INTEGRAL EQUATIONS OF THE FIRST KIND OF SONINE TYPE

By S. Samko, R. Cardoso. 2003

111 citations recorded by the source corpus.

A Volterra integral equation of the first kind Kϕ(x) :≡ � x −∞ k(x −t)ϕ(t)dt = f( x) with a locally integrable kernel k(x) ∈ L loc (R 1) is called Sonine equation if there exists another locally integrable kernel �(x) such thatx 0 k(x − t)�(t)dt ≡ 1( lo- cally integrable divisors of the unit, with respect to the operation of convolu- tion). The formal inversion ϕ(x) = (d/dx) � x 0 �(x − t)f (t)dt is well known, but it does not work, for example, on solutions in the spaces X = Lp(R 1 ) and is not defined on the whole range K(X). We develop many properties of Sonine ker- nels which allow us—in a very general case—to construct the real inverse oper- ator, within the framework of the spaces Lp(R 1 ), in Marchaud form: K −1 f( x)= �( ∞)f (x)+ � ∞ 0 � � (t)(f (x −t)−f (x))dt with the interpretation of the convergence of this "hypersingular" integral in Lp-norm. The description of the range K(X) is given; it already requires the language of Orlicz spaces even in the case when X is

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/64c68b830c59b0ec6d4f57a60491449445a42c30

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Citations in the corpus

Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense (2022, background): “In [4], several examples of the Sonine kernels and their properties can be found.”
General Fractional Integrals and Derivatives with the Sonine Kernels (2021, background): “Of course, the class S of the Sonine kernels is not restricted to the functions in form (21) and includes, for instance, some functions with the power-logarithmic singularities at the origin ( [24, 25]).”
A comment on a controversial issue: A generalized fractional derivative cannot have a regular kernel (2020, background): “can be determined by a solving sequence of equations [23, 19].”
Operational Calculus for the General Fractional Derivative and Its Applications (2021, unknown): “As mentioned in [31] (see also [10]), any locally integrable Sonine kernel possesses a singularity at the point zero.”
Positivity, monotonicity, and convexity for convolution operators (2020, methodology): “For other examples, see [36, Section 6] and [33].”
General Fractional Calculus, Evolution Equations, and Renewal Processes (2011, background): “Such kernels were searched for from the 19th century; see [26] for a survey.”