Multivalued Functions and Integral Representations

Integral representations of the de Branges functional for univalent functions

Bulletin de la Classe des sciences ◽

10.3406/barb.2001.28182 ◽

2001 ◽

Vol 12 (1) ◽

pp. 121-135

Author(s):

Pavel G. Todorov

Keyword(s):

Univalent Functions ◽

Integral Representations

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ON THE SUP-MEASURABILITY OF MULTIVALUED FUNCTIONS

Real Analysis Exchange ◽

10.2307/44152736 ◽

1996 ◽

Vol 22 (1) ◽

pp. 102

Author(s):

Kwiecińska

Keyword(s):

Multivalued Functions

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ON p – q DUALITY AND EXPLICIT SOLUTIONS IN c ≤ 1 2D GRAVITY MODELS

International Journal of Modern Physics A ◽

10.1142/s0217751x95000577 ◽

1995 ◽

Vol 10 (08) ◽

pp. 1219-1236 ◽

Cited By ~ 17

Author(s):

S. KHARCHEV ◽

A. MARSHAKOV

Keyword(s):

String Theory ◽

Exact Solutions ◽

2D Gravity ◽

Integral Formula ◽

Integral Representations ◽

Gravity Models ◽

Explicit Solutions ◽

String Equation ◽

Duality Transformation

We study the role of integral representations in the description of nonperturbative solutions to c ≤ 1 string theory. A generic solution is determined by two functions, W(x) and Q(x), which behave at infinity like xp and xq respectively. The integral formula for arbitrary (p, q) models is derived, which explicitly realizes a duality transformation between (p, q) and (q, p) 2D gravity solutions. We also discuss the exact solutions to the string equation and reduction condition and present several explicit examples.

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Integral Representations for the Solution of Dynamic Bending of a Plate with Displacement-Traction Boundary Data

Georgian Mathematical Journal ◽

10.1515/gmj.2003.467 ◽

2003 ◽

Vol 10 (3) ◽

pp. 467-480

Author(s):

Igor Chudinovich ◽

Christian Constanda

Keyword(s):

Existence And Uniqueness ◽

Boundary Data ◽

Boundary Integral Equations ◽

Mixed Boundary ◽

Mixed Boundary Conditions ◽

Boundary Integral ◽

Integral Representations ◽

Transverse Shear Deformation ◽

Uniqueness Of Solutions ◽

Nonstationary Boundary

Abstract The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

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Integral Representations and Continuous Projectors in Some Spaces of Analytic and Pluriharmonic Functions

Georgian Mathematical Journal ◽

10.1515/gmj.2008.739 ◽

2008 ◽

Vol 15 (4) ◽

pp. 739-752

Author(s):

Gigla Oniani ◽

Lamara Tsibadze

Keyword(s):

Integral Representations ◽

Fractional Integrals ◽

Boundary Values ◽

Pluriharmonic Functions

Abstract We consider analytic and pluriharmonic functions belonging to the classes 𝐵𝑝(Ω) and 𝑏𝑝(Ω) and defined in the ball . The theorems established in the paper make it possible to obtain some integral representations of functions of the above-mentioned classes. The existence of bounded projectors from the space 𝐿(ρ, Ω) into the space 𝐵𝑝(Ω) and from the space 𝐿(ρ, Ω) into the space 𝑏𝑝(Ω) is proved. Also, consideration is given to the existence of boundary values of fractional integrals of functions of the spaces 𝐵𝑝(Ω) and 𝑏𝑝(Ω).

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On the Stability in Bonnet's Theorem of the Surface Theory

Georgian Mathematical Journal ◽

10.1515/gmj.2007.543 ◽

2007 ◽

Vol 14 (3) ◽

pp. 543-564

Author(s):

Yuri G. Reshetnyak

Keyword(s):

Differential Operators ◽

Sobolev Class ◽

Integrability Condition ◽

Complete Integrability ◽

Integral Representations ◽

Completely Integrable Systems ◽

Frobenius Theorem ◽

Fundamental Tensor ◽

The Stability ◽

Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

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On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

Symmetry ◽

10.3390/sym13071219 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1219

Author(s):

Marek T. Malinowski

Keyword(s):

Differential Equations ◽

Existence And Uniqueness ◽

Existence Theorems ◽

Type Theorem ◽

Integral Representations ◽

Uniqueness Of The Solution ◽

Nonempty Compact ◽

Picard Type Theorem ◽

Convex Subsets

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.

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