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Significance and Relevances of Functional Equations in Various Fields
Published in Hemen Dutta, Topics in Contemporary Mathematical Analysis and Applications, 2020
B. V. Senthil Kumar, Hemen Dutta
An equation that indicates a function in hidden type is a functional equation. A functional equation communicates the value of a function at a particular point with its values at other points. Since a function is a solution of a functional equation, we can determine the properties of such function which satisfies the equation. It is not so easy to reduce a functional equation to an algebraic equation.
A contribution to best proximity point theory and an application to partial differential equation
Published in Optimization, 2023
Sakan Termkaew, Parin Chaipunya, Dhananjay Gopal, Poom Kumam
Banach fixed point theorem (generalized fixed point theorem) is the most widely used analytical tool in solving various non-linear problems, such as integral equation, differential equation, functional equation etc. Since the solution of such equations can be found as the fixed point of corresponding self operator equation. Thus, it is desirable to have appropriate theory when the corresponding operator become non-self operator. This exactly evolute the concept of best proximity point. Let be two non-empty subsets of a metric space . A best proximity point of a non-self mapping is a point satisfying the equality , where . In the case, when V = U, we get i.e. becomes fixed point of T. Thus, every best proximity point is a natural generalization of fixed point. Hence to this reason many authors have studied on this topic (see for instance [1–15]).
Heuristic solution to the problem of diffraction of a TE-polarized electromagnetic wave on a semitransparent half-plane
Published in Waves in Random and Complex Media, 2021
Michael V. Vesnik, Sergey E. Bankov
A rigorous solution to the problem of scattering by a semitransparent half-plane with boundary conditions (8) was obtained as follows. Due to boundary conditions (8), the field on the half-plane has a discontinuity in the tangential magnetic field, which is equivalent to the surface electric current flowing along the half-plane. The field excited by the indicated current is found using the Green’s function of free space, for which it is convenient to use the representation in the form of the Fourier integral. Then the found fields are summed up with the incident wave field and substituted into the boundary conditions (8). As a result, we obtain an integral equation with respect to the unknown current, given on a semi-infinite interval. The solution of such an equation by the Wiener–Hopf method was first considered in the work of T. Senior [31]. It describes a method for solving a functional equation, to which the original integral equation is reduced using the Fourier transform.
Disintegration property of coherent upper conditional previsions with respect to Hausdorff outer measures for unbounded random variables
Published in International Journal of General Systems, 2021
Let F and G be two σ-fields of subsets of Ω with contained in and let X be an integral non-negative random variable. Let P be a probability measure on ; defined a measure ν on by . This measure is finite and absolutely continuous with respect to P. Thus there exists a non-negative function, the Radon–Nikodym derivative denoted by , defined on Ω, i) G-measurable, ii) integrable and satisfying the functional equation: