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Some classes of integral and integro-differential equations of convolutional type

Part of the Lecture Notes in Mathematics book series (LNM,volume 827)

Abstract

Starting with classical convolutional integral equations on ℝn, translation-invariant operators and their symbol representation according to Hörmander are introduced. The various generalizations concerning the domains G of integration lead to Wiener-Hopf integral and integro-differential equations on ℝ n+ and on cones. Compound integral and integro-differential equations of the principal value and L1-kernel type are discussed on ℝn using results by Rakovshčik. Simonenko's theory of local type operators permits us to investigate generalized translation invariant operators. Wiener-Hopf integral equations with strongly singular kernels correspond to equations with piecewise continuous symbols in both variables. Convolutional equations on quadrants and wedges are studied via the theorey of operators of bi-local type.

Keywords

  • Toeplitz Operator
  • Singular Integral Equation
  • Singular Integral Operator
  • Riemann Boundary
  • Translation Invariant Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Extended version of a General Lecture at the Dundee Conference on Differential Equations, 31st March 1978.

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References

  1. Agranovič, M.S. Elliptic singular integro-differential operators. Russ. Math. Surveys 20, 1–121 (1965)

    CrossRef  MathSciNet  Google Scholar 

  2. Atkinson, F.V. The normal solubility of linear equations in normed spaces (russ.). Mat. Sbornik N.S. 28 (70), 3–14 (1951)

    MathSciNet  Google Scholar 

  3. Bancuri, R.D. Integro-differential equation on the half-axis with kernels depending on the difference of arguments (russ.). Sakharth. SSR Mecn. Akad. Moambe 54, 17–20 (1969)

    MathSciNet  Google Scholar 

  4. Betz, A. Tragflügeltheorie. Ber.u.Abhandlg. d. Wiss. Ges. f. Luftfahrt, Heft 2 (1920)

    Google Scholar 

  5. Boutet de Monvel, L. Opérateurs pseudo différentiels analytiques et problèmes aux limites elliptiques. Ann.Inst. Fourier 19, 163–268 (1969)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Boutet de Monvel, L. Boundary problems for pseudo-differential operators Acta Math. 126, 11–51 (1971)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Calderón, A.P. Singular integrals. Bull.Amer. Math. Soc. 72, 427–465 (1966)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Calderón, A.P., Zygmund, A. On singular integrals. Amer. J. Math. 78, 289–309 (1956)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Calderón, A.P., Zygmund, A. Singular integral operators and differential equations. Amer. J. Math. 79, 901–921 (1957)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Colella,P., Cordes, H.O. The C*-algebra of the elliptic boundary problem SFB 72, preprint no. 160, Univ. Bonn 1977

    Google Scholar 

  11. Cordes, H.O. Pseudo-differential operators on a half-line. Journ. Math. Mech. 18, 893–908 (1969)

    MathSciNet  MATH  Google Scholar 

  12. Cordes, H.O. Lecture notes on Banach algebra methods in partial differential equations. Univ. Lund 1970/71

    Google Scholar 

  13. Cordes, H.O. On compactness of commutators of multiplications and convolutions, and boundedness of pseudo-differential operators. Journ. Funct.Anal. 18, 115–131 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Cordes, H.O. A global parametrix for pseudo-differential operators over ℝn, with applications. SFB 72 preprint no.90, Univ. Bonn 1976

    Google Scholar 

  15. Cordes, H.O., Herman, E. Singular integral operators on a half-line. Proc. Nat. Acad. Sci. USA 56, 1668–1673 (1966)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Devinatz, A., Shinbrot, M. General Wiener-Hopf operators. Trans. Amer. Math. Soc. 145, 467–494 (1969)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Dikanskii, A.S. Problems adjoint to elliptic pseudodifferential boundary value problems. Soviet Math. Dokl. 12, 1520–1525 (1971)

    MATH  Google Scholar 

  18. Dikanskii, A.S. Conjugate problems of elliptic differential and pseudo-differential boundary value problems in a bounded domain. Math. USSR Sbornik 20, 67–83 (1973)

    CrossRef  MATH  Google Scholar 

  19. Donig, J. Zur Theorie einer Klasse elliptischer singulärer Integro-Differentialoperatoren in Grund-und Distributionenräumen. Diss. Univ. Tübingen 1973

    Google Scholar 

  20. Donig, J. Das Cauchy-Problem für eine parabolische Gleichung mit singulären Koeffizienten. Vortr. Tagg. Math. Meth. d. Phys., Oberwolfach 1974

    Google Scholar 

  21. Donig, J. ‘A priori’ Abschätzungen für eine Klasse elliptischer Pseudo-Differentialoperatoren im Raum LP(ℝn). Function theoretic methods for partial differential equations: Proc. Int. Symp., Darmstadt, 171–191. Springer Lecture Notes 561, Berlin-Heidelberg 1976

    Google Scholar 

  22. Donig, J. Ober ein Transmissionsproblem mit Integro-Differentialbedingungen. manuscripta math. 23, 1–18 (1977)

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Douglas, R.G., Howe, R. On the C*-algebra of Toeplitz operators on the quarter-plane. Trans. Amer. Math. Soc. 158, 203–217 (1971)

    MathSciNet  MATH  Google Scholar 

  24. Dudučava, R.V. Wiener-Hopf integral operators with discontinuous symbols. Sov. Math. Dohl. 14, 1001–1005 (1973)

    MATH  Google Scholar 

  25. Dudučava, R.V. On convolution integral operators with discontinuous coefficients. Sov. Math. Dohl. 15, 1302–1306 (1974)

    MATH  Google Scholar 

  26. Dudučava, R.V. On Wiener-Hopf integral operators (russ.). Mathem. Nachr. 65, 59–82 (1975)

    CrossRef  MATH  Google Scholar 

  27. Dudučava, R.V. On bisingular integral operators and convolution operators on a quadrant. Sov. Math. Dohl. 16, 330–334 (1975)

    MATH  Google Scholar 

  28. Dudučava, R.V. Bisingular integral operators and boundary value problems of the theory of analytic functions in spaces of generalized functions. Sov. Math. Dohl. 16, 1324–1328 (1975)

    MATH  Google Scholar 

  29. Dudučava, R.V. Convolution integral operators on a quadrant with discontinuous symbols. Math. USSR Izvestija 10, 371–392 (1976)

    CrossRef  MATH  Google Scholar 

  30. Èskin, G.I. Boundary value problems and the parametrix for systems of elliptic pseudo-differential equations. Trans. Moscow Math. Soc. 28, 74–115(1973)

    MATH  Google Scholar 

  31. Èskin, G. Boundary problems for elliptic pseudo-differential operators (russ.). ‘Nauka', Moscow 1973

    Google Scholar 

  32. Feldman, I.A. An equation of radiation energy transfer and Wiener-Hopf operator equations (russ.). Functional. Anal. i. Priložen. 5, 106–108 (1971)

    Google Scholar 

  33. Feldman, I.A. On an iteration method for the equation of radiant energy transfer. Sov.Math.Dohl. 12, 1034–1038 (1971)

    Google Scholar 

  34. Feldman, I.A. Operator Wiener-Hopf equations and their applications to the transport equation (russ.). Matem. Issled. 6, 115–132 (1971)

    Google Scholar 

  35. Friedman, A. Partial differential equations. Holt, Rinehart & Winston, New York 1969

    MATH  Google Scholar 

  36. Friedrichs, K.O. Pseudo-differential operators, an introduction. Lecture Notes, Courant Inst. Math. Sci, New York Univ. 1970

    Google Scholar 

  37. Gakhov, F.D. Boundary value problems. Pergamon Press, Oxford et al. 1966

    MATH  Google Scholar 

  38. Gerlach, E. Zur Theorie einer Klasse von Integro-Differentialgleichungen. Diss. TU Berlin 1969

    Google Scholar 

  39. Gerlach, E., Kremer, M. Singuläre Integraloperatoren in Lp-Räumen. Zeitschr. Angew. Math. Mech. 53, T 158–159 (1973)

    MathSciNet  MATH  Google Scholar 

  40. Gerlach, E., Kremer, M. Singuläre Integraloperatoren in Lp-Räumen. Math. Ann. 204, 285–304 (1973)

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. Gerlach, E., Latz, N. Zur Spektraltheorie bei Faltungsoperatoren. Z.Angew. Math. Mech. 57, T 231–232 (1977)

    MathSciNet  MATH  Google Scholar 

  42. Gohberg, I.Z., Krein, M.G. The basic propositions on defect numbers, root numbers and indices of linear operators. Uspehi Mat. Nauk 12 (74), 44–118 (1957) (russ.). AMS Transl. (2) 13, 185–264 (1960)

    MathSciNet  Google Scholar 

  43. Gohberg, I.Z., Krein, M.G. Systems of integral equations on a half-line with kernels depending on the difference of arguments. AMS Transl. (2) 14, 217–287 (1960)

    MathSciNet  Google Scholar 

  44. Gohberg, I.Z., Krupnik, N.A. The spectrum of one-dimensional singular integral operators with piece-wise continuous coefficients. Matem. Issled. 3 16–30 (1968) (russ.). AMS transl., ser. 2, 103, 181–193 (1974)

    MathSciNet  Google Scholar 

  45. Gohberg I.Z., Krupnik, N.A. Algebra generated by one-dimensional singular integral operators with piece-wise continuous coefficients. Funct. Anal. and its Applic. 4, 193–201 (1970)

    CrossRef  MathSciNet  Google Scholar 

  46. Gohberg I.Z., Krupnik, N.A. Singular integral operators with piece-wise continuous coefficients and their symbols. Math. USSR Izvestija 5, 955–979 (1971)

    CrossRef  MathSciNet  Google Scholar 

  47. Gohberg, I.Z., Krupnik, N.A. Introduction to the theory of one-dimensional singular integral operators (russ.). Štiincie, Kishinjow 1973

    Google Scholar 

  48. Goldenstein, L.S., Gohberg, I.Z. On a multidimensional integral equation on a half-space whose kernel is a function of the difference of the arguments and on a discrete analogue of this equation. Sov. Math.Dokl. 1, 173–176 (1960)

    MathSciNet  Google Scholar 

  49. Grabmüller, H. On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem. Proc.Roy.Soc.Edinburgh 76A, 110–137 (1976)

    MathSciNet  Google Scholar 

  50. Grabmüller, H. Asymptotic behavior of solutions of linear integro-differential equations. J. Integral Equs. and Op. Theory 1, 28–56 (1978)

    CrossRef  MATH  Google Scholar 

  51. Hörmander, L. Estimates for translation invariant operators in Lp-spaces. Acta Math. 104, 93–140 (1960)

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. Kohn, J.J., Nirenberg, L. An algebra of pseudo-differential operators. Comm.Pure Appl.Math. 18, 269–305 (1965)

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. Koppelman, W., Pincus, J.D. Spectral representations for finite Hilbert transformations. Math. Zeitschr. 71, 399–407 (1959)

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. Krein, M.G. Integral equations on a half-line with a kernel depending upon the difference of the arguments. AMS Transl. 22, 163–288 (1962)

    MATH  Google Scholar 

  55. Kremer, M. Über eine Klasse singulärer Integralgleichungen vom Faltungstyp. Diss. TU Berlin 1969

    Google Scholar 

  56. Kremer, M. Über eine Algebra ‘nicht normaler’ Wiener-Hopf-Operatoren I & II. Math. Ann. 220, 77–86 & 87–95 (1976)

    CrossRef  MathSciNet  MATH  Google Scholar 

  57. Kumano-Go, H. Pseudo-differential operators (japan.), Iwanami-Shoten Publish., Tokyo 1974

    MATH  Google Scholar 

  58. Meister, E., Speck, F.-O. Some multi-dimensional Wiener-Hopf equations with applications. FB Mathem., TH Darmstadt, preprint Nr. 373 (1977); to appear in Proc. 2nd Sympos. on Trends in Appl. Pure Math. Mech. Kozubnik, Sept. 1977 Pitman Monographs, vol. 2 (1978)

    Google Scholar 

  59. Meister, E., Speck, F.-O. Wiener-Hopf operators on three-dimensional wedgeshaped regions. FB Mathematik, TH Darmstadt, preprint Nr. 389 (1978), to appear in Applicable Analysis.

    Google Scholar 

  60. Michlin, S.G. Singular integral equations. Uspehi mat. Nauk (N.S.) 3, 29–112 (1948) (russ.). AMS transl. 24, 84–198 (1950)

    Google Scholar 

  61. Michlin, S.G. Multidimensional singular integrals and integral equations. Pergamon Press, Oxford et al. 1965

    Google Scholar 

  62. Mushkelishvili, N.I. Singular integral equations. Noordhoff, Groningen 1953

    Google Scholar 

  63. Nickel, K. Lösung eines Integralgleichungssystems aus der Tragflügeltheorie. Math. Zeitschr. 54, 81–96 (1951)

    CrossRef  MathSciNet  MATH  Google Scholar 

  64. Peetre, J. Mixed problems for higher order elliptic equations in two variables I. Annali di Scuola norm.Pisa (3) 15, 337–353 (1961)

    MathSciNet  MATH  Google Scholar 

  65. Pellegrini, V. General Wiener-Hopf operators and the numerical range of an operator. Proc. Amer. Math. Soc. 38, 141–146 (1973)

    CrossRef  MathSciNet  MATH  Google Scholar 

  66. Pilidi, V.S. On multidimensional bisingular operators. Sov. Math. Dokl. 12, 1723–1726 (1971)

    MathSciNet  MATH  Google Scholar 

  67. Pilidi, V.S., Sazanov, L.I. A priori estimates for characteristic bisingular integral operators. Sov. Math. Dokl. 15, 1064–1067 (1974)

    MATH  Google Scholar 

  68. Prössdorf, S. Operators admitting unbounded regularization (russ.). Vestn. Leningrad Univ. 20, 59–67 (1965)

    Google Scholar 

  69. Prössdorf, S. Eindimensionale singuläre Integralgleichungen und Faltungsgleichungen nicht normalen Typs in Tokalkonvexen Räumen. Habil.schrift, Karl-Marx-Stadt (1967)

    Google Scholar 

  70. Prössdorf, S. Zur Theorie der Faltungsgleichungen nicht-normalen Typs. Math. Nachr. 42, 103–131 (1969)

    CrossRef  MathSciNet  MATH  Google Scholar 

  71. Prössdorf, S. Einige Klassen singulärer Gleichungen. Birkhäuser Verlag, Basel-Stuttgart 1974

    CrossRef  MATH  Google Scholar 

  72. Rabinovič, V.S. Pseudo-differential equations in unbounded regions with conical structure at infinity. Math. USSR Sbornik 9, 73–92 (1969)

    CrossRef  Google Scholar 

  73. Rabinovič, V.S. Pseudo-differential equations in unbounded regions. Sov. Math. Dokl. 12, 452–456 (1971)

    Google Scholar 

  74. Rabinovič, V.S. Pseudo-differential operators on a class of non-compact manifolds. Math. USSR Sbornik 18, 45–59 (1972)

    CrossRef  Google Scholar 

  75. Rakovshčik, L.S. On the theory of integral equations of convolution type (russ.). Uspehi Matem. Nauk 18, 171–178 (1963)

    Google Scholar 

  76. Reeder, J. On the invertibility of general Wiener-Hopf operators. Proc.Amer. Math. Soc. 27, 72–76 (1971)

    CrossRef  MathSciNet  MATH  Google Scholar 

  77. Šahbagjan, R.L. Convolution equations in a half-space. AMS Translat., Ser. 2, 75, 117–148 (1968)

    Google Scholar 

  78. Samko, S.G The general singular equation in the exceptional case. Diff. Equat. 1, 867–874 (1965)

    MathSciNet  Google Scholar 

  79. Schüppel, B. Regularisierung singulärer nicht elliptischer Integralgleichungen mit unstetigen Koeffizienten. Diss. Univ. München 1973

    Google Scholar 

  80. Schüppel, B. Regularisierung singulärer Integralgeichungen vom nichtnormalen Typ mit stückweise stetigen Koeffizienten. 'Function theoretic methods for partial differential equations': Proc. Int. Symp., Darmstadt; 430–442, Springer Lecture Notes 561, Berlin-Heidelberg 1976

    Google Scholar 

  81. Schwartz, J.T. Some results on the spectra and spectral resolutions of a class of singular integral operators. Comm. Pure Appl. Math. 15, 75–90 (1962)

    CrossRef  MathSciNet  MATH  Google Scholar 

  82. Seeley, R.T. Integro-differential equations on vector bundles. Trans. Amer. Math. Soc. 117, 167–204 (1965)

    CrossRef  MathSciNet  MATH  Google Scholar 

  83. Seeley, R.T. Topics in pseudo-differential operators. Pseudo-Diff.Operators (C.I.M.E., Stresa 1968). Edizioni Cremonese, Roma, 169–305 (1969)

    Google Scholar 

  84. Shamir, E. Évaluation dans Ws,p pour des problèmes au limites elliptiques mixtes dans le plan. C.R. Acad. Sci. Paris 254, 3621–3623 (1962)

    MathSciNet  MATH  Google Scholar 

  85. Shamir, E. Mixed boundary value problems for elliptic equations in the plane. The Lp theory. Annali Scuola Norm.Sup. Pisa (3) 17, 117–139 (1963)

    MathSciNet  MATH  Google Scholar 

  86. Shamir, E. Reduced Hilbert transforms and singular integral equations. Journ. Analyse Math. (Jerusalem) 12, 277–304 (1964)

    CrossRef  MathSciNet  MATH  Google Scholar 

  87. Shamir, E. Wiener-Hopf type problems for elliptic systems of singular integral equations. Bull.Amer.Math.Soc. 72, 501–503 (1966)

    CrossRef  MathSciNet  MATH  Google Scholar 

  88. Shamir, E. Elliptic systems of singular integral operators. I. The halfspace case. Trans. Amer. Math. Soc. 127, 107–124 (1967)

    CrossRef  MathSciNet  MATH  Google Scholar 

  89. Shinbrot, M. On singular integral operators. Journ. Math. Mech. 13, 395–406 (1964)

    MathSciNet  MATH  Google Scholar 

  90. Šubin, M.A. Factorization of matrices depending on a parameter and elliptic equations in a half-space. Math. USSR Sbornik 14, 65–84 (1971)

    CrossRef  MATH  Google Scholar 

  91. Simonenko, I.B. A new general method of investigating linear operator equations of the type of singular integral equations. Sov. Math. Dokl. 5, 1323–1326 (1964)

    MATH  Google Scholar 

  92. Simonenko, I.B. A new general method of investigating linear integral operator equations of the type of singular equations. (russ.). Izvest. Akad., Nauk, Ser. Matem., 29 567–586 (1965)

    MathSciNet  Google Scholar 

  93. Simonenko, I.B. Convolution operators in cones. Sov. Math. Dokl. 8, 1320–1323 (1967)

    MATH  Google Scholar 

  94. Simonenko, I.B. Operators of convolution type in cones. Math. USSR Sbornik 3, 279–293 (1967)

    CrossRef  MATH  Google Scholar 

  95. Söhngen, H. Zur Theorie der endlichen Hilbert-Transformation. Math. Zeitschr. 60, 31–51 (1954)

    CrossRef  MathSciNet  MATH  Google Scholar 

  96. Speck, F.-O. Über verallgemeinerte Faltungsoperatoren und eine Klasse von Integrodifferentialgleichungen. Dissert. TH Darmstadt 1974

    Google Scholar 

  97. Speck, F.-O. Über verallgemeinerte Faltungsoperatoren und ihre Symbole. Function theoretic methods for partial differential equations: Proc. Int. Symp., Darmstadt; 459–471, Springer Lecture Notes 561, Berlin-Heidelberg 1976

    Google Scholar 

  98. Speck, F.-O. Eine Erweiterung des Satzes von Rakovshčik und ihre Anwendung in der Simonenko-Theorie. Math. Ann. 228, 93–100 (1977)

    CrossRef  MathSciNet  MATH  Google Scholar 

  99. Strang, G. Toeplitz operators in a quarter-plane. Bull. Amer. Math. Soc. 76, 1303–1307 (1970)

    CrossRef  MathSciNet  MATH  Google Scholar 

  100. Talenti, G. Sulle equazioni integrali di Wiener-Hopf, Boll. Un. Mat. Ital. 7, Suppl. fasc. 1, 18–118 (1973)

    MathSciNet  MATH  Google Scholar 

  101. Titchmarsch, E.C. Theory of Fourier Integrlas. 2nd ed., Clarendon Press., Oxford 1948

    Google Scholar 

  102. Tricomi, F. On the finite Hilbert-transformation. Quart. J. Math. (2), 2 199–211 (1951)

    CrossRef  MathSciNet  MATH  Google Scholar 

  103. Vishik, M.I., Èskin, G.I. Equations in convolutions in a bounded region. Russ. Math. Surveys 20, 85–151 (1965)

    CrossRef  MATH  Google Scholar 

  104. Vishik, M.I., Èskin, G.I. Normally solvable problems for elliptic systems of equations in convolutions (russ.). Mat. Sbornik 74, (116), 326–356 (1967)

    Google Scholar 

  105. Vishik, M.I., Èskin, G.I. Elliptic equations in convolution in a bounded domain and their applications. Russ. Math. Surveys 22, 13–75 (1967)

    CrossRef  MATH  Google Scholar 

  106. Widom, H. Singular integral equations in Lp. Trans. Amer. Math. Soc. 97, 131–160 (1960)

    MathSciNet  MATH  Google Scholar 

  107. Wiener, N., Hopf, E. Über eine Klasse singulärer Integralgleichungen. Sitzg.-ber. Preuss. Akad. Wiss.; Phys.-Math. Kl. 30–32, 696–706 (1931)

    MATH  Google Scholar 

  108. Zabreyko, P.P. et al. Integral Equations — a reference text. Noordhoff Intern. Publ., Leyden 1975

    CrossRef  Google Scholar 

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Meister, E. (1980). Some classes of integral and integro-differential equations of convolutional type. In: Everitt, W.N. (eds) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091381

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