Functional Analysis in Interdisciplinary Applications

TheAbiev, N. aim of this paper is to discuss some results of [2, 3] relating to the study of the evolution of invariant Riemannian metricsRiemannian metriconGeneralized wallach space generalized Wallach spacesWallach space with $$a_1=a_2=a_3=a$$a1=a2=a3=a, where $$a\in (0,1/2)$$a∈(0,1/2). We proved that for the Wallach spacesWallach space$$SU(3)/T_{\max }$$SU(3)/Tmax, $$Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$$Sp(3)/Sp(1)×Sp(1)×Sp(1), and $$F_4/Spin(8)$$F4/Spin(8), the normalized Ricci flowNormalized Ricci flow evolves all generic invariant Riemannian metricsRiemannian metric with positive sectional curvatureSectional curvature into metrics with mixed sectional curvatureSectional curvature. Moreover, we obtained general results concerning the evolution of invariant Riemannian metricsRiemannian metriconGeneralized wallach space generalized Wallach spacesWallach space with $$a\in (0,1/2)\setminus \{1/4\}$$a∈(0,1/2)\{1/4} under the normalized Ricci flowNormalized Ricci flow. The very special case $$a=1/4$$a=1/4 is also considered.

In thisBalgimbayeva, S.A paper we obtain estimates sharp in order hyperbolic crossHyperbolic cross approximation w.r.t. $$\texttt {d}-$$d-multiple wavelet system with compact supportsWavelet system with compact supports$$\psi ^{(\texttt {d})}$$ψ(d) of the Nikol’skii – Besov and Lizorkin – Triebel type classes associated with this system in the space $$L_q([0,1]^d)$$Lq([0,1]d) for a number of relations between the parameters of the classes and the space.

In this paper we obtain estimates (sharp in order) for hyperbolic crossHyperbolic crossapproximationApproximation w.r.t. multiple ($$\texttt {n}-$$n-fold) Haar system $$\chi ^{(\texttt {n})}$$χ(n)ofThe Nikol’skii – Besov and the Lizorkin – Triebel spaces associated with the multiple Haar system the Nikol’skii – Besov and Lizorkin – Triebel type classes associated with this Haar system in the space $$L_r([0,1]^n)$$Lr([0,1]n) for a number of relations between the parameters of the classes and the space.

ThisBokayev, N.paperMatin, D. is dedicated to a sufficient condition for compactnessCompactness of CommutatorsCommutators for singular integralsSingular integrals$$\left[ b,T\right] $$b,T in theGeneralized Morrey space generalized Morrey spaceMorrey space$$M_{p}^{w}$$Mpw.

TheKussainova, L. aim of the paper is to obtain descriptions of multipliers actingMyrzagaliyeva, A. from weighted Sobolev spaces $$W_{p,\rho }^{l} $$Wp,ρl to $$L_{q,\omega }.$$Lq,ω. The space $$W_{p,\rho }^{l}$$Wp,ρl is defined as the completion of the set $$C_0^{\infty }$$C0∞ in the following finite norm $$\Vert u;\, W_{p,\rho }^{l}\Vert = \Vert \rho |\nabla _{l} u|\Vert _{p} + \Vert u\Vert _{p},$$‖u;Wp,ρl‖=‖ρ|∇lu|‖p+‖u‖p, where $$\rho $$ρ is a weight on $$R^{n};$$Rn;$$L_{q,\omega }$$Lq,ω denotes the Lebesgue space.

This paperNursultanov, E.isTleukhanova, N. devoted to the studySarybekova, L. of Fourier series and Fourier transform multipliersFourier transform multipliers and contains introduction, which put some new results into a general frame. In the following Sections several further examples and results are presented and discussed. In Section 2 we present some important results (including the most early papers we know) concerning Fourier series multipliers of particular interest for the investigations. The corresponding result for Fourier transform multipliers can be found in Section 3. In Section 4 we give some applications and in Section 5 we describe shortly the main results: A generalization and sharpening of the Lizorkin theoremLizorkin theorem concerning Fourier transform multipliersFourier transform multipliers between $$L_p$$Lp and $$L_q$$Lq. The Fourier series multipliers in the case with a regular system, which is rather general. A generalization and sharpening of the Lizorkin type theorem concerning Fourier series multipliers between $$L_p$$Lp and $$L_q$$Lq in this general case. A generalization of the Hörmander multiplier theorem for two dimensional Fourier series to the case with a general regular systemGeneral regular system.

Tuǧ and Başar [3] have recently studied the concept of four dimensional generalized difference matrix B(r, s, t, u) and its matrix domainMatrix domain in some double sequence spaces. In this present paper, as a natural continuation of [3], we introduce new almost null and almost convergent double sequence spacesAlmost convergent double sequence space$$B(C_f)$$B(Cf) and $$B(C_{f_0})$$B(Cf0) as the domain of four-dimensional generalized difference matrixFour-dimensional generalized difference matrixB(r, s, t, u) in the spaces $$C_f$$Cf and $$C_{f_0}$$Cf0, respectively. Firstly, we prove that the spaces $$B(C_f)$$B(Cf) and $$B(C_{f_0})$$B(Cf0) of double sequences are Banach spaces under some certain conditions. We give some inclusion relations with some topological properties. Moreover, we determine the $$\alpha -$$α-dual $$\alpha -$$ α-dual, $$\beta (bp)-$$β(bp)-dual and $$\gamma -$$γ-dual $$\gamma -$$ γ-dual of the spaces $$B(C_f)$$B(Cf). Finally, we characterize the classes of four dimensional matrix mappings defined on the spaces $$B(C_f)$$B(Cf) of double sequences.

ThisTleukhanova, N. paper is devoted to the study of Fourier series multipliersMultiplier of Fourier series. An analog of the Marcinkiewicz theorem on multipliers of Fourier seriesMultiplier of Fourier series in weighted Lebesgue spacesWeighted Lebesgue spaces is obtained.

A methodAisagaliev, S. for the study of periodic solutionsPeriodic solution of autonomous dynamic systemsDynamic system described by ordinary differential equationsOrdinary differential equation with phase and integral constraints is supposed. General problem ofPeriodic solution periodic solutionZhunussova Zh. is formulated in the form of the boundary value problemBoundary value problem with constraints. The boundary problem is reduced to the controllability problemControllability problem of dynamic systemsDynamic system with phase and integral constraints by introducing a fictitious control. Solution of the controllability problemControllability problem is reduced to a Fredholm integral equation of the first kind. The necessary and sufficient conditions for existence of the periodic solutionPeriodic solution are obtained and an algorithm for constructing periodic solutionPeriodic solution to the limit points of minimizing sequences is developed. Scientific novelty of the results consists in a completely new approach to the study of periodic solutionsPeriodic solution for linear systems focused on the use of modern information technologies is offered. The existence of periodic solutionPeriodic solution and its construction are solved together.

In this paper some new existence and uniqueness results are proved and maximal regularity estimates of solutions of third-order differential equationDifferential equation with unbounded coefficientsAkhmetkaliyeva, R. are given.

A multipoint-integralAssanova, A.T boundary value problemMultipoint-integral boundary value problem for a third order differential equationThird order differential equation with variable coefficients isImanchiev, A.E considered. The questions of the existence of a unique solution of the considered problem and ways of its construction are investigated. The multipoint-integral boundary value problem for the differential equation of third orderThird order differential equation with variable coefficients is reduced to a multipoint-integral boundary value problem for a system of three differential equations by introducing new functions. To solve the resulting multipoint-integral boundary value problem, a parametrization methodParametrization method is applied. AlgorithmsAlgorithm of finding the approximate solutionApproximate solution to the multipoint-integral boundary value problem for the system of three differential equations are constructed and their convergence is proved. The conditions of the unique solvabilityUnique solvability of the multipoint-integral boundary value problem for the system of three differential equations are established in the terms of initial data. The results are also formulated relative to the original of the multipoint-integral boundary value problem for the differential equation of third orderThird order differential equation with variable coefficients. The obtained results are applied to a two-point boundary value problem for the third order ordinary differential equation.

In thisJenaliyev, M.paperKosmakova, M. we consider the questions of solvabilitySolvability of the nonhomogeneous boundary value problemBoundary value problem for the Burgers equationBurgers equation in infinite angular domainAngular domain. It is reduced to the study of the solvability of a systemRamazanov, M. consisting of two homogeneous integral equationsIntegral equation. We prove some lemmas which establish properties of integral operators in weighted space of essentially bounded functionsWeighted space of essentially bounded functions and prove the existence and properties of non-trivial solutionsNon-trivial solution to the system of homogeneous integral equations. On the basis of Lemmas the solvability theorems of the nonhomogeneous boundary value problem for the Burgers equation in infinite angular domain are established.

In this paperKabdrakhova, Symbat S. we consider a semi-periodical boundary value problemSemiperiodical boundary value problem for a system of nonlinear hyperbolic equationsSystems of nonlinear hyperbolic equation in a rectangular domain. An algorithm for finding of approximate solutionApproximate solution to the semi-periodical boundary value problem for the systems of nonlinear hyperbolic equation is offered. Conditions for the convergence of the approximate solutions to the exact solution of the semi-periodical boundary value problem for the system of nonlinear hyperbolic equations are established.

In this paperOrumbayeva, N. we consider a nonlinear semi-periodic boundary-value problemSemi-periodic boundary value problem for a partial differential equation. By meansSabitbekova, G. of a replacement, the nonlinear problem is reduced to a linear semi-periodic boundary-value problem for hyperbolic equations with a mixed derivative. To solve the obtained problem, partitioning by the first variable is made. Further, in the obtained domains, the parametrization methodParametrization method proposed in the works of D.S. Dzhumabaev for solving a two-point boundary value problem for an ordinary differential equation is applied. A new algorithmAlgorithm for finding the solution to the given problem is proposed. Sufficient conditions for the unique solvabilitySolvability conditions of a semi-periodic boundary-value problem with arbitrary functions for a nonlinear partial differential equation are established.

An analogue ofPolunin, V. the Schwarz problemSchwarz problemforSoldatov, A. the Moisil–Teodorescu systemMoisil–Teodorescu system is considered in a domain D. It is shown that this problem has Fredholm propertyFredholm property in the Hoelder classHoelder class$$C^\mu (\overline{D})$$Cμ(D¯). If the domain D is homeomorphic to a ball, then the problem is investigated in detail. In particular its index is equal to $$-1$$-1 in this case.

In this paper a nonlocal problemNonlocal problem for the Poisson equation in a rectangular domain is considered. It is shown that this problem is ill-posedIll-posed problem as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with deviating argumentEquation with deviating argument establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with the deviating argument.

In the paper certain method for constructing exact solutionsConstructing exact solutions of a class of linear differential equations of fractional orderDifferential equations of fractional order is considered. AlgorithmsTurmetov, B. for constructing solutions of the explicit form are developed forHomogeneous differential equations homogeneous and inhomogeneous differential equationsInhomogeneous differential equations of fractional order. This method is based on construction of normalized systemsNormalized systems associated with fractional differentiation operatorOperator method. 0 - normalized and f - normalized systems are built concerning to the pair of operators connected with the considered equation. Using 0 - normalized systems, linearly independent solutions of the homogeneous equation are constructed. Similarly, with the help of f - normalized systems partial solutions of the inhomogeneous equation are built in the case, where the right side is a quasi-polynomial, analytic function and an arbitrary function from the class of continuous functions.

The common form for degenerate boundaryAkhtyamov, A.M.conditionsDegenerate boundary conditions for the operator $$D^4$$D4 ($$D^n$$Dn) is found. It is shown that the matrixOperator for coefficients of degenerate boundary conditions has a two diagonal form and the elements for one of the diagonal are units. Operator $$D^4$$D4 whose spectrumSpectrum fills the entire complex plane are studied, too. Earlier, examples of eigenvalue problemsEigenvalue problems for the differential operator of even orderDifferential operator of even order with common boundary conditions (not containing a spectral parameter) whose spectrum fills the entire complex plane were given. However, in connection with this, another question arises whether there are other examples of such operators. In this paper we show that such examples exist. Moreover, all eigenvalue boundary problems for the operator $$D^4$$D4 whose spectrum fills the entire complex plane are described. It is proved that the characteristic determinantCharacteristic determinant is identically equal to zero if and only if the matrix of coefficients of boundary conditions has a two diagonal form. The elements of this matrix for one of the diagonal are units, and the elements of the other diagonal are 1, $$-1$$-1 and an arbitrary constant.

In the present studyAshyralyev, A., the problemSarsenbi, A.M. of a hyperbolic equationHyperbolic equation with the involutionInvolution is investigated. The stability estimatesStability estimates in maximum norm in t for the solution of this problem are established.

In this paperAbdrasheva, G.K. we studyBiyarov, B.N. the spectral propertiesSpectral properties of relatively bounded correct perturbationsRelatively bounded perturbations of the correct restrictionsCorrect restrictions and extensions. Method for constructing a class of correct perturbations, which spectra coincide with the spectrum of a fixed boundary correct extensionCorrect extensions, is obtained. Examples illustrating the application of the obtained results are given.

We are studying the issueImanbaev, N.S. of stability and instability of the basis property of the system of eigenfunctions and associated functions of theSturm-Liouville operators Sturm-LiouvilleSadybekov, M.A. operator with an integral perturbation of one boundary condition. This paper is devoted to a spectral problem for operator with an integral perturbation of boundary conditions, which are regular, but not strongly regular. We assume that the unperturbed problem has system of normalized eigenfunctions and associated functions which forms a Riesz basis. We construct a characteristic determinant of the spectral problemSpectral problem with an integral perturbation of the boundary conditions. The present work is the continuation of authors’ researchers on stability (instability) of basis property of root vectors of a differential operator with nonlocal perturbation of one of boundary conditions. The work includes a more detailed exposition of some previous results of authors in this directive, and there are given new results.

In this paperKal’menov, T.Sh. we construct aNon-local boundary conditions nonlocal integral boundary condition of theSabitbek, B. volume potentialVolume potential for second order strongly elliptic differential equations, which generalizes previous known results. We also review similar results for polyharmonic operatorsPolyharmonic operators.

Estimates in variousKritskov, L.V. Lebesgue spaces $$L_s(G)$$Ls(G), $$1\le s\le \infty $$1≤s≤∞, are obtained for the root functions of an operator which relates to the differential operation $$-u''+p(x)u'+q(x)u$$-u′′+p(x)u′+q(x)u, $$x\in G=(a,b)$$x∈G=(a,b), with complex-valued singular coefficients. Among these estimates there are also the so-called anti-a priori estimatesAnti-a priori estimates that link the root functions in the same chain. It is supposed that p(x) and q(x) belong locally to the spaces $$L_2$$L2 and $$W_2^{-1}$$W2-1, respectively, may have singularities at the end-points of G, and $$q(x)=q_1(x)+Q'(x)$$q(x)=q1(x)+Q′(x) while $$Q(x), p(x), Q^2(x)w(x), p^2(x)w(x), q_1(x)w(x)$$Q(x),p(x),Q2(x)w(x),p2(x)w(x),q1(x)w(x) are integrable on the whole interval G with $$w(x)=(x-a)(b-x)$$w(x)=(x-a)(b-x).

In this workMuratbekov, M.B.aUnique solvability unique solvabilityMuratbekov Ma. of a class of hyperbolic type partialDadaeva, A. differential equations with unbounded coefficients is proved in $$\mathbb {R}^2$$R2. The estimates of the weight normsEstimation of the norm of the solution u and its partial derivatives $$u_x$$ux and $$u_y$$uy are derived.

In this paperTokmagambetov, N. we extend results on regularized traceNalzhupbayeva, G. formulae which were established in [9, 10] for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order $$m\in \mathbb N$$m∈N. By using techniques of Sadovnichii and Lyubishkin [21], the authors in the papers [9, 10] described regularized trace formulae in the spatial dimension $$d=2$$d=2. In this remark one is to be claimed that the formulae are also valid in the higher spatial dimensions, namely, $$2\le d \le 2m$$2≤d≤2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ‘nonharmonic’ analysis.

We considerNurakhmetov, D.B. well-posedness issues of problems of the Laplace operator in theAniyarov, A.A. unit circle with two internal points. For boundary value problems, one of the main issues is the well-posedness of the problem. When the problem is considered in a non-simply-connected domain, there usually appear additional conditions depending on the features of the domain under consideration. If for the well-posedness of the problem, in addition to the boundary conditions, one requires to take into account the internal communications of the domain, then such problems are called internal boundary value problems. For such problems there is written out a class of functions in which there exist such kinds of well-posed problems. A constructive method for constructing solutions to such problems is developed. As an illustration, examples are considered.

In this paper we investigate a singular high-order differential operator with rapidly growing intermediate coefficients. We give sufficient conditions for complete continuity of its resolvent in the space $$L_{2} (-\infty ,\, +\infty )$$L2(-∞,+∞). Furthermore, we show that this resolvent belongs to the Schatten class $$\sigma _{p} $$σp, $$1<p<\infty $$1<p<∞, and give the uniform estimate for the resolvent norm.

In thisOspanov, M.N. paper we give sufficient conditions for complete continuity of resolvent of a differential operator corresponding to a system of first order singular differential equationsSingular differential equation. Using coercive estimates for the solution of the above differential equation, we obtain the main result.

InTomar, M. this paper, we aimAgarwal, P. to present the improved version of the reverse Hölder type inequalitiesJain, Sh. by taking $$(k,s)-$$(k,s)-Riemann-Liouville fractional integrals. Furthermore, we also discuss some applicationsMilovanović, G. of Theorem 1 using some types of fractional integrals.

In this work we suggest a new method for investigating the model Volterra type integral equation with super-singularity, theSuper-singular kernels kernel of which consists of a composition of polynomial functions with super-singularity and functions with super-singular points. The problem of investigating this type of integral equation for $$n=2m$$n=2m is reduced to m Volterra type integral equation for $$n=2$$n=2, and for $$n=2m+1$$n=2m+1 it is reduced to m Volterra integral equation for $$n=2$$n=2 and one integral equation for $$n=1$$n=1.

In this paperRuzhansky, M. we review our previous isoperimetricSuragan,D. results for the logarithmic potential and Newton potential operators. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problemsGeometric extremum problem, is that they produce a priori bounds for spectral invariants of operators on arbitrary domains. We demonstrate these in explicit examples.

We consider a problem on finding a solution of an initial-boundary value problem for a heat equation with regular, but not strongly regular boundary conditions. It is shown that in the case of the potential parity $$q(x)=q(1-x)$$q(x)=q(1-x) the researched class of problems can always be reduced to a sequential solution of two analogous problems, but with strongly regular boundary conditions. Herewith the proof does not depend on whether the system of eigen- and associated functions of a corresponding spectral problem for an ordinary differential equation arising in applying the Fourier method forms a basis. The suggested way of the problem solution can be applied for constructing as classical, and for various types of generalized solutions. The solution method earlier suggested by the author is modernized. Due to this fact input data of the problem do not require an additional smoothness.

In this paperKassymov, A. we prove that theSuragan, D. first s-number of the Cauchy-Dirichlet heat operator is minimized in the equilateral cylinder among all Euclidean triangular cylindric domains of a given volume as well as we obtain spectral geometric inequalities of the Cauchy-Dirichlet-Neumann heat operator in the right and equilateral triangular cylinder. It is also established that maximum of the second s-number of the Cauchy-Neumann heat operator is reached by the equilateral triangular cylinder among all triangular cylinders of given volume. In addition, we prove that the second s-number of the Cauchy-Neumann heat operator is maximized in the circular cylinder among all cylindrical Lipschitz domains of fixed volume.

We describe different aspects of the theory of pseudo-differential equations on manifolds with non-smooth boundaries. Using a concept of special factorization for an elliptic symbol we consider distinct variants of this approach including asymptotic and discrete situations.

The mathematical modelKassabek, S.describingKharin, S the axisymmetric electromagnetic field and the constriction resistance of the semispace with AC electrical current passing through a ring-shaped contact is presented. It is based on the system of the Maxwell equations with the special boundary conditions. The analytical formulas for the electric and magnetic fields are obtained. The asymptotic expression for the constriction resistance is found and the corresponding expression for the DC current may be derived from this general expression as a special case. Comparison of this expression with the well known classical formula shows very good approximation.

In thisKhompysh, Kh. work we consider an inverse problem of finding a coefficient of right hand side of pseudo-parabolic equation. By successive approximation method the existence and uniqueness of a strong solution are proved. Under the integral overdetermination condition, which has important applications in various areas of applied science and engineering.

The purpose ofSarsengeldin, M.M this studyNauryz, T. is to test the elaborated theory for inverse twoBizhigitova, N. phase sphericalOrinbasar, A. Stefan problem by using Integral Error Function and check effectiveness of the suggested solution form for engineering purposes. It was shown that by collocation method we can achieve small error which doesn’t exceed 8 percent for three points, which substantially eases calculations. Investigation of such problems enables one to analyse diverse electric contact phenomena.

We present an alternative proof for the classification of semistable representations of a linear quiver and of a circular quiver with three vertices and briefly discuss the meaning of this result for the study of quiver sheaves.

The force function isTleubergenov, M.I constructed for the givenAzhymbaev, D.T properties of motion, independent from velocities. Previously the stochastic Ito equation is built for a given integral manifold by quasi-inversion method. Further, the equivalent equation of Lagrangian structure is built according to stochastic Ito equation, and then the force function is defined by Lagrange’s function.

In thisYuldasheva, A.V. paper an inverse problem of finding the time-dependent coefficient of heat capacity together with solution of high-order heat equation with nonlocal boundary and integral overdetermination conditions is considered. The existence and uniqueness of a solution of the inverse problem are proved by using the Fourier method and the iteration method. Continuous dependence upon the data of the inverse problem is shown.

HereZakiryanova, G.K. the two-component medium of M. Biot consisting of solid and fluid components is considered under action of moving loads. The fundamental and generalized solutions of Biot equations have been constructed for subsonic and supersonic velocities of loads.

The methodologyZhumatov, S.S. of stability analysis is expounded to the systems automatic control feedback at presence of non-linearity. The conditions of asymptotically instability of the basic control systems are considered in the neighborhood of a program manifold. Nonlinearity satisfies to generalized conditions of local quadratic relations. The sufficient conditions of instability of the program manifold have been obtained relatively to a given vector-function by means of construction of Lyapunov function, in the form “quadratic form plus an integral from nonlinearity”. It is solved more general inverse problem of dynamics: not only builds the corresponding system of differential equations, but also investigates the instability, which is very important for a variety of mathematical models mechanics.

Some generalizationsZhunussova, Z.Zh. of the Landau-Lifschitz equation are integrable, admit physically interesting exact solutions and these integrable equations are solvable by the inverse scattering method. Investigations of the integrable spin equations in (1+1)-, (2+1)-dimensions are topical both from the mathematical and physical points of view. Integrable equations admit different kinds of physically interesting equations as domain wall solutions. We consider an integrable spin equation. There is a corresponding Lax representationLax representation. Moreover the equation allows an infinite number of integrals of motion. We construct a surface corresponding to domain wall solution of the equation. Further, we investigate some geometrical features of the surface.