Linear relations generated by an integral equation with Nevanlinna operator measure

We define families of maximal and minimal linear relations generated by an integral equation with Nevanlinna operator measure and prove their holomorphic property. We also prove that if a restriction of a maximal relation is continuously invertible, then the operator inverse to this restriction is integral. We apply the obtained results for proving the constancy of deficiency indices of some integral and differential equations.     

On linear relations generated by an integro-differential equation with Nevanlinna measure in the infinite-dimensional case

A system of integral equations that can be reduced to an integro-differential equation with Nevanlinna measure is considered. The families of maximal and minimal linear relations are defined and their holomorphy is established. It is proved that the operators inverse to continuously invertible restrictions of the maximal relations are integral.     

Solutionof an integral equation by consistency method

We consider the problem to invert an integral equation obtained from integral geometry. We apply a consistencymethodsuggestedbytheauthortosolvetheproblemandfindarepresentationforthegeneratingdensityofcentrally symmetrical convex bodies. Finally, we check the representation.     

Solvability of an implicit fractional integral equation via a measure of noncompactness argument

In this paper, we study the existence of solutions to an implicit functional equation involving a fractional integral with respect to a certain function, which generalizes the Riemann-Liouville fractional integral and the Hadamard fractional integral. We establish an existence result to such kind of equations using a generalized version of Darbo's theorem associated to a certain measure of noncompactness. Some examples are presented.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

Invertible changes of variables generated by Bäcklund transformations

Institute of Mathematics, Bashkir Scientific Center, Urals Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 3, pp. 368–375, December, 1990.     

On rank variation of block matrices generated by Nevanlinna matrix functions

Within the framework of the multiple Nevanlinna–Pick matrix interpolation and its related matrix moment problem, we study the rank of block moment matrices of various kinds, generalized block Pick matrices and generalized block Loewner matrices, as well as their Potapov modifications, generated by Nevanlinna matrix functions, and derive statements either on rank (or inertia) invariance in different senses or on rank variation of such types of block matrices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

A solution of an integral equation

In this paper an integral equation $$\int\limits_0^\infty {f(u)f\left( {\tfrac{x}{u}} \right)\tfrac{{du}}{u} = g(x)} $$ is studied.     

A problem with integral conditions for an elliptic-parabolic equation

Mathematical Notes - A problem with integral conditions for a differential equation of elliptic-parabolic type in a mixed domain consisting of a rectangle and a half-disk is posed. The unique...     

Spectral synthesis in the sobolev space generated by an integral metric

Let l ∈ ℕ, A ⊂ ℝⁿ . The main goal of this paper is to describe (in inner terms) the closure of the set {f ∈ W ₁ ^l : f=0 in a neighborhood of the set A} with respect to the norm of the space W ₁ ^l (ℝⁿ). Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 92–111.     

On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument

We study the existence of monotonic solutions of a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument. The quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases. We show that the quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument has a monotonic solution in the Banach space of all real functions defined and continuous on a bounded and closed interval. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.