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Applications of Linear One-Dimensional Inequalities

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Integral and Discrete Inequalities and Their Applications
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Abstract

In this section, we give some applications of Theorems 1.1.2, 2.1.13–2.1.14 to sublinear perturbations of the differential equation y (n) = 0 and of the analogous difference equation, which is due to Mate and Nevai [392].

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Bibliography

  1. R.P. Agarwal, On finite systems of difference inequalities. J. Math. Phys. Sci. 10 (4), 277–288 (1976)

    MathSciNet  MATH  Google Scholar 

  2. R.P. Agarwal, E. Thandapani, On some new discrete inequalities. Appl. Math. Comput. 7, 205–244 (1980)

    MathSciNet  MATH  Google Scholar 

  3. R.P. Agarwal, E. Thandapani, On discrete generalizations of Gronwall’s inequality. Bull. Inst. Math. Acad. Sin. 9, 235–248 (1981)

    MathSciNet  MATH  Google Scholar 

  4. V.M. Alekseev, An estimate for the perturbation of the solutions of ordinary differential equations. Vestnik Moskov. Univ. Ser. I. Mat. Meh. (Russian) 2, 28–36 (1961)

    Google Scholar 

  5. S. Asirov, Ja. D. Mamedov, Investigation of solutions of nonlinear Volterra-Fredholm operator equations. Dokl. Akad. Nauk SSSR, 229, 982–986 (1976)

    Google Scholar 

  6. F.V. Atkinson, Discrete and Continuous Boundary Problems (Academic, New York/London, 1964)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  8. R.E. Bellman, The boundedness of solutions of linear differential equations. Duke Math. J. 14, 83–97 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  9. R.E. Bellman, Stability Theory of Differential Equations (McGraw-Hill, New York, 1953)

    MATH  Google Scholar 

  10. M.L. Boas, R.P. Boas, N. Levinson, The growth of solutions of a differential equation. Duke Math. J. 9, 847–853 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Brauer, Perturbations of nonlinear systems of differential equations II. J. Math. Anal. Appl. 17, 418–434 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Brauer, A nonlinear variation of constants formula for Volterra equations. Math. Syst. Theory 6, 226–234 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. F. Brauer, A. Strauss, Perturbations of nonlinear systems of differential equations III. J. Math. Anal. Appl. 31, 37–48 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Brunner, On collocation approximations for Volterra equations with weakly singular kernels, in Treatment of Integral Equations by Numerical Methods, ed. by C.T.H. Baker, G.F. Miller (Academic, New York, 1982), pp. 409–420

    Google Scholar 

  15. T.S. Chihara, P.G. Neval, Orthogonal polynomials and measures with finitely many point masses. J. Approx. Theory 35, 370–380 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. W.A. Coppel, Stability and Asymptotic Behaviour of Differential Equations (Heath, Boston, 1965)

    MATH  Google Scholar 

  17. W.A. Day, Entropy and hidden variables in continuum thermodynamics. Arch. Rational Mech. Anal. 62, 367–389 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Denche, H. Khellaf, Integral inequalities similar to Gronwall inequality. Electron. J. Differ. Equ. 176, 14 (2007)

    MathSciNet  MATH  Google Scholar 

  19. J. Douglas Jr., On the relation between stability and convergence in the numerical solution of linear parabolic and hyperbolic differential equations. J. Soc. Ind. Appl. Math. 3, 42–65 (1955)

    Article  Google Scholar 

  20. J. Douglas Jr., On the errors in analogue solutions of heat conduction problems. Q. Appl. Math. 14, 333–335 (1956)

    MathSciNet  MATH  Google Scholar 

  21. N. Dunford, J. Schwartz, Linear Operator, Part I (Interscience, New York/London, 1958)

    MATH  Google Scholar 

  22. W. Feller, Generalized second order differential operators and their lateral conditions. Ill. J. Math. 1, 459–504 (1957)

    MathSciNet  MATH  Google Scholar 

  23. R.A.C. Ferreira, D.F.M. Torres, Retarded integral inequalities of Gronwall-Bihari type (2008). arXiv:0806.4709

    Google Scholar 

  24. R.A.C. Ferreira, D.F.M. Torres, Generalizations of Gronwall-Bihari inequalities on time scales. J. Differ. Equ. Appl. 15 (6), 529–539 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. K.O. Friedrichs, H. Lewy, Über die Eindeutigkeit und das Abhängigkeitsgebiet der Lsosungen beim Anfangöwertproblem linear hyperbolischer Döifferentialgleichungen. Math. Ann. 98, 192–204 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  26. J.S. Geronimo, K.M. Case, Scattering theory and polynomials orthogonal on the real line. Trans. Am. Math. Soc. 258, 467–494 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Ghizzfiti, Un teorema sul comportamento asintotico degli integrali delle equazioni differenziali lineari omogenee. Rend. Mat. 8, 28–42 (1949)

    MathSciNet  MATH  Google Scholar 

  28. J. Groh, Übereine Klasse eindimensionaler Markovprozesse. Math. Nachr. 65, 125–136 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Groh, Optimal control of one dimensional non-conservative quasi-diffusion processes, in Transactions of the Eighth Prague Conference on Information Theory etc., vol. A (Springer, Netherlands, 1978), pp. 263–274; Stoch. Process Appl. 10 (3), 271–297 (1980)

    Google Scholar 

  30. O. Haupt, Über das asymptotische verhalten der Lösungen gewisser linearer gewöhnlicher differentialgleichungen. Math. Z. 48, 289–292 (1942–1943)

    Google Scholar 

  31. E. Hille, Non-oscillation theorems. Trans. Am. Math. Soc. 64, 234–252 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  32. F. John, On integration of parabolic equations by difference methods. Commun. Pure Appl. Math. 5, 155–270 (1952)

    Article  MATH  Google Scholar 

  33. 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 

  34. Q. Kong, B. Zhang, Some generalization of Gronwall-Bihari integral inequalities and their applications. Chin. Ann. Math. 10B (3), 371–385 (1989)

    MathSciNet  MATH  Google Scholar 

  35. P.D. Lax, R.D. Richtmyer, Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Lees, Energy inequality for the solution of differential equations. Trans. Am. Math. Soc. 94, 58–73 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Lin, P.P. Varaiya, Bounded-input bounded-output stability of nonlinear time-varying discrete control systems. IEEE Trans. Autom. Control Ac-12, 423–427 (1967)

    Google Scholar 

  38. P. Mandl, Analytical Treatment of One-Dimensional Markov Processes (Springer/Academia, New York/Prague, 1968)

    MATH  Google Scholar 

  39. A. Mate, P. Nevai, Sublinear perturbations of the differential equation y (n) = 0 and of the analogous difference equation. J. Differ. Equ. 52, 234–257 (1984)

    Google Scholar 

  40. M. Medved̆, Integral inequalities and global solutions of semilinear evolution equations. J. Math. Anal. Appl. 267 (2), 643–650 (2002)

    Google Scholar 

  41. R.K. Miller, J.A. Nohel, J.S.W. Wong, A stability theorem for nonlinear mixed integral equations. J. Math. Anal. Appl. 25, 446–449 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  42. L.M. Milne-Thomson, The Calculus of Finite Differences (MacMillan and Co., London, 1933)

    MATH  Google Scholar 

  43. A. Morro, A Gronwall-like inequality and its application to continuum thermodynamics. Boll. Un. Mat. Ital. B 6, 553–562 (1982)

    MathSciNet  MATH  Google Scholar 

  44. V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations (Princeton University Press, Princeton, 1960)

    MATH  Google Scholar 

  45. P.G. Neval, Orthogonal polynomials defined by a recurrence relation. Trans. Am. Math. Soc. 250, 369–384 (1979)

    Article  MathSciNet  Google Scholar 

  46. N.E. Norlund, Vorlesungen über differenzenrechnung (Springer, Berlin/New York, 1924)

    Book  MATH  Google Scholar 

  47. B.G. Pachpatte, On perturbed stochastic discrete systems. Bull. Aust. Math. Soc. 11, 385–393 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  48. B.G. Pachpatte, Perturbations of nonlinear systems of differential equations. J. Math. Annl. Appl. 51, 550–556 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  49. B.G. Pachpatte, On some fundamental finite difference inequalities. Univ. Beograd Publ. Elek. Fak. Ser. Mat. Fiz. 577–598, 65–73 (1977)

    MathSciNet  MATH  Google Scholar 

  50. B.G. Pachpatte, Finite difference inequalities and discrete time control systems. Indian J. Pure Appl. Math. 9, 1282–1290 (1978)

    MathSciNet  MATH  Google Scholar 

  51. B.G. Pachpatte, On the existence and uniqueness of solutions of Volterra-Fredholm integral equations. Math. Semin. Notes 10, 733–742 (1982)

    Google Scholar 

  52. B.G. Pachpatte, A note on certain integral inequality. Tamkang J. Math. 33, 353–358 (2002)

    MathSciNet  MATH  Google Scholar 

  53. B.G. Pachpatte, On some retarded integral inequalities and applications. J. Inequal. Pure Appl. Math. 3, Art.18 (2002)

    Google Scholar 

  54. S.B. Pachpatte, B.G. Pachpatte, Inequalities for terminal value problems for differential equations. Tamkang J. Math. 33 (3), 199–208 (2002)

    MathSciNet  MATH  Google Scholar 

  55. H. Poincaré, Sur les équations linéaires aux différentielles et aux différences finies. Am. J. Math. Soc. 7, 203–258 (1885)

    Article  MATH  Google Scholar 

  56. V.S.H. Rao, Stability of motion under impulsive perturbation. Ph.D. Theory, I. I. T. Kanpur, 1976

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. J.A. Shohat, J.D. Tamarkin, The Problem of Moments, Mathematical Surveys, No. 1 (American Mathematical Society, Providence, 1943)

    Google Scholar 

  59. M.H. Stone, Linear Transformation in Hilbert Space and Their Applications to Analysis. American Mathematical Society Colloquium Publications, vol. XV (American Mathematical Society, Providence, 1932)

    Google Scholar 

  60. G. Szegö, Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. XXIII, 4th edn. (American Mathematical Society, Providence, 1975)

    Google Scholar 

  61. O. Taussky, A recurring theorem on determinants. Am. Math. Mon. 55, 672–676 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  62. J.E. Wilkins, On the growth of solutions of linear differential equations. Bull. Am. Math. Soc. 50, 388–394 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  63. D. Willett, A linear generalization of Gronwall’s inequality. Proc. Am. Math. Soc. 16, 774–778 (1965)

    MathSciNet  MATH  Google Scholar 

  64. V.A. Yakubovic̆, On the asymptotic behavior of the solutions of systems of differential equations. Dokl. Akad. Nauk. SSSR 63, 363–366 (1948)

    Google Scholar 

  65. E.H. Yang, On some new discrete inequalities of the Bellman-Bihari type. Nonlinear Anal. 7, 1237–1246 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Qin, Y. (2016). Applications of Linear One-Dimensional Inequalities. In: Integral and Discrete Inequalities and Their Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-33301-4_4

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