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We study the existence and uniqueness of C 1 + α -strict solutions for a general class of abstract differential equations with state dependent delay. We also study the local well-posedness of this type of problems on subspaces of C 1 + α ( [ − p , 0 ] ; X ) . Some examples involving partial differential equations are presented.

In this paper, we study the existence of solutions for a class of abstract fractional non-autonomous differential equations with applications to nonlinear partial differential equations with fractional temporal derivatives. Some examples are presented.

Let ( X , ‖ ⋅ ‖ ) be a Banach space and ω ∈ R . A bounded function u ∈ C ( [ 0 , ∞ ) ; X ) is called S -asymptotically ω -periodic if lim t → ∞ [ u ( t + ω ) − u ( t ) ] = 0 . In this paper, we establish conditions under which an S -asymptotically ω -periodic function is asymptotically ω -periodic and we discuss the existence of S -asymptotically ω -periodic and asymptotically ω -periodic solutions for an abstract integral equation. Some applications to partial differential equations and partial integro-differential equations are considered.

Made available in DSpace on 2015-04-27T11:56:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2011 Made available in DSpace on 2015-04-27T11:56:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2011 Made available in DSpace on 2015-04-27T11:56:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2011 By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered. Universidade Estadual Paulista Júlio de Mesquita Filho, Instituto de Biociências Letras e Ciências Exatas de São José do Rio Preto, Sao Jose do Rio Preto, Rua Cristovão Colombo, 2265, Jardim Nazareth, CEP 15054-000, SP, Brasil Universidade Estadual Paulista Júlio de Mesquita Filho, Departamento de Matemática, Instituto de Biociências Letras e Ciências Exatas de São José do Rio Preto

We study the existence of weighted -asymptotically -periodic mild solutions for a class of abstract fractional differential equations of the form , where is a linear sectorial operator of negative type.

This paper is devoted to the study of the class of continuous and bounded functions f : [ 0 , ∞ ) → X for which exists ω > 0 such that lim t → ∞ ( f ( t + ω ) − f ( t ) ) = 0 (in the sequel called S-asymptotically ω-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically ω-periodic functions. We also study the existence of S-asymptotically ω-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces.

We introduce a class of abstract differential equation with non instantaneous impulses which extend and generalize some recent models considered in the literature. We study the existence of mild and classical solution and present some applications involving partial differential equations with non-instantaneous impulses.

We characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a UMD space.

In [Haiyin Gao, Ke Wang, Fengying Wei, Xiaohua Ding, Massera-type theorem and asymptotically periodic Logistic equations, Nonlinear Analysis: Real World Applications 7 (2006) 1268–1283, Lemma 2.1] it is established that a scalar S -asymptotically ω -periodic function (that is, a continuous and bounded function f : [ 0 , ∞ ) → R such that lim t → ∞ ( f ( t + ω ) − f ( t ) ) = 0 ) is asymptotically ω -periodic. In this note we give two examples to show that this assertion is false.