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A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation


Author: David Lowell Lovelady
Journal: Trans. Amer. Math. Soc. 182 (1973), 71-83
MSC: Primary 45J05; Secondary 34K10, 47H10
DOI: https://doi.org/10.1090/S0002-9947-1973-0336263-9
MathSciNet review: 0336263
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Abstract: A fixed point theorem is obtained for an equation of the form $ u = T[p,f + G[u]]$. This theorem is then applied to a functionally perturbed ordinary differential equation of the form $ u'(t) = f(t) + A(t,u(t)) + G[u](t);u(0) = p$, and, as a consequence of this, Fredholm integrodifferential equations of the form $ z'(t) = f(t) + \phi (t,z(t)) + \int_0^\infty {\alpha (t,s)\omega (z(s))ds,z(0) = p} $. Applications are also made to a multivariable Volterra integral system

\begin{displaymath}\begin{array}{*{20}{c}} {{u_1}(s,t) = {g_1}(s,t) + \int_0^s {... ...),{u_2}(x,t),{u_3}(x,t)} \right]} \;dx,} \hfill \\ \end{array} \end{displaymath}

and, as a corollary to this, a differential equation of the form

\begin{displaymath}\begin{array}{*{20}{c}} \hfill {\frac{{{\partial ^2}}}{{\part... ...\ \hfill {u(s,0) = \phi (s),u(0,t) = \psi (t).} \\ \end{array} \end{displaymath}

These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as

\begin{displaymath}\begin{array}{*{20}{c}} {\frac{{{\partial ^2}}}{{\partial s\p... ...u(s,0,z) = \sigma (s,z),u(0,t,z) = \tau (t,z).} \\ \end{array} \end{displaymath}


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0336263-9
Keywords: Fixed point theorems, perturbed differential equations, multivariable Volterra integral equations, hyperbolic equations in function spaces, hyperbolic integrodifferential equations
Article copyright: © Copyright 1973 American Mathematical Society

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