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A General Class of Autoconvolution Equations of the Third Kind
Abstract
Continuing recent investigations by L. Berg and L. v. Wolfersdorf on a model integral equation of autoconvolution type of the third kind, two existence theo-rems for a general class of such equations are derived. Further, an existence theorem is proved for the model equation with data and solutions of a general logarithmic form. Moreover, a singular perturbation problem for a related integrodifferential equation of first order to the model equation is studied which could serve as a basis for its regularization by the Lavrentiev method.
... which has been studied recently by L. Berg, J. Janno and the second author in [2] and [14]. More precisely, we focus on the case that the function x is a probability density function of a non-negative absolutely continuous random variable, i.e., x(t) = 0 (−∞ < t < 0), x(t) ≥ 0 (0 ≤ t < ∞) and ...
... Remark 3. 2 We immediately emphasize that as a consequence of a theorem [12, p. 289] proven by Janno the functions z K ∈ C[0, T ] and hence the solutions x K of (1.2) in the context of Theorem 3.1 depend continuously in the norm of C[0, T ] on the data K and A, B occurring in (3.2) and (3.1). For more details we refer to [12] and [14]. ...
... There exist further and sharper existence theorems under additional assumptions on the function B in (3.1) for the case n = 1 (cf. [2], [14]). In the case n ∈ N, ...
We deal with a modification of the well-known ill-posed autoconvolution equation xx = y on a finite interval, e.g., analyzed in [88.
R. Gorenflo and B. Hofmann ( 1994 ). On autoconvolution and regularization . Inverse Problems 10 : 353 – 373 [CROSSREF] [CSA] [CrossRef], [Web of Science ®]View all references]. In this paper, we focus on solutions that are probability density functions and assume to have data of the autoconvolution coefficient k of the density function x, which we define as the quotient of the autoconvolution function xx and x itself. The corresponding inverse problem leads to the nonlinear integral equation kx − xx = 0 of the third kind. For this equation, we give results on existence and make notes on uniqueness and stability. We show the ill-posedness of the equation by an example and make assertions on its regularization by Tikhonov's method. In this context, we prove the weak closedness of the forward operator for some appropriate domain.
... In continuation of our paper [17] (which we cite as Part I of this paper in the following) we deal with two types of autoconvolution equations of the third kind whose free terms possess nonzero values at x = 0. The first type of equations has a coefficient k(x) of the unknown function with asymptotics k(x) ∼ Ax as x → 0. It comprises the well-known equation of Bernstein–Doetsch [4,9] as an important special case. The second type of equations has a coefficient k(x) with k(x) ∼ Ax 1/2 as x → 0. We derive existence theorems for a one-parametric family of solutions and an additional solitary solution for both types of equations. ...
... Eq. (2.1) in the case γ = 0 (with ω = 0 and r(x) = O(x) as x → 0) has been considered in [9]. We make the first ansatz in Eq. (2.1) ...
The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived.
Existence results of Part I of the paper are generalized to two types of autoconvolution equations of the third kind having free terms with nonzero values at x = 0 like the well-known Bernstein-Doetsch equation for the Jacobian theta zero functions. Also uniqueness results for the linear convolution equations in Part I of the paper are extended to more general function spaces. Further, a special class of integro-differential equations with autoconvolution integral and two classes of the linear singular Abel-Volterra equations are dealt with.
In this chapter, we outline the mathematical theory of direct regularization methods for in general nonlinear and ill-posed inverse problems. One focus is on Tikhonov regularization in Hilbert spaces with quadratic misfit and penalty terms. Moreover, recent results of an extension of the theory to Banach spaces are presented concerning the variational regularization with convex penalty term. Five examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.
In this paper a class of autoconvolution equations of the third kind with additional fractional integral is investigated. Two general existence theorems are proved, and a new type of solutions is shown for an exceptional equation of this class.
Existence and uniqueness theorems for a basic class of autoconvolution equations of the third kind with power-logarithmic coefficeint and free term are derived. Under suitable assumptions the existence of a solitary solution and of a one-parametric family of solutions is proved.
Two classes of first order integro-differential equations with autoconvolution integral are studied generalizing an equation of J. M. Burgers from the turbulence theory. General existence and stability theorems in a finite interval are proved and the asymptotic behavior of the solutions at infinity is discussed.
For confluent hypergeometric functions, a class of autoconvolution equations of the first kind is derived, which behave like equations of the third kind as x0. Moreover, a further class of autoconvolution equations of the first kind with Mittag-Leffler type functions as solutions is treated, which generalizes a related class of equations for the error functions in part I of the paper. Finally, two examples of autoconvolution equations of the second kind are listed, which possess Bessel functions as solutions.
This article attends to the specific inverse problem of determining the difference kernel k of a second kind linear Volterra integral equation of convolution type from a prescribed quotient μ of resolvent r and kernel k of the equation. A problem of this type arises, for example, in the theory of viscoelasticity and leads to a nonlinear integral equation of the third kind , for which several existence theorems are proved. Moreover, the applicability of Tikhonov's regularization method is studied. Finally, the case of non-convolution Volterra equations is also briefly discussed.
By means of the contraction principle we prove existence, uniqueness and stability of solutions for nonlinear equations u+G 0 [D,u]+L(G 1 [D,u],G 2 [D,u])=f in a Banach space E, where G 0 , G 1 , G 2 satisfy Lipschitz conditions in scales of norms, L is a bilinear operator and D is a data parameter. The theory is applicable for inverse problems of memory identification and generalized convolution equations of the second kind.
Existence, uniqueness, smoothness, and asymptotics of solutions to a class of generalized autoconvolution equations with outer and inner coefficient are investigated. Further, holomorphic and asymptotic solutions of the equations are derived, and a numerical procedure for an approximate solving of the equations is tested by an example.
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