Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

A Countable System of Fractional Inclusions with Periodic, Almost, and Antiperiodic Boundary Conditions

This article is dedicated to the existence results of solutions for boundary value problems of inclusion type. We suggest the infinite countable system to fractional differential inclusions written by D α ABC ν i t ∈ Y i t , ν i t i = 1 ∞ . The mappings y i t , ν i t i = 1 ∞ are proposed to be Lipschitz multivalued mappings. The results are explored according to boundary condition σ ν i 0 = γ ν i ρ , σ , γ ∈ ℝ . This type of condition is the generalization of periodic, almost, and antiperiodic types.

A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function(FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar .

On solvability of one class of third order differential equations

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

Some properties of shift operators on algebras generated by $*$-polynomials

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

FROM LINGUISTIC REPRESENTATION TO FUZZY MATHEMATICS IN GROWN UP PEOPLE

The aim of this note is to give some critical examples where even the use of the same clustering rules lead to fuzziness. It starts from poor numerical systems and compares them with the expanded Sergeyev model, where the grossone is used, as an infinite terminal element. It can be compared with terminal elements of the ancient languages, such as the Greek myriad and the Chinese wan. On them some propositions that hold in the arithmetic of the grossone are similar, while they are not meaningful for the countable system of infinity. The note shows that both the upward and downward trend are actually present in human language and in conceptual arrangements. The note then goes on to sketch the model of evolution of Bak-Sneppen, showing two significant applications: the case of the evolution and study of foreign languages and, according to the model of Lloyd, the territorial analysis. In both cases it is highlighted how the Bak-Sneppen model becomes more stable when the universe is segmented, as already proven by the authors in previous works. The third part examines some cases of false probabilistic intuition due to incomplete perception of the phenomena, what could therefore be defined as hidden conditional probability. Interesting is the classic application of the theory of games to lotteries and ternary games, such as Chinese morra.

EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

We present a construction that enables one to find Banach spaces$X$whose sets$\operatorname{NA}(X)$of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,$X$does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2,Israel J. Math.(to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces,J. Funct. Anal. 272(2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space$X$where the set$\operatorname{NA}(X)$for the original norm is not “too large”. The construction can be applied to every Banach space containing$c_{0}$and having a countable system of norming functionals, in particular, to separable Banach spaces containing$c_{0}$. We also provide some geometric properties of the norms we have constructed.

On the well-posedness of boundary value problems for nonclassical differential equations of higher order

This paper investigates a high even-order nonclassical differential equation with a spectral parameter. We proved that this equation has a countable system of nontrivial solutions if spectral parameter is negative. We consider two cases, one where the spectral parameter is equal to eigenvalues and one where the spectral parameter is not equal to eigenvalues. In both cases, we proved the existence of regular solutions of boundary value problems for this equation. To do this, we combined the Fourier method and the method of a priori estimates. Moreover, we found some conditions for unsolvability of boundary value problems. In addition, for adjoint problems, we proved that there is no complex eigenvalues.