Nonlocal boundary value problems for elliptic equations arise in many applied models; however, their numerical analysis in Banach spaces, particularly under the Samarskii–Ionkin condition, remains insufficiently studied. Motivated by this gap, the present work develops first- and second-order two-step difference schemes for the numerical solution of nonlocal boundary value problems associated with elliptic equations involving a positive definite operator in a Banach space. Coercive stability estimates for the proposed schemes are established. Furthermore, the abstract results are applied to the construction of stable difference schemes for multidimensional elliptic partial differential equations. Numerical experiments for two- and three-dimensional problems confirm the efficiency and applicability of the proposed method.
This paper presents a unified approach to investigate a coupled system of Caputo fractional differential equations with three types of boundary conditions: multi-point, infinite-point, and Riemann–Stieltjes integral boundary conditions. Under a generalized Lipschitz condition, we first prove the uniqueness of solutions for the system with multi-point boundary conditions via the Banach contraction mapping principle. By taking limits, we obtain the existence of solutions for the remaining two boundary conditions directly without requiring any additional proof. The feasibility of this limiting argument lies in overcoming the complexity of the associated space and operator. Finally, numerical examples verify the theoretical findings.
A frame can be thought of as a redundant coordinate system in the sense that it can contain more vectors that are necessary to represent each vector in a Hilbert space. In this paper, we aim to study the stability of *-K-modular frames under some class of closed range operators.
This paper investigates a priori estimate, existence, and uniqueness of solutions for a third-order differential system with coupled integral boundary conditions. The Leray–Schauder continuation principle forms the basis of our approach to establishing the existence result. To demonstrate the applicability of the obtained results, illustrative examples are provided.
This paper mainly utilizes the nondecomposition method to examine a problem of polynomial synchronization for the quaternion-valued memristive neural networks. First, to avoid considering the “magnitude” of a quaternion, the nonreduced-order and nondecomposition methods are applied to maintain computational efficiency without introducing new variables. Then, under a simple controller, the system’s global polynomial synchronization is guaranteed by formulating a suitable Lyapunov functional and using quaternion properties. Finally, an illustrative numerical example is included to substantiate the conclusion.
