Abstract
In this paper, we investigate the existence and uniqueness of S-asymptotically ω-periodic solutions to fractional differential equations of order qin(0, 1) with finite delay in a Banach space X. Existence and uniqueness theorems, which are new even in the case of X=mathbf{R}^{n} or A=0, are established. As examples of applications of our existence and uniqueness results, we obtain the S-asymptotically ω-periodic solutions for the fractional-order autonomous neural networks with delay.
Highlights
As one important branch of the research on evolution equations, the study of fractional differential equations in Banach spaces is very active recently due to its strong background in physics, chemistry, engineering, biology, financial sciences, etc
Where δ >, q ∈ (, ) and the fractional derivative is understood here in the Caputo sense, A : D(A) ⊂ X → X is the generator of an analytic semigroup on a Banach space X, f is a given function, ut : [–δ, ] → X is defined by ut(θ ) = u(t + θ ) for θ ∈ [–δ, ], and φ ∈ C([–δ, ], X)
While the almost periodic, almost automorphic, and weighted pseudo almost periodic solutions to various evolution equations are investigated by many scholars, the S-asymptotically ω-periodic solutions to some evolution equations are studied by some researchers
Summary
As one important branch of the research on evolution equations, the study of fractional differential equations in Banach spaces is very active recently due to its strong background in physics, chemistry, engineering, biology, financial sciences, etc. (cf., e.g., [ – ] and the references therein). We note that some papers about the existence of S-asymptotically ωperiodic solutions of fractional differential equations focus on the order q ∈ ( , ) ([ , ] and references therein). Motivated by all this work, we pay attention in this paper to the study of the existence of S-asymptotically ω-periodic (mild) solutions for differential equation of fractional order of type ). In Section , we apply our result to a study of the existence and uniqueness of S-asymptotically ω-periodic solution for the fractional-order neural network with finite delay. ). The following definition of S-asymptotically ω-periodic functions taking values in a Banach space X is from [ ]. Let SAPω(X) represent the space of all the X-valued S-asymptotically ω-periodic functions endowed with the uniform convergence norm denoted by · ∞.
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