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

Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where ${\displaystyle Au:={\sum }_{i,j=1}^m{a}_{ij}\left(x\right)\frac{\partial ²u}{\partial x_i\partial x_j}}$, ${\displaystyle x=(x_1,...,x_m)\in G\subset {ℝ}^m}$, G is a bounded domain with ${\displaystyle C^{2+\alpha }}$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real ${\displaystyle L^p(G̅)}$ 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 ${\displaystyle Au+f(x,u\left(x\right),{\int }_{G}u\left(x\right)dx)=0}$. Interesting results about existence and uniqueness of solutions for this equation were obtained by H. Ugowski [17].