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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 $x=(x₁,...,x_m)\in G\subset {ℝ}^m$, G is an open and bounded domain with $C^{2+\alpha }$ (0 < α ≤ 1) boundary, the operator     Az := ∑j,k=1m ajk(x) (∂²z/(∂xj ∂xk)) is...

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.

Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where $Au:={\sum }_{i,j=1}^m{a}_{ij}\left(x\right)\left(\partial ²u\right)/\left(\partial x_i\partial x_j\right)$, $x=(x_1,...,x_m)\in G\subset {ℝ}^m$, G is a bounded domain with $C^{2+\alpha }$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real $L^p\left(G̅\right)$ 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...

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.