A New Approach to Almost Periodicity

Bochner, S.

doi:10.1073/pnas.48.12.2039

Cited by 326 publications

(175 citation statements)

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“…1-3.) If f is a solution of [3] and 5' is an index then, as in (1), there is a lesser 5 for which g = Aj -f is a solution of [2], so that, for an even lesser 6, Aag = g. Thus, by induction on n, (Ab)nf = (A6)Y-'f + g, and hence, for all integers n, ng = (A6)nf -f. [4] Thus, for every semi-norm, IIgI < 1/n2 11f11 for all n, so that jjg = 0. Thusg = 0,orAbf=f.…”

Section: Strong Solutionsmentioning

confidence: 94%

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General almost automorphy

Bochner

1975

Proc. Natl. Acad. Sci. U.S.A.

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We will propose a very general setting for the following proposition. I a linear operator Lf is almost automorphic and all bounded solutions of the homogeneous equation Lg = 0 are almost automorphic then, for almost automorphic so, all bounded solutions of the inhomogeneous equation Lf = so are also almost automorphic.The almost automorphy previously introduced by us will be generalized and diversified into a property which we will name "A-invariance"; and we will deal with both strong solutions and weak solutions. We will also compare our A-invariance with minimality, and propose a rather thinned-out almost periodicity that will be associated with our A-invariance.We deal throughout with convergence by sequences and not by general nets, and for the sake of weak solutions, which will require a Lagrange adjoint operator L*X(t), we will favor a Euclidean setting. But strong solutions can be easily presented in an abstract setting, and we will begin with such a one. I. Strong solutionsWe stipulate the following data.(a) A linear vector space F over real or complex numbers with a "sufficient" number of pseudo-norms.(b) A partially ordered index set {a;

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Section: Strong Solutionsmentioning

confidence: 94%

“…If 5 < ' and A af is defined then so is A4 and the two are equal. For any pseudo-norm on F, we have IIAWfII < Ifll [1] and A (af + bg) = aA f + bA g whenever all terms are defined.…”

Section: Strong Solutionsmentioning

confidence: 99%

General almost automorphy

Bochner

1975

Proc. Natl. Acad. Sci. U.S.A.

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“…The concept of almost automorphy, which is an important generalization of the classical almost periodicity, was first introduced in the literature [1][2][3][4] by Bochner in relation to some aspects of differential geometry. Since then, this pioneer work has attracted more and more attention and has been substantially extended in several different directions.…”

Section: Introductionmentioning

confidence: 99%

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

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This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations D ( ) = ( ) + D −1 ( , ( ), ( )), ∈ R, where 1 < < 2, is a linear densely defined operator of sectorial type on a complex Banach space and is a bounded linear operator defined on , is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

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“…In 1962, Bochner [1] introduced the concept of almost automorphy, which is an important generalization of almost periodicity. The concept of almost periodic functions was introduced by Bohr [2].…”

Section: Introductionmentioning

confidence: 99%

“…v k is stated to be a mild solution to (1) indicate that the function 1 ! KT l ðtÀ1q CðlÞ uð1; tð1ÞÞ is integrable on ðÀ1; tÞ; 1 !…”

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confidence: 99%

Symmetric-periodic solutions for some types of generalized neutral equations

2016

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The existence of symmetric-periodic outcomes for a class of fractional differential equations has been increasingly studied. Such study has used various methods such as fixed point theory, critical point theory, and approximation theory. In this work, we study the m-pseudo almost automorphic (m-PKK) outcomes for a category of fractional neutral differential equations. To satisfy this aim, we introduce composition results under suitable conditions and employ them to establish some extant outcomes using interpolation theory mixed with fixed point technique. Examples are illustrated.

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A New Approach to Almost Periodicity

Cited by 326 publications

References 1 publication

General almost automorphy

General almost automorphy

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Symmetric-periodic solutions for some types of generalized neutral equations

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