–213.Google Scholar
  6. [6]
    Vainikko, G., and Pedas, A. — The properties of solutions of weakly singular integral equations, Jl. Austral. Math. Soc. (Series B) 22(1981), 419–430.Google Scholar
  7. [7]
    Vainikko, G. and Uba, P. — A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, Jl. Austral. Math. Soc. (Series B) 22(1981), 431–438.Google Scholar
  8. [8]
    Richter, G. — On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl. 55 (1976) 32–42.Google Scholar
  9. [9]
    Graham, I. — Singularity expansions for the solution of second king Fredholm integral equations with weakly singular convolution kernels, J. of Integral Eqns. 4, (1982), 1–30.Google Scholar
  10. [10]
    Graham, I. and Schneider, C. — Product integration for weakly singular integral equations, inConstructive Methods for the Practical Treatment of Integral Equations, ed. by G. Hammerlin and K. Hoffmann, ISNM 73, Birkhauser, (1985), 156–164.Google Scholar
  11. [11]
    Krasnoselskii, M, and Zabreiko, P. et-al. —Approximate Solution of Operator Equations, Noordhoff (1976).Google Scholar
  12. [12]
    Krasnoselskii, M and Zabreiko, P. —Geometrical Methods of Nonlinear Analysis, Norfhoff (1984).Google Scholar
  13. [13]
    Chandler, G.A. — Galeskin method for boundry integral equations on polygonal domains, J. Austral. Math. Soc. Set B, 26(1984), 1–13.Google Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • Hideaki Kaneko
    • 1
  • Richard Noren
    • 1
  • Yuesheng Xu
    • 1
  1. 1.Department of Mathematics & StatisticsOld Dominion UniversityNorfolk

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