Pseudo Asymptotic Behavior of Mild Solution for Nonautonomous Integrodifferential Equations with Nondense Domain

Xia, Zhinan

doi:10.1155/2014/419103

Cited by 3 publications

(2 citation statements)

References 26 publications

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“…We would like to note that the statements of [55,Lemma 17,Theorem 18], concerning the existence and uniqueness of almost automorphic solutions of the problem (3.9), can be straightforwardly reformulated for almost periodicity by replacing the assumptions (H4) and (H5) with the corresponding almost periodicity assumptions as well as by assuming that the function Γ(t, s) from the condition (H3) of this paper is (R, B)-almost periodic with R being the collection of all sequuences in A := {(a, a) : a ∈ R} and X ∈ B. Similarly, the statements of [55, Lemma 20,Theorem 21], concerning the existence and uniqueness of almost automorphic solutions of the problem (3.10), can be straightforwardly reformulated for almost periodicity; see also [116,Theorem 26,Theorem 27], where the same comment can be given and the recent result of J. Cao, Z. Huang and G. M. N'Guérékata [23,Theorem 3.6], where a similar modification of condition (H3) for bi-almost periodicity on bounded subsets can be made.…”

Section: Examples and Applications To The Abstract Volterra Integro-d...mentioning

confidence: 94%

Multi-dimensional almost periodic type functions and applications

Chávez¹,

Khalil²,

Kostić³

et al. 2020

Preprint

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In this paper, we analyze multi-dimensional (R X , B)-almost periodic type functions and multi-dimensional Bohr B-almost periodic type functions. The main structural characterizations and composition principles for the introduced classes of almost periodic functions are established. Several applications of our abstract theoretical results to the abstract Volterra integrodifferential equations in Banach spaces are provided, as well. Examples and applications to the abstractVolterra integro-differential equations 3.1. Application to nonautonomous retarded functional evolution equations 4. Appendix 4.1. n-Parameter strongly continuous semigroups 4.2. Multivariate trigonometric polynomials and approximations of periodic functions of several real variables References 2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. (R, B)-Multi-almost periodic type functions, (R X , B)-multi-almost periodic type functions, Bohr B-almost periodic type functions, composition principles, abstract Volterra integro-differential equations. Marko Kostić is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia. Manuel Pinto is partially supported by Fondecyt 1170466.The Euler Gamma function is denoted by Γ(•). If t 0 ∈ R n and ǫ > 0, then we set B(t 0 , ǫ)Now we are ready to briefly explain the organization and main ideas of this paper. In Subsection 1.1, we recall the basic facts and definitions about vectorvalued almost periodic functions of several real variables; in Subsection 1.2, we recall some applications of vector-valued almost periodic functions of several real variables made so far. Definition 2.1 and Definition 2.2 introduce the notion of (R, B)-multi-almost periodicity and the notion of (R X , B)-multi-almost periodicity for a continuous function F : I × X → Y, respectively. The convolution invariance of space consisting of all (R X , B)-multi-almost periodic functions is stated in Proposition 2.5, while the supremum formula for the class of (R, B)-multi-almost periodic functions is stated in Proposition 2.6.The notion of Bohr B-almost periodicity and the notion of B-uniform recurrence for a continuous function F : I × X → Y are introduced in Definition 2.9, provided that the region I satisfies the semigroup property I + I ⊆ I. Numerous illustrative examples of Bohr B-almost periodic functions and B-uniformly recurrent functions are presented in Example 2.12 and Example 2.13. In Definition 2.14, we introduce the notion of Bohr (B, I ′ )-almost periodicity and (B, I ′ )-uniform recurrence, provided that ∅ = I ′ ⊆ I ⊆ R n , F : I × X → Y is a continuous function and I + I ′ ⊆ I. After that, we provide several examples of Bohr (B, I ′ )-almost periodic functions and (B, I ′ )-uniformly recurrent functions in Example 2.15. The relative compactness of range F (I × B) for a Bohr B-almost periodic function F : I × X → Y,where B ∈ B, is analyzed in Proposition 2.16. The Bochner criterion for Bohr B-almost periodic functions is sta...

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Section: Examples and Applications To The Abstract Volterra Integro-d...mentioning

confidence: 94%

Multi-dimensional almost periodic type functions and applications

Chávez¹,

Khalil²,

Kostić³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…to mention a recent research study [11] by A. Chávez, M. Pinto and U. Zavaleta, where the authors systematically analyzed the notion of bi-almost automorphy in the study of abstract nonlinear integral equations that are simultaneously of advanced and delayed type, as well as the research studies [14] by Y.-K. Chang, S. Zheng, [27] by Z. Hu, Z. Jin, [44] by Z. Xia and [45] by Z. Xia, D. Wang.…”

Section: Introductionmentioning

confidence: 99%

$({\mathrm R},{\mathcal B})$-Multidimensional-almost automorphic functions with applications to integral and partial differential equations

Chávez,

Khalil,

Kostić

et al. 2021

Preprint

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Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

2021

Advances in Mathematical Physics

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Through the use of the measure theory, evolution family, “Acquistapace–Terreni” condition, and Hölder inequality, the core objective of this work is to seek to analyze whether there is unique μ -pseudo almost periodic solution to a functional evolution equation with Stepanov forcing terms in a Banach space. Certain adequate conditions are derived guaranteeing there is unique μ -pseudo almost periodic solution to the equation by Lipschitz condition and contraction mapping principle. Finally, an example is used to demonstrate our theoretical findings.

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Pseudo Asymptotic Behavior of Mild Solution for Nonautonomous Integrodifferential Equations with Nondense Domain

Cited by 3 publications

References 26 publications

Multi-dimensional almost periodic type functions and applications

Multi-dimensional almost periodic type functions and applications

$({\mathrm R},{\mathcal B})$-Multidimensional-almost automorphic functions with applications to integral and partial differential equations

Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

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