Abstract
There are Gronwall type inequalities in which the unknown function is not a function on R n, rather in some other space. This Chapter is devoted to these kinds; on discrete, functional and then abstract spaces.
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References
Rabczuk, R., “Elemnty nierównósci vóiniczkowych,” Warszawa, 1976.
Grace, S.R., B.S. Lalli and C.C. Yeh, Comparison theorems for difference inequalities, J. Math. Anal. Appl. 113 (1986), 468–472.
Pachpatte, B.G. and V.M. Singare, Comparison theorems on difference equations of the Volterra type, Math. Education Sect. A. 7 (1973), 45–48.
Redheffer, R. and W. Walter, A comparison theorem for difference inequalities, J. Differential Equations 44 (1982), 111–117.
Beesack, P.R., Comparison theorems and integral inequalities for Volterra integral equations, Proc. Amer. Math. Soc. 20 (1969), 61–66.
Jones, G., Fundamental inequalities for discrete and discontinuous functional equations, J. Soc. Ind. Appl. Math. 12 (1964), 43–57.
Hull, T. and W. Luxemburg, Numerical methods and existence theorems for ordinary differential equations, Numer. Math. 2 (1960), 30–41.
Filatov, A. and L. Šarova, “Integral’nye neravenstva i teorija nelineinyh kolebanii,” Moskva, 1976.
Willett, D. and J. Wong, On the discrete analogues of some generalizations of Gronwall’s inequality, Monatsh. Math. 69 (1965), 362–367.
Lees, M., Approximate solutions of parabolic equations, SIAM Journal 7 (1959), 167–183.
Singare, V.M. and B.G. Pachpatte, On some discrete generalizations of an inequality of Gronwall, Chinese J. Math. 6 (1978), 121–135.
Singare, V.M. —, Wendroff type discrete inequalities and their applications, J. Math. Phys. Sci. 13 (1979), 149–167.
Yeh, C.C., Discrete inequalities of the Gronwall-Bellman type in n independent variables I, II, J. Math. Anal. Appl. 105 (1985), 322–332; 106 (1985), 282-285.
Pachpatte, B.G., On some fundamental finite difference inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No 577–598 (1977), 65–73.
Popenda, J., Finite difference inequalities, Ann. Soc. Math. Polonae. Ser. I: Comm. math 26 (1986), 89–96.
Yang, E.H., On some new discrete generalizations of Gronwall’s inequality, J. Math. Anal. Appl. 119 (1988), 505–516.
Thandapani, E. and R.P. Agarwal, On some new integro-differential inequalities; theory and applications, Tamkang J. Math. 11 (1980), 169–184.
Pachpatte, B.G. and V.M. Singare, Discrete generalized Gronwall inequalities in three independent variables, Pac. J. Math. 82 (1979), 197–210.
Pachpatte, P.G., On some nth order finite difference inequalities, Proc. Nat. Acad. Sci. India 40 (A) (1970), 235–240.
Pachpatte, P.G. —, Discrete generalizations of Gronwall’s inequality, Math. Education 8 (1979), A73–A75.
Pachpatte, P.G. —, On discrete inequalities related to Gronwall’s inequality, Proc. Indian. Acad. Sci. 85 (A) (1977), 26–40.
Pachpatte, P.G. —, A note on some new nth order discrete inequalities, Tamkang J. Math. 9 (1978), 29–39.
Pachpatte, P.G. —, On some nth order differential inequalities and their discrete analogues, Proc. Nat. Acad. Sci. India. Sect A 49 (1979), 7–12.
Pachpatte, P.G. —, On some fundamental nth order finite difference inequalities, Ann. Ştinş. Univ. “Al. I. Cuza” Iaşi Sect. Ia Mat. (N.S.) 26 (1980), 361–367.
Agarwal, R.P. and E. Thandapani, On some new integro-differential inequalities, An. Ştinş, Univ. “Al. I. Cuza” Iaši. N.S. 28 (1982), 123–126.
Novotná, J., Variations of discrete analogues of Wirtinger’s inequality, Čas. Pěsto. Mat. 105 (1980), 278–285.
Das, P. and R. Sharma, Some Stieltjes integral inequalities, J. Math. Anal. Appl. 73 (1980), 423–433.
Mingarelli, A.B., On a Stieltjes version of Gronwall’s inequality, Proc. Amer. Math. Soc. 82 (1981), 249–252.
Mamedov, Ja. D. and S.A. AŠirov, “Nelineĭnie uravnenija Volterra,” Ašhabad, YLYM, 1977.
Mamedov, Ja.D., S. AŠirov and S. Atdaev, “Teoremy o neravenstvah,” Ašhabad, YLYM, 1980.
Mamedov, Ja. D. and G.M. Novruzov, K teorii resenii differencial’nyh uravneniĭ s vol’terrovskim operatorom, Differencial’nye uravnenija 8 (1972), No 6.
Walter, W., “Differential and integral inequalities,” New York-Heidelgerg-Berlin, 1970.
Bertolino, M., “Diferencijalne jednacine,” Beograd, 1980.
Kurpel’, N.S, and B.A. Shuvar, “Dvustoronnie operatornye neravensta i 16 primenenija,” Kiev, 1980.
Hille, E., “Lectures on ordinary differential equations,” Reading, Mass., 1969.
Hille, E. —, “Methods in classical and functional analysis,” Reading. Mass., 1972.
Chandra, F. and B. Fleishman, Bounds and maximal solutions of nonlinear functional equations, Bull. Amer. Math. Soc. 74 (1968), 512–516.
Chandra, F. —, On a generalization of the Gronwall-Bellman lemma in partially ordered Banach spaces, J. Math. Anal. Appl. 31 (1970), 668–681.
Losonczi, L. A generalization of the Gronwall-Bellman lemma and its applications, J. Math. Anal. Appl. 44 (1973), 701–709.
Boyd, D. and J. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464.
Geraghty, M., On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604–608.
Kannan, R., Construction of fixed points of a class of nonlinear mappings, J. Math. Anal. Appl. 41 (1973), 430–438.
Martynyuk, A.A., V. Lakshmikantham, and S. Leela, “Motional Stability: The Method of Integral Inequalities,” Kiev, 1989.
Moore, R.E., “Methods and applications of interval analysis,” SIAM Studies in Appl. Math., Philadelphia, 1979, 250 pp.
Moore, R.E. —, Set-valued extensions of integral inequalities, J. of Integral Equations 5 (1983), 187–198.
Moore, R.E. —, Set-valued extensions of integral inequalities, Trends in theory and practice of nonlinear differential equations 80 (1984), 383–388.
Caprani, O., K. Madsen, and L.L. Rall, Integration of integral functions, SIAM J. Math. Analysis 12 (1981), 321–341.
Alefeld, G. and J. Herzberger, “Introduction to interval computations,” New York, 1983.
Agarwal, R.P. and E. Thandapani, On nonlinear discrete inequalities of Gronwall type, An. Ştinţ. Univ. “Al. I. Cuza” Iaşi. N.S. Sect. Ia (1981), 139–149.
Agarwal, R.P. —, On discrete generalizations of Gronwall’s inequality, Bull. Inst. Math. Acad. Sinica 9 (1981), 235–248.
Agarwal, R.P. —, On discrete inequalities in n independent variables, Riv. Mat. Univ. Parma 4 (1987), 241–256.
Agarwal, R.P. —, Linear and nonlinear discrete inequalities in n independent variables, General Inequalities 5 (Oberwolfach, 1986), Basel-Boston (1987), 303–318.
Agarwal, R.P. —, Systems of multi-dimensional discrete inequalities, J. Math. Anal. Appl. 140 (1989), 241–250.
AŠirov, S. and Ja.D. Mamedov, Issledovanie resenii nelineinyh operatornyh uravnenii Vol’terra-Fredgol’ma, DAN SSSR, 229 (1976), No 2.
Atdaev, S., K teorii rešeniĭ nelineĭnyh mnogomernyh operatornyh uravneniĭ Vol’terra, Izvestija AN TSSR. Ser. FTN i GN, 1976, No 6.
Atdaev, S. —, K teorii nelineĭnyh operatornyh uravnenii Vol’terra, Izvestijaan Tssr. Ser. Ftn i GN, 1977, No 5.
Atdaev, S. and S. AŠirov, Nekotorye iteracionnye processy dlja resenija nelineĭnyh operatornyh uravneniĭ na poluosi, Izvestija AN TSSR Ser. FTN i GN, 1975, No. 6.
Azbelev, N.V. and Z.B. Caljuk, O zadace Čaplygina. Ukrain. Mat. Ž. 19 (1958), No 1.
Beesack, P.R., Lower bounds from discrete inequalities of Gollwitzer-Langenhop type, An. Ştinţ. Univ. “Al. I. Cuza” Iaşi Sect Ia Mat. (N.S.) 30 (1984), 25–30.
Beesack, P.R. —, More generalised discrete Gronwall inequalities, Z. Angew. Math. Mech. 65 (1985), 583–595.
Beesack, P.R. —, On integral inequalities of Bihari type, Acta Math. Acad. Sci. Hungar. 28 (1976), 81–88.
Beesack, P.R. —, On Lakshmikantham’s comparison methods for Gronwall inequalities, Annal. Polon. Math. 25 (1977), 187–222.
Bykov, Ja.V. and V. Linenko, Kačestvennoe issledovanie traektoriĭ nelineĭnyh sistem raznostnyh uravneniĭ, Izv. Vyss. Ucebn. Zaved. Matematika 56 (1967), 21–29.
Bykov, Ja.V. —, O periodičeskom predel’nom režime dlja sistem raznostnyh uravneniĭ, Izv. Vysš. Učebn. Zaved. Matematika 57 (1967), 9–19.
Caljuk, Z.B., Funkcional’nye neravenstva Vol’terra, Izvestija VUZov. Matematika, 1969, No 3.
Chandra, J. and V. Lakshmikantham, Some pointwise estimates for solutions of a class of nonlinear functional-integral inequalities, Ann. Mat. Pura. Appl. (4) 94 (1972), 63–74.
Chandra, J. —, A discrete integral inequality of the convolution type, Bul. Inst. Politehn. Iaşi. Sect. Ia 33(37) (1987), 35–40.
Das, P.C. and R. Sharma, Some Stieltjes integral inequalities, J. Math. Anal. Appl. 73 (1980), 423–433.
DemidoviČ, V., Ob asimptotičeskom povedenii rešenii konačnoraznostnyh uravnenii. I, Obščie položenija, Differencial’nie uravnenija 10 (1974), 2267–2278.
DemidoviČ, V. —, Ob odnom priznake ustoicivosti raznostnyh uravnenij, Differencial’nye uravnenija 5 (1969), 1247–1255.
Dragomir, S.S., On a Volterra integral equation, An. Univ. Timişoara Ser. St. Mat. 24 (1986), 3–18.
Dragomir, S.S. —, On Volterra integral equation with kernels of (L)-type, An. Univ. Timisoara Ser. St. Mat. 25 (1987), 21–41.
Konstantinov, M., Ob odnom funkcional’nom neravenstve, Ukrain. Mat. Ž. 30 (1978), 226–231.
Kvapis, M., O suščestvovanii i edinstvennosti rešenija i shodimosti posledovatel’nyh priblizeniĭ dlja funkcional’nyh uravnenii tipa Vol’terra v Banahovom prostranstve, Tr. sem. po teor. diff. uravn. s otkl. argumentom. M. Univ. druž. nar. im. P. Lumumby, 4 (1967).
Makhmudov, A.P. and R.Z. Abbasov, Theorems on integral inequalities in Riemann-Stieltjes integrals and their applications (Russian), Dokl. Akad. Nauk. SSSR 265 (1982), No. 5, 1069–1073.
Mamedov, Ja.D. and A.I. Perov, O mnogomernyh integral’nyh uravnenijah i neravenstvah tipa Vol’terra, Izv. AN AzSSR. Ser. FT i MN, 1965, No. 1.
Mamedov, Ja.D. and S.M. Sardarly, K teorii rešeniĭ nelineĭnyh operatornyh uravneniĭ Vol’terra na poluosi, Sib. Mat. Ž. 12 (1971), No 3.
Martynjuk, D., “Lekcii po kacestvennoi teorii raznostnyh uravnenii,” Kiev, 1972.
Mckee, S., Generalised discrete Gronwall lemmas, Z. Angew. Math. Mech. 62 (1982), 429–434.
Numirow, T. and I. Chorshanbaev, Some integral inequalities with singular kernels (Russian), Dokl. AN UzSSR, 1983, No 1, 8–10.
Pachpatte, B.G., On the discrete generalizations of Gronwall’s inequality, J. Indian Math. Soc. N.S. 37 (1973), 147–156.
Pachpatte, B.G. —, Finite difference inequalities and their applications, Proc. Natn. Acad. Sci., India, A43 (1973), 348–356.
Pachpatte, B.G. —, A note on some finite difference inequalities, Math. Student. 43 (1975), 405–416.
Pachpatte, B.G. —, On some discrete inequalities of Bellman-Bihari type, Indian. J. Pure. Appl. Math. 6 (1975), 1479–1487.
Pachpatte, B.G. —, On a class of integral inequalities of Bellman-Bihari type, Indian. J. Pure. Appl. Math. 7 (1976), 1166–1177.
Pachpatte, B.G. —, A note on some new discrete inequalities, Proc. Nat. Acad. Sci. India Sect. A 46 (1976), 190–198.
Pachpatte, B.G. —, On some new discrete inequalities and their applications, Proc. Nat. Acad. Sci. India Sect. A 46 (1976), 255–262.
Pachpatte, B.G. —, On some nonlinear discrete inequalities of Gronwall type, Bull. Inst. Math. Acad. Sinica 5 (1977), 305–315.
Pachpatte, B.G. —, On some new integral inequalities and their discrete analogues, Indian J. Pure Appl. Math. 8 (1977), 1093–1107.
Pachpatte, B.G. —, A note on discrete inequalities of the Gronwall-Bellman type, J. Math. Phys. Sci. 17 (1977), 115–124.
Pachpatte, B.G. —, On some new generalizations of Gronwall’s inequality and their discrete analogues, J. of M.A.C.T. 10 (1977), 55–69.
Pachpatte, B.G. —, A note on some fundamental discrete inequalities of the Gronwall-Bellman type, Bull. Inst. Math. Acad. Sinica 5 (1977), 121–128.
Pachpatte, B.G. —, On some new discrete inequalities involving partial sums, Indian J. Pure Appl. Math. 9 (1978), 640–647.
Pachpatte, B.G. —, Finite difference inequalities and discrete time control systems, Indian J. Pure Appl. Math. 9 (1978), 1282–1290.
Pachpatte, B.G. —, On some new discrete inequalities and their applications to a class of sum-difference equation, Ann. Ştinţ. Univ. “Al I. Cuza” Iaşi Sect Ia Mat. (N.S.) 24 (1978), 315–326.
Pachpatte, B.G. —, On some fundamental discrete inequalities in two independent variables, Tamkang J. Math. 12 (1981), 22–33.
Pachpatte, B.G. —, On some integral inequalities and their application to higher order differential equations, Soochow J. Math. 14 (1988), 221–230.
Pachpatte, B.G. —, On a class of new discrete inequalities, Tamkang J. Math. 20 (1989), 19–28.
Pelczar, A., On invariant points of monotone transformations in partially ordered spaces, Ann. Polon. Math. 17 (1965), 49–53.
Perov, A.I., Ob integral’nyh neravenstvah, Tr. sem. po Funkcional’nomu Analizu. Voronež, Vyp. 5 (1975), 87–97.
Popenda, J., Finite difference inequalities, Fasc. Math. 13 (1981), 79–87.
Popenda, J. —, On the discreteanalogy of Gronwall lemma, Demonstratio Math. 16 (1983), 11–25.
Popenda, J. —, On the discrete analogy of Gronwall-Wendroff inequality, Demonstratio Math. 18 (1985), 1083–1103.
Popenda, J. —, On some discrete Gronwall type inequalities, Fasc. Math. 17 (1985), 109–114.
Popenda, J. —, On the discrete inequalities of Gronwall-Bellman type, An. Ştinţ. Univ. “Al. I. Cuza” Iaşi Ser. Ia Mat. 33 (1987), 37–52.
Popenda, J. —, On the discrete analogy of Gronwall inequality, Fasc. Math. 19 (1987), 63–78.
Popenda, J and J. Werbowski, On the discrete analogy of Gronwall lemma. Fasc. Math. 11 (1979), 143–154.
Singare, V.M. and B.G. Pachpatte, On some fundamental discrete inequalities of the Wendroff type, An. Ştinţ. Univ. “Al. I. Cuza” Iaşi Sect Ia Mat. (N.S.) 26 (1980), 87–94.
Singare, V.M. —, On certain discrete inequalities of the Wendroff type, Indian J. Pure Appl. Math. 11 (1980), 727–736.
Sree Hari Rao, V., Gronwall-Bellman type inequalities to distributions, Proc. Conf. Math. Math, in Physics (Mysore, 1978), Part B, pp. 60–64, Matscience Rep., 96, Inst. Math. Sci., Madras, 1979.
Sree Hari Rao, V. —, Integral inequalities of Gronwall type for distributions. J. Math. Anal. Appl. 72 (1979), 545–550.
Thandapani, E., Discrete inequalites in n independent variables of Gronwall-Bellman type, Appl. Anal. 39 (1988), 189–199.
Thandapani, E. —, Discrete inequalities in n independent variables of Gronwall-Bellman type, Appl. Anal. 39 (1988), No 1-3, 189–199.
Thieme, H.R., The asymptotic behavior of solutions of nonlinear integral equations, Math. Z. 157 (1977), 141–154.
Vidyasagar, M., Nonlinear functional inequalities of Gronwall-Bihari type, private communication.
Wirth, T., Abschätzungen von Lösungen Voltenascher und Fredhomscher Integralungleichungen, Diss. Dokt. Naturwiss. Fak. Math, und Naturwiss. Tehn. Univ. Hannover, 1976, 55 pages.
Wu, J. and G.S. Yang, On discrete Gronwall’s inequalities. Tamkang J. Math. 12 (1981), 161–170.
Yang, E.H., On some new discrete inequalities of the Bellman-Bihari type. Nonlinear Anal. 7 (1983), 1237–1246.
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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Gronwall Inequalities on Other Spaces: Discrete, Functional and Abstract. 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_14
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