Existence and stability of a class of nonlinear Volterra integral equations

Grossman, Stanley I.

doi:10.1090/S0002-9947-1970-0265886-8

Existence and stability of a class of nonlinear Volterra integral equations
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Trans. Amer. Math. Soc. 150 (1970), 541-556 Request permission

Abstract:

In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation $x = f + {a_1}{g_1}(x) + \cdots + {a_n}{g_n}(x)$, where the ${a_i}$ are continuous linear operators mapping a Fréchet space $\mathcal {F}$ into itself and the ${g_i}$ are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of $\mathcal {F}$ having a stronger topology. The equations are studied first when the ${g_i}$ are of the form ${g_i}(x) = x + {h_i}(x)$ where ${h_i}(x)$ is “small", and then when the ${g_i}$ are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.

References

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 175672, DOI 10.1016/0022-247X(65)90120-4
C. Corduneanu, Some perturbation problems in the theory of integral equations, Math. Systems Theory 1 (1967), 143–155. MR 213919, DOI 10.1007/BF01705524
R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl. 22 (1968), 319–340. MR 227707, DOI 10.1016/0022-247X(68)90176-5
R. K. Miller, On the linearization of Volterra integral equations, J. Math. Anal. Appl. 23 (1968), 198–208. MR 230070, DOI 10.1016/0022-247X(68)90127-3
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 240573, DOI 10.1016/0022-247X(69)90265-0
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
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, DOI 10.1090/coll/019
I. W. Sandberg, On the boundedness of solutions of nonlinear integral equations, Bell System Tech. J. 44 (1965), 439–453. MR 199663, DOI 10.1002/j.1538-7305.1965.tb01671.x
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923

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