Factorization method for solving nonlocal boundary value problems in Banach space

This article deals with the factorization and solution of nonlocal boundary value problems in a Banach space of the abstract form B1u = Au − SΦ(u) − GΨ(A0u) = f, u ∈ D(B1),where A, A0 are linear abstract operators, S, G are vectors of functions, Φ, Ψ are vectors of linear bounded functionals, and u, f are functions. It is shown that the operator B1 under certain conditions can be factorized into a product of two simpler lower order operators as B1 = BB0. Then the solvability and the unique solution of the equation B1u = f easily follow from the solvability conditions and the unique solutions of the equations Bv = f and B0u = v. The universal technique proposed here is essentially different from other factorization methods in the respect that it involves decomposition of both the equation and boundary conditions and delivers the solution in closed form. The method is implemented to solve ordinary and partial Fredholm integro-differential equations.

Difference methods for solving nonlocal boundary value problems for fractional-order differential convection-diffusion equations with memory effect

In the present paper, in a rectangular domain, we study nonlocal boundary value problems for one-dimensional in space differential equations of convection-diffusion of fractional order with a memory effect, in which the unknown function appears in the differential expression and at the same time appears under the integral sign. The emergence of the integral term in the equation is associated with the need to take into account the dependence of the instantaneous values of the characteristics of the described object on their respective previous values, i.e. the effect of its prehistory on the current state of the system. For the numerical solution of nonlocal boundary value problems, two-layer monotone difference schemes are constructed that approximate these problems on a uniform grid. Estimates of solutions of problems in differential and difference interpretations are derived by the method of energy inequalities. The obtained a priori estimates imply the uniqueness, as well as the continuous and uniform dependence of the solution on the input data of the problems under consideration and, due to the linearity of the problem under consideration, the convergence of the solution of the difference problem to the solution of the corresponding differential problem with the rate $O(h^2+\tau^2)$.

Existence and Multiplicity Results for Nonlocal Boundary Value Problems with Strong Singularity

In this paper, we study singular φ -Laplacian nonlocal boundary value problems with a nonlinearity which does not satisfy the L 1 -Carathéodory condition. The existence, nonexistence and/or multiplicity results of positive solutions are established under two different asymptotic behaviors of the nonlinearity at ∞.