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Abstract comparison principles and multivariable Gronwall-Bellman inequalities. (English) Zbl 0613.47037
The paper is a continuation of the previous papers by the author [Atti Accad. Naz. Lincei 8, 66, 189-193 (1979; Zbl 0452.47062), Bull. Acad. Polon. Sci. Ser. Sci. Math. 30, 161-166 (1982; Zbl 0497.47038), Demonst. Math. 15, 145-153 (1982; Zbl 0509.47045)]. The paper divided into four chapters contains seven theorems, four lemmas, many remarks, and examples. In the first chapter the author studies compactness of increasing sequences of elements in an ordered metrizable uniform space X with denumerable sufficient family of semi-metrics. In the second chapter the author studies comparison theorems between the solutions in Y of an operator inequality (O.I.) $x\le Tx$ and associated equation (O.E.) $x=Tx$ where T:Y$\to Y$ is increasing mapping on some ordered closed subset Y of X. The statement is e.g.: to any solution u of (O.I.) there corresponds solution v of (O.E.) such that $u\le v$ and if there exists other solution w of (O.I.) such that $v\le w$ then $v=w.$ In the next two chapters applying the above described theorems the author studies multivariable (and scalars) Gronwall type inequalities $$ x(t)\le p(t)+\int\sp{t}\sb{0}k(x)(s)ds,\quad t\in R\sp n\sb+ $$ and many of its particular cases. Here $x,p\in X\sp m\sb n$, $k:X\sp m\sb n\to X\sp m\sb n$ is an increasing mapping and $X\sp m\sb n$ denotes the class of all continuous functions from $R\sp n\sb+$ to $R\sp m$.
Reviewer: J.Popenda

MSC:
47B60Operators on ordered spaces
26D10Inequalities involving derivatives, differential and integral operators
WorldCat.org
Full Text: DOI
References:
[1] Abian, A.: A fixed point theorem for nonincreasing mappings. Boll. un. Mat. ital. 2, 200-201 (1969) · Zbl 0175.01301
[2] Abian, S.; Brown, A. B.: A theorem on partially ordered sets with applications to fixed point theorems. Canad. J. Math. 13, 78-82 (1961) · Zbl 0098.25502
[3] Abramovich, J.: On Gronwall and Wendroff type inequalities. Proc. amer. Math. soc. 87, 481-486 (1983) · Zbl 0521.26006
[4] Ashirov, S.; Mamedov, Ya.D: Studies on the solutions of nonlinear Volterra-Fredholm operator equations. Dokl. akad. Nauk SSSR 229, 265-268 (1976)
[5] Azbelev, N. V.; Tsaljuk, Z. B.: On the Chaplygin’s problem. Ukrain. mat. Zh. 10, 3-12 (1958) · Zbl 0080.10501
[6] Bakhtin, I. A.: On the existence of generalized fixed points for abelian families of noncontinuous operators. Sibirsk. mat. Zh. 13, 243-251 (1972) · Zbl 0235.47033
[7] Beckenbach, E. F.; Bellman, R.: Inequalities. (1961) · Zbl 0097.26502
[8] Bellman, R.; Cooke, K. L.: Differential-difference equations. (1963) · Zbl 0105.06402
[9] Onesti, N. Berruti: A proposito del lemma di Gronwall perle funzioni di due variabili. Rend. ist. Lombardo sci. Lett. (A) 95, 119-126 (1961) · Zbl 0101.04601
[10] Bielecki, A.: Une remarque sur l’application de la théorie de Banach-cacciopoli-Tikhonov dans la théorie de l’équation $s = f(x, y, z, p, q)$. Bull. acad. Polon. sci. (Ser. Sci. math.) 4, 265-268 (1956) · Zbl 0070.09004
[11] Bihari, I.: A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations. Acta math. Acad. sci. Hungar. 7, 71-94 (1956) · Zbl 0070.08201
[12] Bondge, B. K.; Pachpatte, B. G.: On Wendroff type integral inequalities in n independent variables. Chinese J. Math. 7, 37-46 (1979) · Zbl 0423.26007
[13] Bondge, B. K.; Pachpatte, B. G.: On nonlinear integral inequalities of the Wendroff type. J. math. Anal. appl. 70, 161-169 (1979) · Zbl 0423.26006
[14] Bondge, B. K.; Pachpatte, B. G.; Walter, W.: On generalized Wendroff type inequalities and their applications. Nonlinear anal. TMA 4, 491-495 (1980) · Zbl 0471.26007
[15] Chandra, J.; Davis, P. W.: Linear generalizations of Gronwall’s inequality. Proc. amer. Math. soc. 60, 156-160 (1976) · Zbl 0347.34009
[16] Chandra, J.; Fleishman, B. A.: On a generalization of the Gronwall-Bellman lemma in partially ordered Banach spaces. J. math. Anal. appl. 31, 668-681 (1970) · Zbl 0179.20302
[17] Chu, S.; Metcalf, F. T.: On Gronwall’s inequality. Proc. amer. Math. soc. 18, 439-440 (1967) · Zbl 0148.28902
[18] Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations. (1955) · Zbl 0064.33002
[19] Conlan, J.; Diaz, J. B.: Existence of solutions of an nth order hyperbolic partial differential equation. Contrib. differential equations 2, 277-289 (1963)
[20] Corduneanu, A.: A note on the Gronwall inequality in two independent variables. J. integral equtions. 4, 271-276 (1982) · Zbl 0526.26008
[21] Corduneanu, C.: Sur certains équations fonctionnelles de Volterra. Funkcial. ekvac. 9, 119-127 (1966) · Zbl 0173.17202
[22] Corduneanu, C.: Principles of differential and integral equations. (1977) · Zbl 0208.10701
[23] Defranco, R. J.: Gronwall’s inequality for systems of multiple Volterra integral equations. Funkcial. ekvac. 19, 1-9 (1976) · Zbl 0339.45003
[24] Dieudonne, J.: Foundations of modern analysis. (1960)
[25] Dugundji, J.; Granas, A.: ”Fixed point theory,” vol. I. Monograf. mat. 61 (1982) · Zbl 0483.47038
[26] Ghoshal, S. K.; Masood, M. A.: Gronwall’s vector inequality and its application to a class of non self-adjoint linear and nonlinear hyperbolic partial differential equations. J. indian math. Soc. 38, 383-394 (1974) · Zbl 0348.35061
[27] Headley, V. B.: A multidimensional nonlinear Gronwall inequality. J. math. Anal. appl. 47, 250-255 (1974) · Zbl 0289.34018
[28] Krasnoselskii, M. A.: Positive solutions of operator equations. (1962)
[29] Krasnoselskii, M. A.; Sobolev, A. V.: On the fixed points of non-continuous operators. Sibirsk. mat. Zh. 14, 674-677 (1973)
[30] Kurepa, D.: Fixpoints of decreasing mappings of ordered sets publ. Inst. math. (N. S.). Publ. inst. Math. (N. S.) 18, No. 32, 111-116 (1975) · Zbl 0339.54037
[31] Lakshmikantham, V.; Leela, S.: Differential and integral inequalities. (1969) · Zbl 0177.12403
[32] Mamedov, Ya.D; Ashirov, S.; Atdaev, S.: Theorems about inequalities. (1980)
[33] Matkowski, J.: Fixed point theorems for mappings with a contractive iterate at a point. Proc. amer. Math. soc. 62, 344-348 (1977) · Zbl 0349.54032
[34] Nachbin, L.: Topology and order. (1965) · Zbl 0131.37903
[35] Nagumo, M.; Simoda, S.: Note sur l’inégalité differentielle concernant LES équations du type parabolique. Proc. Japan acad. 27, 536-539 (1951) · Zbl 0044.09902
[36] Oberg, R. J.: On the local existence of solutions of certain functional-differential equations. Proc. amer. Math. soc. 20, 295-302 (1969) · Zbl 0182.12602
[37] Pachpatte, B. G.: On some new integral and integrodifferential inequalities in two independent variables and their applications. J. differential equations 33, 249-272 (1979) · Zbl 0406.45016
[38] Pachpatte, B. G.: On some new integrodifferential inequalities of the Wendroff type. J. math. Anal. appl. 73, 491-500 (1980) · Zbl 0451.26011
[39] Pachpatte, B. G.: On comparison method for some fundamental partial integral inequalities. J. math. Phys. sci. 15, 341-357 (1981) · Zbl 0502.26007
[40] Pachpatte, B. G.: On some partial integral inequalities in n independent variables. J. math. Anal. appl. 79, 256-272 (1981) · Zbl 0455.26009
[41] Pelczar, A.: On invariant points of monotone transformations in partially ordered sets. Ann. polon. Math. 17, 49-53 (1965) · Zbl 0128.02601
[42] Pelczar, A.: Some functional differential equations. Dissertationes math. 100 (1973) · Zbl 0224.35065
[43] Rasmussen, D. L.: Gronwall’s inequality for functions of two independent variables. J. math. Anal. appl. 55, 407-417 (1976) · Zbl 0333.26006
[44] Seda, V.: Antitone operators and ordinary differential equations. Czech math. J. 31, No. 106, 531-553 (1981) · Zbl 0491.34022
[45] Shastri, R. P.; Kasture, D. Y.: Wendroff type inequalities. Proc. amer. Math. soc. 72, 248-250 (1978) · Zbl 0359.26005
[46] Shih, M. H.; Yeh, C. C.: Some integral inequalities in n independent variables. J. math. Anal. appl. 84, 569-583 (1981) · Zbl 0485.26010
[47] Singare, V. M.; Pachpatte, B. G.: Lower bounds on some integral inequalities in n independent variables. Indian J. Pure math. 12, 318-331 (1981) · Zbl 0485.26009
[48] Skripnik, V. P.: Systems with transformed argument; boundary value problems and Cauchy problems. Mat. sb. 62, No. 104, 385-396 (1963) · Zbl 0125.04403
[49] Smithson, R. E.: Fixed points of order preserving multifunctions. Proc. amer. Math. soc. 28, 304-310 (1971) · Zbl 0238.06003
[50] Snow, D. R.: Gronwall’s inequality for systems of partial differential equations in two independent variables. Proc. amer. Math. soc. 33, 46-54 (1972) · Zbl 0232.35005
[51] Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math. 5, 285-309 (1955) · Zbl 0064.26004
[52] Taskovic, M. R.: Partially ordered sets and some fixed point theorems. Publ. inst. Math. (N. S.) 27, No. 41, 241-247 (1980) · Zbl 0461.06005
[53] Tricomi, F. G.: Integral equations. (1957) · Zbl 0078.09404
[54] Turinici, M.: Nonlinear contractions and applications to Volterra functional equations. An. stiint. Univ. ”al. I. cuza” iaşi sect. Ia, mat. (N.S.) 23, 43-50 (1977)
[55] Turinici, M.: Abstract monotone mappings and applications to functional differential equations. Atti accad. Naz. lincei 66, 189-193 (1979) · Zbl 0452.47062
[56] Turinici, M.: Constant and variable drop theorems on metrizable locally convex spaces. Comment. math. Univ. carolin. 23, 383-398 (1982) · Zbl 0497.47030
[57] Turinici, M.: Mapping theorems via contractor directions in metrizable locally convex spaces. Bull. acad. Polon. sci. (Sér. Sci. math.) 30, 161-166 (1982) · Zbl 0497.47038
[58] Turinici, M.: Abstract Gronwall-Bellman inequalities on ordered metrizable uniform spaces. J. integral equations 6, 105-117 (1984) · Zbl 0555.47025
[59] Viswanatham, B.: A generalization of Bellman’s lemma. Proc. amer. Math. soc. 14, 15-18 (1963) · Zbl 0118.08001
[60] Vulikh, B. Z.: An introduction to the theory of partially ordered spaces. GoS izd. Fiz.-mat. Lit. (1961)
[61] Wallace, A. D.: A fixed point theorem. Bull. amer. Math. soc. 51, 413-416 (1945) · Zbl 0060.40104
[62] Walter, W.: Differential- und integral- ungleichungen. (1964) · Zbl 0119.12205
[63] Jr., L. E. Ward: Partially ordered topological spaces. Proc. amer. Math. soc. 5, 144-161 (1954) · Zbl 0055.16101
[64] Westphal, H.: Zur abschätzung der lösungen nichtlinearer parabolischer differentialgleichungen. Math. Z. 51, 690-695 (1949) · Zbl 0035.06501
[65] Yeh, C. C.: Bellman-bihari integral inequalities in several independent variables. J. math. Anal. appl. 87, 311-321 (1982) · Zbl 0514.26006
[66] Yeh, C. C.; Shih, M. H.: The Gronwall-Bellman inequality in several variables. J. math. Anal. appl. 86, 157-167 (1982) · Zbl 0507.26007
[67] Young, E. C.: Gronwall’s inequality in n independent variables. Proc. amer. Math. soc. 41, 241-244 (1973) · Zbl 0269.35061