Nonlinear volterra integrodifferential equations in a Banach space |
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Authors: | Ronald Grimmer Marvin Zeman |
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Institution: | (1) Department of Mathematics, Southern Illinois University, 62901 Carbondale, IL, USA |
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Abstract: | We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right)
+ \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right),
whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D
s), whereD
s is the differentiation operator, withF bounded linear andK andD
sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy
problem. |
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Keywords: | |
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