Skip to main content
SpringerLink
Log in

Linear Multi-Dimensional Discontinuous Integral Inequalities

  • Chapter
  • First Online:
Integral and Discrete Inequalities and Their Applications

  • 871 Accesses

Abstract

In this chapter, we shall introduce some multi-dimensional linear discontinuous integral inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. R.P. Agarwal, On an integral inequality in n independent variables. J. Math. Anal. Appl. 85, 192–196 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Bainov, A. Myshkis, A. Zahariev, On an abstract analogy of the Bellmann-Gronwall inequality. Publ. Res. Inst. Math. Sci. 20 (5), 903–911 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Bainov, P. Simeonov, Integral Inequalities and Applications (Kluwer Academic Publishers, Dordrecht, 1992)

    Book  MATH  Google Scholar 

  4. E.F. Beckenbach, R. Bellman, Inequalities (Springer, Berlin, 1961)

    Book  MATH  Google Scholar 

  5. P.R. Beesack, Riemann’s function has an exponential bound. Proc. Am. Math. Soc. 88, 313–316 (1983)

    MathSciNet  MATH  Google Scholar 

  6. P.R. Beesack, On some Gronwall-type integral inequalities in n independent variables. J. Math. Anal. Appl. 100, 393–408 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. P.R. Beesack, Systems of multidimensional Gronwall integral inequalities. J. Math. Phys. Sci. Nonlinear Anal. 18 (3), 215–231 (1984)

    MathSciNet  MATH  Google Scholar 

  8. P.R. Beesack, Systems of multidimensional Volterra integral equations and inequalities. Nonlinear Anal. 9, 1451–1486 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. R.E. Bellman, The stability of solutions of linear differential equations. Duke. Math. J. 10, 643–647 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  10. R.E. Bellman, Upper and lower bounds for solution of the matrix Riccati equation. J. Math. Anal. Appl. 17, 373–379 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Bihari, A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hung. 7, 71–94 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  12. B.K. Bondge, B.G. Pachpatte, On Wendroff type integral inequalities in n independent variables. Chin. J. Math. 7 (1), 37–46 (1978)

    MathSciNet  MATH  Google Scholar 

  13. B.K. Bondge, B.G. Pachpatte, On some fundamental integral inequalities in n independent variables. Bull. Inst. Math. Acad. Sinca 8, 553–560 (1980)

    MathSciNet  MATH  Google Scholar 

  14. D.W. Boyd, J.S.W. Wong, On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Chandra, P.W. Davis, Linear generalizations of Gronwall’s inequality. Proc. Am. Math. Soc. 60, 157–160 (1976)

    MathSciNet  MATH  Google Scholar 

  16. J. Chandra, B.A. Fleishman, Bounds and maximal solutions for nonlinear functional equations. Bull. Am. Math. Soc. 74, 512–516 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Chandra, B.A. Fleishman, On a generalization of the Gronwall-Bellman lemma in partially ordered spaces. J. Math. Anal. Appl. 31, 668–681 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  18. S.C. Chu, F.T. Metcalf, On Gronwall’s inequality. Proc. Am. Math. Soc. 18, 439–440 (1967)

    MathSciNet  MATH  Google Scholar 

  19. J. Conlan, C.L. Wang, Gronwall-Bellman type inequalities for systems. J. Math. Anal. Appl. 120, 622–630 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. K.M. Das, A note on an inequality due to Greene. Proc. Am. Math.Soc. 77, 424–425 (1979)

    Google Scholar 

  21. P.C. Das, R.R. Sharma, Some stieltjes integral inequalities. J. Math. Anal. Appl. 73, 423–433 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. K. Deiming, Nonlinear Functional Analysis (Springer, Berlin, 1985)

    Book  Google Scholar 

  23. N. Dunford, J. Schwartz, Linear Operators. Spectral Theory, Self-Adjoint Operators in Hilbert Space, Part II (Interscience, New York/London, 1963)

    Google Scholar 

  24. H. Erbe, Q.K. Kong, Stieltjes integral inequalities of Gronwall type and applications. Ann. Mat. Pura Appl. 157, 77–97 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. D.E. Greene, An inequality for a class of integral systems. Proc. Am. Math. Soc. 62, 101–104 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Groh, A nonlinear Volterra-Stieltjes integral equation and a Gronwall inequality in one dimension. Ill. J. Math. 24, 244–263 (1980)

    MathSciNet  MATH  Google Scholar 

  27. T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20, 292–296 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  28. I. Györi, A generalization of Bellman’s inequality for Stieltjes integrals and uniqueness theorem. Stud. Sci. Math. Hung. 6, 137–145 (1971)

    MATH  Google Scholar 

  29. I. Györi, L. Horváth, Gronwall-Bellman type integral inequalities for abstract Lebesgue integral. Math. Comput. Model 25 (1), 51–58 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. P.R. Halmos, A Hilbert Space Problem Book (Van Nostrand, Princeton, 1967)

    MATH  Google Scholar 

  31. D.L. Hanson, P. Waltman, A note on a functional equation. J. Math. Anal. Appl. 10, 330–333 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  32. T.H. Hildebrandt, On systems of linear differentio-Stieltjes integral equations. Ill. J. Math. 3, 352–373 (1959)

    MathSciNet  MATH  Google Scholar 

  33. H. Hochstadt, Integral Equations (Wiley, New York, 1973)

    MATH  Google Scholar 

  34. G. Hoheisel, Integral Equations (Ungar, New York, 1968)

    MATH  Google Scholar 

  35. L. Horváth, Gronwall-Bellman type integral inequalities in measure spaces. J. Math. Anal. Appl. 202, 183–193 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. S.G. Hristova, D.D. Bainov, Integral inequalities for functions of many variables. Mathematica (Cluj) 23 (46), 205–210 (1982)

    Google Scholar 

  37. S.G. Hu, Generalized Volterra-type integral equations and inequalities. J. Huazhong Univ. Sci. Tech. 21 (5), 164–170 (1993)

    MathSciNet  MATH  Google Scholar 

  38. G.S. Jones, Fundamental inequalities for discrete and discontinuous functional equations. J. Soc. Ind. Appl. Math. 12, 43–57 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  39. M.A. Krasnoselskii, Positive Solution of Operator Equations (Noordhoff, Groningen, 1964)

    Google Scholar 

  40. M. Kwapisz, Bielecki’s method, existence and uniqueness results for Volterra integral equations in L p spaces. J. Math. Anal. Appl. 154, 403–416 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, vols. 1–2 (Academic, New York, 1969)

    MATH  Google Scholar 

  42. S.G. Mikhlin, Linear Integral Equations (Hindustan, Delhi, 1960)

    MATH  Google Scholar 

  43. R.K. Miller, Nonlinear Volterra Integral Equations (Benjamin, Menlo Park, 1971)

    MATH  Google Scholar 

  44. B.G. Pachpatte, On some integral inequalities in n independent variables. J. Math. Anal. Appl. 79, 256–272 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Pelczar, On invariant points of monotone transformations in partially ordered spaces. Ann. Pol. Math. 17, 49–53 (1965)

    MathSciNet  MATH  Google Scholar 

  46. V.S.H. Rao, Integral inequalities of Gronwall type for distributions. J. Math. Anal. Appl. 72, 545–550 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Ronkov, D. Bainov, Integral equations and inequalities of Volterra type for functions defined in partially ordered spaces. J. Math. Anal. Appl. 125, 483–507 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  48. A. Ronkov, D. Bainov, Nonlinear integral inequalities for functions defined in partially ordered topological spaces. Nonlinear Anal. TMA 11, 297–304 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  49. W.W. Schmaedeke, G.R. Sell, The Gronwall inequality for modified Stieltjes integrals. Proc. Am. Math. Soc. 19, 1217–1222 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  50. B.B. Shinde, B.G. Pachpatte, On multidimensional simultaneous integral inequalities. J. Math. Phys. Sci. 17, 219–243 (1983)

    MathSciNet  MATH  Google Scholar 

  51. P.S. Simeonov, D.D. Bainov, On some generalizations of the Gronwall-Bellman integral inequality for scalar functions of many variables. Rend Circ. Mat. Palermo 32 (2), 27–38 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  52. V.M. Singare, B.G. Pachpatte, Lower bounds on some integral inequalities in n independent variables. Indian J. Pure Appl. Math. 12, 318–331 (1981)

    MathSciNet  MATH  Google Scholar 

  53. L.J. Slater, Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966)

    MATH  Google Scholar 

  54. G.W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973)

    MATH  Google Scholar 

  55. E. Thandapani, R.P. Agarwal, On some new inequalities in n independent variables. J. Math. Anal. Appl. 86, 542–561 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  56. F.G. Tricomi, Integral Equations (Interscience, New York, 1957)

    MATH  Google Scholar 

  57. B. Viswanatham, A generalization of Bellman’s lemma. Proc. Am. Math. Soc. 14, 15–18 (1963)

    MathSciNet  MATH  Google Scholar 

  58. C.L. Wang, A short proof of a Greene theorem. Proc. Am. Math. Soc. 69, 357–358 (1978)

    MathSciNet  MATH  Google Scholar 

  59. C.C. Yeh, Wendroff’s inequality in n independent variables. J. Math. Anal. Appl. 78, 78–87 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  60. C.C. Yeh, On some integral inequalities in n independent variables and their applications. J. Math. Anal. Appl. 85, 387–410 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. E.C. Young, Gronwall’s inequality in n independent variables. Proc. Am. Math. Soc. 41, 241–244 (1973)

    MATH  Google Scholar 

  62. E.C. Young, Functional integral inequalities in N variables. Chin. J. Math. 10, 1–7 (1982)

    MathSciNet  MATH  Google Scholar 

  63. E.C. Young, On integral inequalities of Gronwall-Bellman type. Proc. Am. Math. Soc. 94 (4), 636–640 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics College of Science, Donghua University, Shanghai, China

    Yuming Qin

Authors
  1. Yuming Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Qin, Y. (2016). Linear Multi-Dimensional Discontinuous Integral Inequalities. In: Integral and Discrete Inequalities and Their Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-33301-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation