Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type

Stanisław Brzychczy

Annales Polonici Mathematici, Tome 58 (1993), p. 139-146 / Harvested from The Polish Digital Mathematics Library

Résumé

Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where $A u : = \sum_{i, j = 1}^{m} a_{i j} (x) (\partial ² u) / (\partial x_{i} \partial x_{j})$ , $x = (x_{1}, . . ., x_{m}) \in G \subset ℝ^{m}$ , 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 $L^{p} (G ̅)$ 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 $A u + f (x, u (x), \int_{G} u (x) d x) = 0$ . Interesting results about existence and uniqueness of solutions for this equation were obtained by H. Ugowski [17].

Publié le : 1993-01-01

Zbl 0787.35114

EUDML-ID : urn:eudml:doc:262341

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     author = {Stanis\l aw Brzychczy},
     title = {Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type},
     journal = {Annales Polonici Mathematici},
     volume = {58},
     year = {1993},
     pages = {139-146},
     zbl = {0787.35114},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv58z2p139bwm}
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Stanisław Brzychczy. Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type. Annales Polonici Mathematici, Tome 58 (1993) pp. 139-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv58z2p139bwm/

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