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Existence and uniqueness of solutions of nonlinear infinite systems of parabolic differential-functional equations

100%
Brzychczy S.
Annales Polonici Mathematici
|
2001
|
tom 77
|
nr 1
1-9
EN
We consider the Fourier first initial-boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations of parabolic type. The right-hand sides of the system are functionals of unknown functions. The existence and uniqueness of the solution are proved by the Banach fixed point theorem.
2
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On the stability of solutions of nonlinear parabolic differential-functional equations

100%
Brzychczy S.
Annales Polonici Mathematici
|
1996
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tom 63
|
nr 2
155-165
EN
We consider a nonlinear differential-functional parabolic boundary initial value problem (1) ⎧A z + f(x,z(t,x),z(t,·)) - ∂z/∂t = 0 for t > 0, x ∈ G, ⎨z(t,x) = h(x)     for t > 0, x ∈ ∂G, ⎩z(0,x) = φ₀(x)     for x ∈ G, and the associated elliptic boundary value problem with Dirichlet condition (2) ⎧Az + f(x,z(x),z(·)) = 0  for x ∈ G, ⎨z(x) = h(x)    for x ∈ ∂G ⎩ where $x = (x₁,..., x_m) ∈ G ⊂ ℝ^m$, G is an open and bounded domain with $C^{2+α}$ (0 < α ≤ 1) boundary, the operator     Az := ∑_{j,k=1}^m a_{jk}(x) (∂²z/(∂x_j ∂x_k)) is uniformly elliptic in G̅ and z is a real $L^p(G)$ function. The purpose of this paper is to give some conditions which will guarantee that the parabolic problem has a stable solution. Basing on the results obtained in [7] and [5, 6], we prove that the limit of the solution of the parabolic problem (1) as t → ∞ is the solution of the associated elliptic problem (2), obtained by the monotone iterative method. The problem of stability of solutions of the parabolic differential equation has been studied by D. H. Sattinger [14, 15], H. Amann [3, 4], O. Diekmann and N. M. Temme [8], and J. Smoller [17]. Our results generalize these papers to encompass the case of differential-functional equations. Differential-functional equations arise frequently in applied mathematics. For example, equations of this type describe the heat transfer processes and the prediction of ground water level.
3
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Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type

100%
Brzychczy S.
Annales Polonici Mathematici
|
1993
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tom 58
|
nr 2
139-146
EN
Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where $Au := ∑_{i,j=1}^m a_{ij}(x) (∂²u)/(∂x_i ∂x_j)$, $x=(x_1,...,x_m) ∈ G ⊂ ℝ^m$, G is a bounded domain with $C^{2+α}$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real $L^p(G̅)$ function. For the equation (1) we consider the Dirichlet problem with the boundary condition (2) u(x) = h(x) for x∈ ∂G. We use Chaplygin's method [5] to prove that problem (1), (2) has at least one regular solution in a suitable class of functions. Using the method of upper and lower functions, coupled with the monotone iterative technique, H. Amman [3], D. H. Sattinger [13] (see also O. Diekmann and N. M. Temme [6], G. S. Ladde, V. Lakshmikantham, A. S. Vatsala [8], J. Smoller [15]) and I. P. Mysovskikh [11] obtained similar results for nonlinear differential equations of elliptic type. A special case of (1) is the integro-differential equation $Au + f(x,u(x), ∫_G u(x)dx) = 0$. Interesting results about existence and uniqueness of solutions for this equation were obtained by H. Ugowski [17].
4
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Existence of solutions and monotone iterative method for infinite systems of parabolic differential-functional equations

100%
Brzychczy S.
Annales Polonici Mathematici
|
1999
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tom 72
|
nr 1
15-24
EN
We consider the Fourier first boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations. To prove the existence and uniqueness of solution, we apply a monotone iterative method using J. Szarski's results on differential-functional inequalities and a comparison theorem for infinite systems.
5
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Approximate iterative method and the existence of solutions of nonlinear parabolic differential-functional equations

75%
Brzychczy S.
Annales Polonici Mathematici
|
1983
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tom 42
|
nr 1
37-43
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