The Perron integral and existence and uniqueness theorems for a first order nonlinear differential equation

Author:
Manoug N. Manougian

Journal:
Proc. Amer. Math. Soc. **25** (1970), 34-38

MSC:
Primary 34.04

DOI:
https://doi.org/10.1090/S0002-9939-1970-0255881-2

MathSciNet review:
0255881

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Perron integral is used to establish an existence and uniqueness theorem concerning the initial value problem , and , for on the interval . The existence and uniqueness of the solution is obtained by use of a generalized Lipschitz condition, and a Picard sequence which is equiabsolutely continuous on . Also, we prove a theorem on the uniqueness of solution by a generalization of Gronwall's inequality.

**[1]**Hans Bauer,*Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen*, Monatsh. Math. Phys.**26**(1915), no. 1, 153–198 (German). MR**1548647**, https://doi.org/10.1007/BF01999447**[2]**H. J. Ettlinger,*Linear derivative inequalities and differential equations*, National Math. Mag.**11**(1936), 1-5.**[3]**Louis Gordon and Sim Lasher,*An elementary proof of integration by parts for the Perron integral*, Proc. Amer. Math. Soc.**18**(1967), 394–398. MR**0210841**, https://doi.org/10.1090/S0002-9939-1967-0210841-2**[4]**T. H. Gronwall,*Note on the derivatives with respect to a parameter of the solutions of a system of differential equations*, Ann. of Math. (2)**20**(1919), no. 4, 292–296. MR**1502565**, https://doi.org/10.2307/1967124**[5]**E. Kamke,*Das Lebesgue-Stieltjes-Integral*, B. G. Teubner Verlagsgesellschaft, Leipzig, 1956 (German). MR**0081330****[6]**Manoug N. Manougian,*An existence and uniqueness theorem for a nonlinear differential equation*, J. Differential Equations**8**(1970), 367–373. MR**0265733**, https://doi.org/10.1016/0022-0396(70)90011-2**[7]**Manoug N. Manougian,*On the convergence of a sequence of Perron integrals*, Proc. Amer. Math. Soc.**23**(1969), 320–322. MR**0247009**, https://doi.org/10.1090/S0002-9939-1969-0247009-1**[8]**Edward James McShane,*Integration*, Princeton University Press, Princeton, N. J., 1944 1957. MR**0082536****[9]**R. A. Northcutt,*Perron type differential and integral inequalities*, Dissertation, The University of Texas, Austin, Texas, 1968.**[10]**Stanisław Saks,*Theory of the integral*, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964. MR**0167578****[11]**G. Vitali,*Sul integrazione per serie*, Rend. Circ. Mat. Palermo**23**(1907), 137-155.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
34.04

Retrieve articles in all journals with MSC: 34.04

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1970-0255881-2

Keywords:
Initial value problem,
Lebesgue integral,
Perron integral,
bounded variation,
Picard sequence,
locally absolutely continuous,
equicontinuous,
equiabsolutely continuous,
Cauchy-Euler meth[ill]d,
Gronwall inequality

Article copyright:
© Copyright 1970
American Mathematical Society