# Asymptotic Behavior of Neutral Stochastic Partial Functional Integro--Differential Equations Driven by a Fractional Brownian Motion

Volume 7, Issue 6, pp 407--421
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### Authors

Tomás Caraballo - Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla. Apdo. de Correos, 1160, 41080-Sevilla, Spain. Mamadou Abdoul Diop - Département de Mathématiques, Université Gaston Berger de Saint-Louis, , UFR SAT, 234, Saint-Louis, Sénégal. Abdoul Aziz Ndiaye - Département de Mathématiques, Université Gaston Berger de Saint-Louis, UFR SAT 234, Saint-Louis, Sénégal.

### Abstract

This paper deals with the existence, uniqueness and asymptotic behavior of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion BH, with Hurst parameter $H \in ( \frac{1}{2} , 1)$. The main tools for the existence of solution is a fixed point theorem and the theory of resolvent operators developed in Grimmer [R. Grimmer, Trans. Amer. Math. Soc., 273 (1982), 333-349.], while a Gronwall-type lemma plays the key role for the asymptotic behavior. An example is provided to illustrate the results of this work.

### Keywords

• Resolvent operators
• $C_0$-semigroup
• Wiener process
• Mild solutions
• Fractional Brownian motion
• Exponential decay of solutions.

•  60H15
•  60G15
•  60J65

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