Existence and stability of a class of nonlinear Volterra integral equations

Author:
Stanley I. Grossman

Journal:
Trans. Amer. Math. Soc. **150** (1970), 541-556

MSC:
Primary 45.30

MathSciNet review:
0265886

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Abstract: In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation , where the are continuous linear operators mapping a Fréchet space into itself and the are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of having a stronger topology. The equations are studied first when the are of the form where is ``small", and then when the are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.

**[1]**V. E. Beneš and I. W. Sandberg,*On the response of time-variable nonlinear systems to almost periodic signals*, J. Math. Anal. Appl.**10**(1965), 245–268. MR**0175672****[2]**C. Corduneanu,*Some perturbation problems in the theory of integral equations*, Math. Systems Theory**1**(1967), 143–155. MR**0213919****[3]**R. K. Miller,*On Volterra integral equations with nonnegative integrable resolvents*, J. Math. Anal. Appl.**22**(1968), 319–340. MR**0227707****[4]**R. K. Miller,*On the linearization of Volterra integral equations*, J. Math. Anal. Appl.**23**(1968), 198–208. MR**0230070****[5]**R. K. Miller, J. A. Nohel, and J. S. W. Wong,*Perturbations of Volterra integral equations*, J. Math. Anal. Appl.**25**(1969), 676–691. MR**0240573****[6]**John A. Nohel,*Remarks on nonlinear Volterra equations*, Proc. U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, Minn., 1967) Benjamin, New York, 1967, pp. 249–266. MR**0222586****[7]**Raymond E. A. C. Paley and Norbert Wiener,*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142****[8]**I. W. Sandberg,*On the boundedness of solutions of nonlinear integral equations*, Bell System Tech. J.**44**(1965), 439–453. MR**0199663****[9]**David Vernon Widder,*The Laplace Transform*, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR**0005923****[10]**G. Zames,*On the input-output stability of time-varying nonlinear feedback systems*, IEEE Trans. Automatic Control**AC-11**(1966), 228-238, 465-476.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0265886-8

Keywords:
Volterra integral equation,
nonlinear integral equation,
applications of contraction map,
integral equation resolvent,
convolution equations,
Fréchet space,
completely monic function,
Laplace-Stieltjes transform,
multiple resolvent,
slope restrictions,
nonlinear network

Article copyright:
© Copyright 1970
American Mathematical Society