• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.57-69
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• 2012

In this paper, we prove the existence of mild solutions for a first order impulsive neutral differential inclusion with state dependent delay. We assume that the state-dependent delay part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed point theorem for condensing multi-valued maps, a main existence theorem is established.

• Journal of the Korean Mathematical Society
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• v.56 no.1
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• pp.149-167
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• 2019

In this work, we study the existence, uniqueness and stability in the ${\alpha}$-norm of solutions for some stochastic partial functional integrodifferential equations. We suppose that the linear part has an analytic resolvent operator in the sense given in Grimmer [8] and the nonlinear part satisfies a $H{\ddot{o}}lder$ type condition with respect to the ${\alpha}$-norm associated to the linear part. Firstly, we study the existence of the mild solutions. Secondly, we study the exponential stability in pth moment (p > 2). Our results are illustrated by an example. This work extends many previous results on stochastic partial functional differential equations.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.915-927
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• 2000

This paper is concerned with degenerate Volterra equations Mu(t) + ∫(sub)0(sup)t k(t-s) Lu(s)ds = f(t) in Banach spaces both in the hyperbolic case, and the parabolic one. The key assumption is played by the representation of the underlying space X as a direct sum X = N(T) + R(T), where T is the bounded linear operator T = ML(sup)-1. Hyperbolicity means that the part T of T in R(T) is an abstract potential operator, i.e., -T(sup)-1 generates a C(sub)0-semigroup, and parabolicity means that -T(sup)-1 generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel k($.$) is degenerated or singular at t=0 using the results of Pruss[8] on analytic resolvents. Finally, we consider the case where $\lambda$ is a pole for ($\lambda$L + M)(sup)-1.