Nonlinear volterra integrodifferential equations in a Banach space
Copyright policy )Nonlinear volterra integrodifferential equations in a Banach space
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(Ds), whereDs is the differentiation operator, withF bounded linear andK andDsK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.
- Southern Illinois University Carbondale United States
Cauchy problem, Banach space, Volterra equation, existence, nonlinear evolution equations, Abstract integral equations, integral equations in abstract spaces, Nonlinear differential equations in abstract spaces, Sobolev space, Integro-ordinary differential equations, continuous dependence, m-accretive, well-posed, Nonlinear accretive operators, dissipative operators, etc.
Cauchy problem, Banach space, Volterra equation, existence, nonlinear evolution equations, Abstract integral equations, integral equations in abstract spaces, Nonlinear differential equations in abstract spaces, Sobolev space, Integro-ordinary differential equations, continuous dependence, m-accretive, well-posed, Nonlinear accretive operators, dissipative operators, etc.
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