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Transactions of the American Mathematical Society

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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 $ 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.

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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