On the asymptotic behavior of the bounded solutions of some integral equations, II

Levin, J. J.; Shea, Daniel F.

doi:10.1016/0022-247x(72)90276-4

Cited by 47 publications

(9 citation statements)

References 32 publications

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“…Finally, the nature of bounded, uniformly continuous solutions of the homogeneous form of equation (1), i.e., with f(t) 0, was investigated by Beurling in [2], [1], and now constitutes one aspect of spectral analysis; see [9, ?8] for a brief discussion which includes references to appropriate literature. 567-568 of [9]. For completeness we include the proof.…”

Section: Proceedings Of the American Mathematical Societymentioning

confidence: 96%

Linear Convolution Integral Equations with Asymptotically Almost Periodic Solutions

Jordan¹,

Madych²,

Wheeler³

1980

Proceedings of the American Mathematical Society

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Section: Proceedings Of the American Mathematical Societymentioning

confidence: 96%

Linear Convolution Integral Equations with Asymptotically Almost Periodic Solutions

Jordan¹,

Madych²,

Wheeler³

1980

Proceedings of the American Mathematical Society

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“…where K(t), X(t), F (t), and K * X(t) have the same meaning as in (2). In Section 2 we deduce the characteristic equation for (2) and give a concrete form of the sought asymptotic expansion for the solution to (2).…”

mentioning

confidence: 98%

“…In this article we propose a new approach to investigating the asymptotic behavior as t → ∞ of the solution X(t) to (2). This approach is based on the Banach space methods of asymptotic analysis and the properties of the Laplace transforms of functions with a given asymptotic behavior at infinity.…”

mentioning

confidence: 99%

“…In Section 2 we deduce the characteristic equation for (2) and give a concrete form of the sought asymptotic expansion for the solution to (2). In Section 3 we give some necessary facts from the theory of Banach algebras of functions with the same asymptotic behavior at infinity.…”

mentioning

confidence: 99%

“…. , m 1 , where X im (t) are unknown functions, A ij ∈ C, while K ij (t) and F im (t) are measurable complex-valued functions such that the integrals ∞ 0 e γt |K ij (t)| dt, ∞ 0 e γt |F im (t)| dt are finite for some γ ∈ R. The systems of type (1) are the traditional objects of mathematical research (for instance, see [1][2][3][4][5][6][7] and the references therein). Studying asymptotics for the solutions plays an important role.…”

mentioning

confidence: 99%

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An asymptotic expansion of the solution to a system of integrodifferential equations with exact asymptotics for the remainder

Сгибнев

2008

Sib Math J

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We obtain an asymptotic expansion of the solution to a system of first order integrodifferential equations taking into account the influence of the roots of the characteristic equation. We establish exact asymptotics for the remainder in dependence on the asymptotic properties of original functions.Keywords: system of integrodifferential equations, asymptotic expansion, characteristic equation, exact asymptotics for the remainder § 1. IntroductionGiven locally integrable functions f (t) and g(t) of t ≥ 0, denote their convolution by f * g(t):This article deals with the system of complex integrodifferential equationswith i = 1, . . . , n and m = 1, . . . , m 1 , where X im (t) are unknown functions, A ij ∈ C, while K ij (t) and F im (t) are measurable complex-valued functions such that the integralsare finite for some γ ∈ R. The systems of type (1) are the traditional objects of mathematical research (for instance, see [1-7] and the references therein). Studying asymptotics for the solutions plays an important role. Write (1) in matrix formwhere A := (A ij ) and K(t) := (K ij (t)) are n × n matrices, while X(t) := (X im (t)), F (t) := (F im (t)), and K * X(t) are n × m 1 matrices; a generic entry of K * X(t) is the sum n j=1 K ij * X jm (t). Let us agree that the operations of passing to the limit, taking derivatives, integrating, and others are performed on matrices entrywise. For instance, X (t) = (X im (t)); a similar agreement applies to the inequalities and set-theoretic operations on matrices. Denote by I the identity n × n matrix; and by 0, the zero matrix of arbitrary size.In this article we propose a new approach to investigating the asymptotic behavior as t → ∞ of the solution X(t) to (2). This approach is based on the Banach space methods of asymptotic analysis and the properties of the Laplace transforms of functions with a given asymptotic behavior at infinity.Novosibirsk.

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Volterra integral equations

Tsalyuk¹

1979

J Math Sci

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One presents a survey of the investigations in the theory of Volterra integral equations, reviewed in Ref. Zh. ~Mat." between 1966 In the last 15-20 years one observes a sharp increase of the interest in the theory of Volterra integral equations.The interest in this theory has been stimulated by the steady extension of the volume of applications as well as by the realization of the fact that the Volterra equations are not only simple special cases of the Fredholm equations but represent a class of equations with their own specific problems.The present survey comprises the investigations referred in the Referativnyi Zhurnal "Matematika" between 1966-1976. Naturally, it has not been possible to proceedwithout reference to earlier investigations; however, the number of these references has been reduced to minimum.The length restriction of this survey has produced rigid demands on the selection of the material. The papers in which one does not consider especially Volterra integral equations, among them papers on Volterra integrodifferential equations, have been left outside the frames of the survey, although in many cases one can extract from these papers useful information regarding integral equations. Similarly, wedo not consider numerous papers having an apptied character. However, even under these conditions the subject is extremely extensive and, obviously, this has caused a fragmentariness in the exposition of a series of sections. As a tradition, we mention that the degree of the minuteness of the exposition of the various sections carries inevitably a subjectivism and that the inclusion or the noninclusion of a result should not be considered as an attempt at appraisal. 1. GENERAL THEORY Linear EquationsWe consider the equationwhere f : [a, b) ~ R n, while Q is a mapping of the square in, b) 2 into the set of n x n-matrices. everywhere that Q(t, s) = 0 for t < s.1.1. Definition. We say that the kernel Q satisfies the Radon condition if Q is measurable in in, b) 2, for every~fixed value of t the function s ~-* Q(t, s) is summable on in, t], and for any c E (a, b) and t o ~ in, c) we have We shall assume

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On the asymptotic behavior of the bounded solutions of some integral equations, II

Cited by 47 publications

References 32 publications

Linear Convolution Integral Equations with Asymptotically Almost Periodic Solutions

Linear Convolution Integral Equations with Asymptotically Almost Periodic Solutions

An asymptotic expansion of the solution to a system of integrodifferential equations with exact asymptotics for the remainder

Volterra integral equations

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