Abstract
Solvability of the main boundary value problems for the nonlocal Poisson equation is studied. Existence and uniqueness theorems for the considered problems are obtained. The necessary and sufficient solvability conditions for all problems are given and integral representations for the solutions are constructed.
Highlights
The concept of a nonlocal operator and the related concept of a nonlocal differential equation appeared in mathematics quite recently
In [20], the notion of nonlocal differential equations incorporated the loaded equations, equations with fractional derivatives of the unknown function, and equations with deviating arguments, or in other words all equations in which the unknown functions and/or their derivatives enter with different values of arguments
A specific type of nonlocal differential equation is formed by equations in which the deviation of arguments has an involutive character
Summary
The concept of a nonlocal operator and the related concept of a nonlocal differential equation appeared in mathematics quite recently. Spectral problems for a firstorder differential equation with involution were studied in [6, 7]. The results of studying the spectral properties for differential equations with involution were used in [1, 14, 31] to solve the related inverse problems. In [12, 27, 28, 32], boundary value problems for second- and fourth-order elliptic equations were studied in the case when an involution appears in the boundary conditions. L we obtain the classical Dirichlet, Neumann, and Robin problems for the conventional Poisson equation. Note that in [22] nonlocal boundary value problems for the classical two-dimensional Laplace equation with the mapping S from Example 1.2 in the boundary condition are studied
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