Displaying similar documents to “On periodic solution of a nonlinear beam equation”

Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation

Věnceslava Šťastnová, Otto Vejvoda (1968)

Aplikace matematiky

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One investigates the existence of an ω -periodic solution of the problem u t = u x x + c u + g ( t , x ) + ϵ f ( t , x , u , u x , ϵ ) , u ( t , 0 ) = h 0 ( t ) + ϵ χ 0 ( t , u ( t , 0 ) , u ( t , π ) ) , u ( t , π ) = h 1 ( t ) + ϵ χ 1 ( t , u ( t , 0 ) , u ( t , π ) ) , provided the functions g , f , h 0 , h 1 , χ 0 , χ 1 are sufficiently smooth and ω -periodic in t . If c k 2 , k natural, such a solution always exists for sufficiently small ϵ > 0 . On the other hand, if c = l 2 , l natural, some additional conditions have to be satisfied.

The existence of a periodic solution of a parabolic equation with the Bessel operator

Dana Lauerová (1984)

Aplikace matematiky

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In this paper, the existence of an ω -periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions f ( t , r ) , h ( t ) , a ( t ) are assumed to be ω -periodic in t , f L 2 ( S , H ) , a , h such that a ' L ( R ) , h ' L ( R ) and they fulfil (3). The solution u belongs to the space L 2 ( S , V ) L ( S , H ) , has the derivative u ' L 2 ( S , H ) and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed.

Polydisc slicing in n

Krzysztof Oleszkiewicz, Aleksander Pełczyński (2000)

Studia Mathematica

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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in n of codimension 1, v o l 2 n - 2 ( D n - 1 ) v o l 2 n - 2 ( H D n ) 2 v o l 2 n - 2 ( D n - 1 ) . The lower bound is attained if and only if H is orthogonal to the versor e j of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector e j + σ e k for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify n with 2 n ; by v o l k ( · ) we denote the usual k-dimensional volume in 2 n . The result is a complex counterpart of Ball’s [B1]...

Boundedness of Marcinkiewicz functions.

Minako Sakamoto, Kôzô Yabuta (1999)

Studia Mathematica

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The L p boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s g * λ -functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts μ S ϱ and μ λ * , ϱ to S and g * λ . The definition of μ S ϱ ( f ) is as follows: μ S ϱ ( f ) ( x ) = ( ʃ | y - x | < t | 1 / t ϱ ʃ | z | t Ω ( z ) / ( | z | n - ϱ ) f ( y - z ) d z | 2 ( d y d t ) / ( t n + 1 ) ) 1 / 2 , where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere S n - 1 , and...