Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems

In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green’s function as a series of functions with the help of spectral theory, and obtain some of its properties. Under suitable conditions on the nonlinear part of the nabla fractional difference equation, we deduce two existence results of the considered nonlinear problem by means of two Leray–Schauder fixed point theorems. We provide a couple of examples to illustrate the applicability of the established results.

Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

Analysis of a coupled system of fractional differential equations with non-separated boundary conditions

Solutions to fractional differential equations is an emerging part of current research, since such equations appear in different applied fields. A study of existence, uniqueness, and stability of solutions to a coupled system of fractional differential equations with non-separated boundary conditions is the main target of this paper. The existence and uniqueness results are obtained by employing the Leray–Schauder fixed point theorem and the Banach contraction principle. Additionally, we examine different types of stabilities in the sense of Ulam–Hyers such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. To prove the effectiveness of our main results, we study a few interesting examples.

Inverse spectral problems for differential operators with non-separated boundary conditions

We give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related to inverse problems for Sturm–Liouville operators with non-separated boundary conditions: periodic, quasi-periodic and Robin-type boundary conditions. At the end, we present the main results on inverse problems for differential pencils with non-separated boundary conditions.

Impulsive boundary value problems containing Caputo fractional derivative of a function with respect to another function

In this paper, we study the existence and uniqueness for a new class of impulsive fractional boundary value problems with separated boundary conditions containing the Caputo fractional derivative of a function with respect to another function. The existence of solutions is established by using the Leray–Schauder nonlinear alternative, and the uniqueness result is proved via Banach’s contraction mapping principle. Some examples are also constructed to demonstrate the application of main results.

The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems

We study the direct and inverse problems for the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as a smooth surface (as the values of a real analytic function of two variables), which has specific properties. We call this function ”the eigenvalues function of the family of Sturm-Liouville operators (EVF)”. From the properties of this function we select those, which are sufficient for a function of two variables be the EVF a family of Sturm-Liouville operators.

Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.