On the structure of the set of solutions of the Darboux problem for hyperbolic equations

Consider the Darboux problemwhere φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:(A1) f(.,.,z) is measurable for every z ∈ Rd;(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.

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Linear Isometries of Spaces of Absolutely Continuous Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-019-7 ◽

1982 ◽

Vol 34 (2) ◽

pp. 298-306 ◽

Cited By ~ 6

Author(s):

V. D. Pathak

Keyword(s):

Real Line ◽

Upper Bound ◽

Continuous Functions ◽

Finite Complex ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

End Points ◽

Arbitrary Compact Subset ◽

Image Position ◽

Complex Valued

Let X be an arbitrary compact subset of the real line R which has at least two points. For each finite complex valued function f on X we denote by V(f; X) (and call it the weak variation of f on X) the least upper bound of the numbers ∑i|f(bi) – f(ai)| where {[ai, bi]} is any sequence of non-overlapping intervals whose end points belong to X. A function f is said to be of bounded variation (BV) on X if V(f; X) < ∞. A function f is said to be absolutely continuous (AC) on X, if given any ∈ > 0 there exists an n > 0 such that for every sequence of non-overlapping intervals {[au bi]} whose end points belong to X, the inequalityimplies that([7], p. 221, 223).

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On Finite Part Integrals and Hadamard-Type Fractional Derivatives

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4037930 ◽

2018 ◽

Vol 13 (9) ◽

Cited By ~ 8

Author(s):

Li Ma ◽

Changpin Li

Keyword(s):

Fractional Derivative ◽

Singular Integral ◽

Fractional Derivatives ◽

Continuous Functions ◽

Finite Part ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Fundamental Properties ◽

Logarithmic Series ◽

The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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The spectral theorem for well-bounded operators

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700031827 ◽

1993 ◽

Vol 54 (3) ◽

pp. 334-351 ◽

Cited By ~ 1

Author(s):

Ian Doust ◽

Qiu Bozhou

Keyword(s):

Functional Calculus ◽

Continuous Functions ◽

Weak Compactness ◽

Compact Interval ◽

Spectral Theorem ◽

Type B ◽

Bounded Operators ◽

Absolutely Continuous Functions ◽

Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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