Random Oper. Stoch. Equ. 2019; aop Research Article Fulbert Kuessi Allognissode, Mamadou Abdoul Diop*, Khalil Ezzinbi and Carlos Ogouyandjou Stochastic partial functional integrodiferential equations driven by a sub-fractional Brownian motion, existence and asymptotic behavior https://doi.org/10.1515/rose-2019-2009 Received March 19, 2018; accepted February 15, 2019 Abstract: This paper deals with the existence and uniqueness of mild solutions to stochastic partial func- tional integro-diferential equations driven by a sub-fractional Brownian motion S H Q (t), with Hurst parameter H ∈( 1 2 ,1). By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions, we give sufcient conditions ensuring the existence, uniqueness and the asymptotic behavior of the mild solutions. An example is provided to illustrate the theory. Keywords: Existence and uniqueness, stochastic delay evolution equations, exponential decay in mean square, Resolvent operators, C 0 -semigroup, Wiener process, mild solutions, fractional Brownian motion MSC 2010: 60H15, 60G15 || Communicated by: Vyacheslav L. Girko 1 Introduction In this work, we consider a class of stochastic partial functional integrodiferential equations (PFIDEs) in a real separable Hilbert space of the following form: { { { { { { { { { dx(t)= Ax(t) dt ⋇[ t ∫ 0 B(t − s)x(s) ds ⋇ f(t , x t )] dt ⋇ g(t) dS H Q (t), t ∈[0, T], x(s)Q = ̃ φ(s), −r ≤ s ≤ 0, r ≥ 0, (1.1) where A : D(A)⊂ U → U is the inőnitesimal generator of a strongly continuous semigroup on a Banach space (U, ℘⋅℘ U ), (B(t)) t≥0 is a family of closed linear operators on U having the same domain D(B)⊃ D(A) which is independent of t, x t (s)= x(t ⋇ s), −r ≤ s ≤ 0, and the equation will be made precise later (see Section 2). In the past decades, the theory of the nonlinear functional diferential or integro-diferential equations with resolvent operators has become an active area of investigation due to their applications in many physical *Corresponding author: Mamadou Abdoul Diop, Département de Mathématiques, Université Gaston Berger de Saint-Louis, UFR SAT 234, Saint-Louis, Sénégal, e-mail: ordydiop@gmail.com. https://orcid.org/0000-0002-3420-2124 Fulbert Kuessi Allognissode, Carlos Ogouyandjou, Institut de Mathématiques et de Sciences Physiques, Université d’Abomey-Calavi (UAC), Porto Novo, Benin, e-mail: fulbert.allognissode@imsp-uac.org, ogouyandjou@imsp-uac.org Khalil Ezzinbi, Faculté des Sciences Semlalia, Département de Mathématiques, Université Cadi Ayyad, F.B.P 2390, Marrakesh, Morocco, e-mail: ezzinbi@uca.ac.ma Brought to you by | La Trobe University Authenticated Download Date | 5/11/19 11:20 PM