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Some nonlinear problem to the equation [Delta]u-cu=f for a sphere

Barański F., Tal-Figiel B.

Fasciculi Mathematici

2002

Nr 33

21-26

The subject of the paper is the construction of a solution of the differential equation delta u (x) - c(x)u(x) = f(x, u(x)), in the spherical domain D = {x = (xt, x2, x,): | x | < R}, satisfying the Dirichlet boundary-value condition u(x) = h(x) for x e B(D). Let (x, y) -ť G(x, y) denote the Green function to the Laplace equation deltaG(x, y) = 0 in the sphere D and to the homogeneous boundary-value condition G(x, y) = 0 for x belong B(D), y belong D. Applying the change of the unknown function x -ť u(x) = u (x) + w(x), where u is a solution to the equation delta u (x, y) = 0 0, x belong D, with the boundary-value condition u(x) = h(x), x belong B(D), and w is a a new unknown function, we obtain the equation deltaw(x) = u + w(x)) + f(x,u(x) + w(x)) or the equation deltaw(x) = F(1)(x,w(x) = c(x)(u(x) + F(x,w(x)) with F(x, w(x)) = c(x)~w(x) + f(x, u(x) + w(x)), where x e D. Inverting the last problem by the Green function G, we obtain the integral equation w(x) = f, (x) + +fffF(y(w(y))G(x,y)dy, xbelongD, with f, (x) = fff c(y)u (y)G(x, y)dy, x belong D, and the homogeneous boundary-value condition w(x) = 0 for x e B(D). Solving by the Banach fixed point method the last equation we obtain w and u = u + w.

Green functions and periodic solution to the m-parabolic equation with boundary-value conditions of Riquier type

Barański F., Zdankiewicz Z.

Fasciculi Mathematici

2001

Nr 32

5-13

The subject of the paper is the construction of the periodic solutions to the w-caloric equation P"'u(x,t) = 0, P=D(2)*(2)-D, P(2) =P(P), Pm = P(P(m~l) in the strip (mathemical formula), satisfying the periodic boundary-value conditions (mathematical formula), where h(11)(t), h1,2(t) are the periodic functions with the period p > 0.

On the Fourier first semilinear iterated problem

Barański F., Byszewski L.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2000

[Z] 40

19-31