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Part of the book series: Mathematics and Its Applications ((MAEE,volume 53))

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Abstract

In quantitative analysis of differential equations the use of differential inequalities is very wide. In this Chapter we shall give various theorems on differential inequalities of the first order, of higher orders, as well as of systems of differential inequalities.

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References

  1. Hille, E., “Lectures on ordinary differential equations,” Reading, Mass., 1969.

    Google Scholar 

  2. Reid, W., Properties of solutions of an infinite system of ordinary linear differential equations of the first order with auxiliary boundary conditions, Trans. Amer. Math. Soc. 32 (1930), 284–318.

    Article  MathSciNet  MATH  Google Scholar 

  3. Szarski, J., “Differential Inequalities,” Warsaw, 1965.

    Google Scholar 

  4. Lakshmikantham, V., Upper and lower bounds of the norm of a solution of differential equations, Proc. Amer. Math. Soc. 13 (1962), 615–616.

    Article  MathSciNet  MATH  Google Scholar 

  5. Beesack, P. R., Gronwall Inequalities, Carleton Un. Math Notes No. 11, (1975).

    Google Scholar 

  6. Čaplygin, S.A., Osnovanija novoga sposoba približennogo integrirovanija differencial’nyh uravneniĭ, Moskva 1919. Sobr. Sočin. Vol. I. Moskva, 1948, 348–368.

    Google Scholar 

  7. Čaplygin, S.A. —, Približennoe integrirovanie obyknovennogo differencial’nogo uravnenija pervogo porjadka, Moskva, 1920. Sobr. Sočin. Vol. I. Moskva, 1948, 402–419.

    Google Scholar 

  8. Peano, G. Sull’ integrabilità délie equazioni differenziali del primo ordine, Atti. R. Accad. Sc. Torino 21 (1885-1886), 677–685. English translation in “Selected Works of Giuseppe Peano,” pp. 51-57, translated and edited by H.C. Kennedy, Toronto, 1973.

    Google Scholar 

  9. Petrovitch, M., Sur une manière d’étendre le théorème de la moyenne aux équations différentielles du premier ordre, Math. Annalen, 54 (1899), 417–436.

    Article  Google Scholar 

  10. Bertolino, M. and D. TrifunoviĆ, Sur le théorème fondamental de S.A. Caplygin sur l’inégalité différentielle du premier ordre, Math. Balkanica 1 (1971), 11–18.

    MathSciNet  MATH  Google Scholar 

  11. Bertolino, M., “Diferencijalne Jednačine,” Beograd, 1980.

    Google Scholar 

  12. Mamedov, Ja.D., S. AŠirov and S. Atdaev, “Teoremy o Neravenstvah,” Ašhabad, YLYM, 1980.

    Google Scholar 

  13. Flett, T. M., “Differential analysis,” Cambridge-London-New York-New Rochelle-Melbourne-Sydney, 1980.

    Google Scholar 

  14. Perov, A.I., K voprosu o strukture integral’noĭ voronki, Nauč. Dokl. Vysšeĭ Školy. Ser FMN, 1959, No. 2.

    Google Scholar 

  15. Adamov, V.G. and T. Sabirov, O differencial’nyh neravenstvah s proizvodnymi čislami, Sb. Rabot. St. I Asp. Po Mat. Voronež. Voron. Un-t 1965.

    Google Scholar 

  16. Berezin, I.S. and N.P. Žitkov, “Metodi vyčisleniĭ II,” Moskva, 1960.

    Google Scholar 

  17. Rabczuk, R., “Elementy nierówności vózniczkowych,” Warszawa, 1976.

    Google Scholar 

  18. LuziŃ, N.N., O metode S.A. Čaplygina s analitičeskoi točki zrenija, Sobr. sočin Vol. 3, Moskva 1953, str. 146–167.

    Google Scholar 

  19. Quade, W., Ein neues Verfahren der schrittweisen Naherungen zur Lösung von y′ = f(x,y), Math. Zeit. 48 (1942), 324–368.

    Article  MathSciNet  Google Scholar 

  20. Čaplygin, S. A., “Novyĭ metod približennogo integrirovanija differencial’nyh uravneniĭ,” Moskva-Leningrad, 1950.

    Google Scholar 

  21. Kasceev, N.A., K voprosu o granice primenimosti teoremy S.A. Čaplygina o diff. ner. k lineĭnnym diff. ur. vysših porjadkov, Uč. zap. Kuib. gos. ped. in-ta 1956, vyp. 14, 53–56.

    Google Scholar 

  22. Kasceev, N.A. —, Tocnaja granica primenimosti teoremi Čaplygina dlja lineĭnogo uravnenija, DAN SSSR 111 (1956), No. 5, 937–940.

    MathSciNet  Google Scholar 

  23. Pólya, G., On the mean-value theorem corresponding to a given linear homogeneous differential equation, Trans. Amer. Math. Soc. 24 (1922), 312–324.

    Article  MathSciNet  Google Scholar 

  24. Mammana, G., Decompositione delle espressioni differenzialli omogene in prodotto di fattori symbolici e applicazione relativa allo studio delle equazioni differenzialli lineari, Math. Zeit. 33 (1931).

    Google Scholar 

  25. Karlin, S. and W.J. Studden, “Tchebycheff systems: with applications in analysis and statistics,” New York-London-Sydney, 1966.

    Google Scholar 

  26. PeČariĆ, J.E., “Konveksne funkcije: Nejednakosti,” Beograd, 1987.

    Google Scholar 

  27. Babkin, B., On a generalization of a theorem of Academician S.A. Čaplygin on a differential inequality (Russian). Molotov. Gos. Univ. Uč. 8 (1953), 3–6.

    MathSciNet  Google Scholar 

  28. Babkin, B. —, Resenie odnoĭ kraevoĭ zadači dlja obyknovennogo differencial’nogo uravnenija vtorogo porjadka metodom Čaplygina, Prikladnaja Mat. I Meh. 18 (1954).

    Google Scholar 

  29. Grace, S.R. and B.S. Lalli, A comparison theorem for general nonlinear ordinary differential equations, J. Math. Anal. Appl. 129 (1986), 39–43.

    Article  MathSciNet  Google Scholar 

  30. Kiguradze, I.T., Oscillation properties of certain ordinary differential equations (Russian), Soviet Math. (Iz.VUZ) 3 (1962), 649–652.

    MATH  Google Scholar 

  31. Atkinson, F. V., On second order differential inequalities, Proc. Roy. Soc. Edinburgh Sect. A 72 (1974), 109–127.

    MathSciNet  MATH  Google Scholar 

  32. Teufel, M., A note on second order differential inequalities and functional differential equations, Pacific J. Math. 41 (1972), 537–541.

    MathSciNet  Google Scholar 

  33. Kartsatos, A.G., On nth-order differential inequalities, J.Math. Anal. Appl. 52 (1975), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  34. Muldowney, J.S., On an inequality of Peano, Canad. Math. Bull. 16(1) (1973), 79–81.

    Article  MathSciNet  MATH  Google Scholar 

  35. Muldowney, J.S. and D. Willett, An elementary proof of the existence of solutions to second order nonlinear boundary value problems, SIAM J. Math. Anal. 5 (1979), 701–707.

    Article  MathSciNet  Google Scholar 

  36. Muldowney, J.S. —, An intermediate value property for operators with applications to integral and differential equations, Can. J. Math. 26 (1979), 27–41.

    Article  MathSciNet  Google Scholar 

  37. Gel’fand, A.V., Priblizennoe integrirovanie sistemy obiknovennyh differencial’nyh uravnenii pervogo porjadka, Izv. AN SSSR. Otd. M i EN, (1938), No. 5–6.

    Google Scholar 

  38. Perov, A.I., Neskoljko zamecanii otnostel’no differencial’nyh neravenstv, Izv. Vuzov. Matematika, (1965), No. 4.

    Google Scholar 

  39. Wazewski, T., Systèmés des équations et des inégalités différentielles ordinaires aux deuxièmes membres monotones et leurs applications, Ann. Sci. Polon. Math. 23 (1950), 112–166.

    MathSciNet  MATH  Google Scholar 

  40. Wazewski, T. —, Certaines propositions de caractère “epidermique” relatives aux inégalités différéntielles, Ann. Sci. Polon. Math. 24 (1952), 1–12.

    MathSciNet  MATH  Google Scholar 

  41. Pachpatte, B.G., On some fundamental integrodifferential and integral inequalities, An. Štinţ. Univ. “Al. I. Cuza” Iaşi. Sect Ia Mat. (N.S.) 23 (1977), 77–86.

    MATH  Google Scholar 

  42. Pachpatte, B.G. —, A note, on some fundamental integro-differential inequalities, Tamkang J. Math. 13 (1982), 63–67.

    MathSciNet  MATH  Google Scholar 

  43. Pachpatte, B.G. —, A note of second order integro-differential inequalities of the Gronwall-Bellman type, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 678–715 (1980), 35–41.

    Google Scholar 

  44. Simeonov, P.S. and D.D. BaïNov, On some generalizations of the Gronwall-Bellman integral inequality for scalar functions of many variables, Rend. Circ. Math. Palermo, II Ser. 32 (1983), 26–38.

    Google Scholar 

  45. Singare, V.M. and B.G. Pachpatte, On some discrete generalizations of an inequality of Gronwall, Chinese J. Math. 6 (1978), 121–135.

    MathSciNet  MATH  Google Scholar 

  46. Shastri, R.P. and D. Y. Kasture, Wendroff type inequalities, Proc. Amer. Math. Soc. 72 (1978), 248–250.

    Article  MathSciNet  MATH  Google Scholar 

  47. Rab, M., Linear integral inequalities, Arch. Math. Brno 15 (1979), 37–46.

    MathSciNet  MATH  Google Scholar 

  48. Margolis, B., An extension of Gronwall inequality, Rev. Un. Math. Argentina 25 (1970/71), 247–251.

    MathSciNet  Google Scholar 

  49. Lu, W. and Y. Wu, Integral inequalities of Gronwall type in n independent variables and their applications to partial differential equations, J. Sichuan Univ. Nat. Sci. Ed. 1982, No. 1, 1–19 (Chinese).

    Google Scholar 

  50. Lu, W., Generalizations of integral inequalities of Gronwall-Wendroff type, J. Sichuan Univ. Nat. Sci. Ed. 1989, No. 2, 12–29 (Chinese).

    Google Scholar 

  51. Abramovich, J., On Gronwall and Wendroff type inequalities, Proc. Amer. Math. Soc. 87 (1983), 481–486.48.

    Article  MathSciNet  Google Scholar 

  52. Aparcin, A.S. and Ten Men Jan, Unimprovable estimates of the solutions of certain integral inequalities (Russian), Sibirsk. Mat. Z. 20 (1979), 192–195, 207.

    Article  MATH  Google Scholar 

  53. Azbelev, N.V., O granicah primenimosti teoremi S.A. Čaplygina, DAN SSSR, 89 (1953), No 4.

    Google Scholar 

  54. Azbelev, N.V. —, Nekotorye uslovija razrešimosti zadači Čaplygina dlja sistemy obyknovecnyh differencial’nyh uravnenii, Naucn. dokl. vyssei skoly. Ser. FMN, 1958, No 6.

    Google Scholar 

  55. Azbelev, N.V. and Z.B. Caljuk, O zadace Čaplygina, Ukrain. mat. Ž. 10 (1958), No 1.

    Google Scholar 

  56. Azbelev, N.V. —, K voprosu ob integral’nyh neravenstvah, Tr. Iževskogo mat. sem. Iževsk. vyp. 1, 1963.

    Google Scholar 

  57. Azbelev, N.V. —, K voprosu ob differencial’nom neravenstve, Differencial’nye uravnenija 1, 1965, No 4.

    Google Scholar 

  58. Azbelev, N.V. —, Ob odnom metode ocenok rešenii uravnenii, Volzskii mat. sb. 1966, vyp. 5.

    Google Scholar 

  59. Azbelev, N.V., A.Ja. Hohrjakov and Z.B. Caljuk, Teoremy o differencial’nom neravenstve dlja kraevyh zadac, Mat. sbornik 59(101), 1962.

    Google Scholar 

  60. Babkin, B. K teoreme S.A. Čaplygina o differencial’nih neravenstvah, Mat. sb. 46(88), 1958, No 4.

    Google Scholar 

  61. Baiada, E., Confronto e dipendenze dai parametri degli integrali delle equazioni differenziali, I, Rend. Accad. Lincei, (8) 3, 258–263 (1974).

    MathSciNet  Google Scholar 

  62. Baluev, A.N., O metode Čaplygina, Vestnik Lening. Univ., 13, (1956) vyp. 3.

    Google Scholar 

  63. Bittner, L. Die elementaren differential-und integralgleichungen mit einem allgemeinen Ungleichungsbegriff, Math. Nach. 38 (1968), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  64. Brauer, F., Bounds for solutions of ordinary differential equations, Proc. Amer. Math. Soc. 14 (1963), 538–545.

    Article  MathSciNet  Google Scholar 

  65. Cafiero, F., Su due teoremi di confronto relativi ad un’equazioni differenziale ordinaria del primo ordine, Boll. Un. Mat. Ital. 3 (1948), 124–128.

    MathSciNet  MATH  Google Scholar 

  66. Caljuk, Z.B., Volterra integral equations (Russian), Math. Analysis Vol. 15, Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1977.

    Google Scholar 

  67. Candirov, G.I., O suščestvovanii (∈,B)-rešenija smešannoi zadači dlja kvazilineĭnogo giperboličeskogo uravnenija, Uč. zap. AGU, Ser. Fiz.— Mat.nauk, 1970, No 2.

    Google Scholar 

  68. Caplygin, S.A., Približennoe integrirovanie sistemy dvuh differencial’nyh uravnenii, Sobr. sočin. Vol. I. Moskva, 1948, 427–444.

    Google Scholar 

  69. Conlan, J. and J. Diaz, Existence of solutions of an nth order hyperbolic partial differential equations, Contrib. to Diff. Equations (1963), 277–289.

    Google Scholar 

  70. Das, P. and R. Sharma, On optimal controls for measure delay differential equations, SIAM J. Control. 9 (1971), 43–61.

    Article  MathSciNet  MATH  Google Scholar 

  71. Grammatikopoulis, M.K., On the effect of deviating argument on the behaviour of bounded solutions of nonlinear differential equations (Russian), Ukrain. Mat. Ž. 30 (1978), 462–473.

    MathSciNet  Google Scholar 

  72. Gronwall, T., On the existence and properties of the solutions of a certain differential equation of the second order, Ann. Math. (2) 28 (1927), 355–364.

    Article  MathSciNet  MATH  Google Scholar 

  73. Gutowski, R., Nekotorye voprosy ustoĭčivosti differencial’nyh uravneniĭ v častnyh proizvodnyh, opisyvajuščih dviženie mehaniceskih sistem v teorii kolebanii, VII über, nichtlineare Schwing. B.1. 1977, 289–305.

    Google Scholar 

  74. Gutowski, R. and B. Radziszewski, Asymptotic behaviour and properties of solutions of a system of non-linear second order ordinary differential equations describing motion of mechanical systems, Arch. Mech. Stos. 22 (1970), 675–694.

    MathSciNet  MATH  Google Scholar 

  75. Gutowski, R. —, Asymptotic behaviour and properties of solutions of nonlinear ordinary differential equations of first order describing the motion of a mechanical system, Arch. Mech. Stos. 23 (1971), 17–25.

    MATH  Google Scholar 

  76. Jackson, L. and K. Schrader, On second order differential inequalities, Proc. Amer. Math. Soc. 17 (1966), No. 5.

    Google Scholar 

  77. Knobloch, H.W., Second order differential inequalities and a nonlinear boundary value problem, J. Different. Equat. 5 (1969), No. 1.

    Google Scholar 

  78. Krasnosel’skiï, M.A. and Ja D. Mamedov, Zamecanie o primenenii differencial’nyh i integral’nyh neravenstv v voprosah o korrektnosti zadači Koši dlja obyknovennyh differencial’nyh uravneniĭ v banahovyh prostranstvah, Naučn. dokl. vysš. šk. ser. FNN, 1959, No. 2.

    Google Scholar 

  79. Krasnosel’skiï, M.A., A. Ju. Levin and Ja.D. Mamedov, Ob ocenkah resenii differencial’nyh uravnenii vtorogo porjadka, Ukrain. mat. ž. 18 (1966), No. 1.

    Google Scholar 

  80. Kurpel’, N.S. and V.I. Ohroncuk, O differencial’nyh neravenstvah v banahavyh prostranstvah, In: “Metody približennogo rešenija differenc. i integral’n. uravneniĭ,” Kiev. Izd. AN USSR, 1973.

    Google Scholar 

  81. LuziŃ, N.N., O metode priblizennogo integrirovanija akademika S.A. Čaplygina, Trudy CAGI, vyp. 141 (1932). Sobr. sočin.T.e, Moskva 1953, str. 145-180.

    Google Scholar 

  82. LuziŃ, N.N. —, O metode priblizennogo integrirovanija akademika S.A. Čaplygina, Uspehi mat. nauk. 6, vyp. 6 Sobr. sočin. T. e, Moskva 1953, str. 181–208.

    Google Scholar 

  83. Mamedov, Ja.D. and P.E. Sobolevskiï, Differencial’noe uravnenie s neograničennympreremennym operatorom v banahovom pronstranstve, Tr. sem. po funkc. anal. Voronež. Voronežki in-t., 1963, vyp. 7.

    Google Scholar 

  84. Martynyuk, A.A. and R. Gutovski, “Integral inequalities and stability of motion” (Russian), Kiev 1979, 272 pages.

    Google Scholar 

  85. Martynyuk, A.A., V. Lakshmikantham, and S. Leela, “Motion Stability, the Method of Integral Inequalities” (Russian), Kiev, 1989.

    Google Scholar 

  86. Mlak, W., Differential inequalities in linear spaces, Ann. Polon. Math. 5, 1958, No. 1.

    Google Scholar 

  87. Mlak, W. —, Differential inequalities with unbounded operators in Banach spaces, Ann. Polon. Math. 9, 1960, No. 1.

    Google Scholar 

  88. Mlak, W. —, Note on abstract linear differential inequalities, Rev. math. pures, appl. 6, 1961, No. 4.

    Google Scholar 

  89. Muldowney, I. S. On an inequality of Čaplygin and Polya, Proc. Roy. Irish Acad.

    Google Scholar 

  90. MyŠkis, A.D., Ob odnom differencial’no-funcional’nom neravenstve, Uspehi mat. nauk 15, 1960, vyp. 4 (94).

    Google Scholar 

  91. MyŠkis, A.D. —, Žamečanije k stat’e P.M. Ždanova “O približennom rešenii sistemy differencial’nyh uravneniĭ pervogo porjadka s zapazdyvajuscim argumentom”. Uspehi mat. nauk 16, 1961, vyp. 2 (98).

    Google Scholar 

  92. Ohroncuk, V., O differencial’nyh neravenstvah vtorogo porjadka v banahovom prostranstve, Ukrain. mat. ž. 27, 1975, No. 1.

    Google Scholar 

  93. Ohroncuk, V. —, O différendjal’nyh neravenstvah dlja nelineiyh differencijal’nyh uravnenija s zapazdyvajušcim argumentom, Ukrain. mat. ž. 27 (1975), 256–262.

    MathSciNet  Google Scholar 

  94. Opoïcev, V.I. and T. A. Hurodze, Some theorems on differential and integral inequalities (Russian), Sakharth. SSR Mecn. Akad. Moambe 87 (1977), 565–568.

    MathSciNet  MATH  Google Scholar 

  95. Pak, S.A. Teoremy o differencial’nyh neravenstvah dlja mnogotočečnyh kraevyh zadač. In: “Priblizennye metody rešenija differencial’nyh uravneniĭ,” Kiev, 1964, vyp. 2.

    Google Scholar 

  96. Petrov, B.N., Granica primenimosti teoremi S.A. Čaplygina o differencial’nyh neravenstvah k lineĭnym uravnenijam s obyknovenym proizvodnymi vtorogo porjadka, DAN SSSR, 51, 1946, No. 4.

    Google Scholar 

  97. Petrov, B.N. —, Neprimenimost’ teoremy o differencial’nom neravenstve S.A. Čaplygina k nekotorym uravnenijam s obyknovennymi proizvodnym vtoroga porjadka, DAN SSSR, 51, 1946, No. 7.

    Google Scholar 

  98. Rahmatullina, L.F., Ob odnom primenenii usloviĭ razresimosti zadači Čaplygina k voprosam ograničennosti i ustoĭčivosti reseniĭ differencial’nyh uravneniĭ, Izv. VUZov Matematika, 1959, No. 2.

    Google Scholar 

  99. Sadovskiï, B.N., Nekotorye teoremy o differencial’nyh neravenstvah, V. sb.: Problemy mat. analiza sloznih sistem. Voronež. Voronežkiĭ Un-t, 1967, vyp. 1.

    Google Scholar 

  100. Schrader, K., A note on second order differential inequalities, Proc. Amer. Math. Soc. 19 (1968), 1007–1012.

    Article  MathSciNet  MATH  Google Scholar 

  101. Slugin, S.N., Približennoe rešenie operatornyh uravneniĭ na osnove metoda S.A. Čaplygina. DAN SSSR, 103, 1955, No. 4.

    Google Scholar 

  102. Sugujama, Shohei, On comparison theorems of nonlinear Volterra integral equations, Kodai Math. Semin. Repts. 27 (1976), 147–154.

    Article  Google Scholar 

  103. Ved’, Ju.A., O vozmuscenijah lineinyh odnorodnyh differencial’nyh uravneniĭ s preremennymi koefficientami. In: “Issledovanija po integrodifferencial’nym uravnenijam v Kirgizii,” vyp. 3, Frunze, “Him”, 1965.

    Google Scholar 

  104. Venckova, M., On the boundedness of solutions of higher order differential equations, Arch. Math. 4, Scripta Fac. Sci. Nat. Ujep Brunensis 13 (1977), 235–242.

    MathSciNet  MATH  Google Scholar 

  105. Westphal, H., Zur Abschätzung der Lösungen nichtlinearer parabplischer Differentialgleichungen. Math. Zeit. 51 (1949), 690–695.

    Article  MathSciNet  MATH  Google Scholar 

  106. Yue-Sheng, L., The bound, stability and error estimates for the solution of nonlinear differential equations, Chinese Math. Acta 3 (1963), 34–41.

    Google Scholar 

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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Inequalities of Čaplygin Type. In: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications, vol 53. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3562-7_11

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