Nonlinear Interpolation and Boundary Value Problems

The following sections are included:

• Dedication
• Preface
• Contents

This chapter is devoted primarily to uniqueness of solutions for boundary value problems for an ordinary differential equation. In particular, focus will be on when uniqueness of solutions for one type of boundary value problem leads to uniqueness of solutions for other types of boundary value problems. In addition to interest in the results themselves, such results sometimes play a role in establishing the existence of solutions (the topic of some of the subsequent chapters of this book). For example, in the case of linear boundary value problems for linear ordinary differential equations, uniqueness of solutions is equivalent to their existence…

This chapter is devoted primarily to existence of solutions for boundary value problems for ordinary differential equations. In particular, focus will be on when uniqueness of solutions for one type of boundary value problem implies existence of solutions for the same type and other types of boundary value problems. Again it is pointed out that in the case of linear boundary value problems for linear ordinary differential equations, uniqueness of solutions is equivalent to their existence…

This chapter is devoted to uniqueness and existence of solutions of nonlocal boundary value problems for (1.1.1). Broadly speaking, a nonlocal linear boundary condition involves a linear condition providing data for a solution at more than one boundary point. Given α₁, … , α_p ∈ ℝ, a ≤ x₁ < … < x_p ≤ b, and A ∈ ℝ,

Known results for ordinary differential equations have long been one of the sources of questions in analogy for finite difference equations. In the context of boundary value problems, a great deal of activity was launched by Hartman [44] in 1978 with his paper devoted to disconjugacy criteria for linear difference equations. Eventually, many researchers turned their attention to questions dealing with boundary value problems for nonlinear difference equations, and papers during the early stages of interest in nonlinear difference equations include those by Agarwal [1, 2], Eloe [22, 23], Hankerson [36, 37] and Peterson [100]. And research devoted to boundary value problems for nonlinear difference equations remains quite high with well over 100 papers published in that area during the years 2012–2015…

In 1990, Hilger published a pioneering paper [74] on the topic of measure chains in which he developed a theory that both unified and extended the theories between the continuous calculus and the discrete calculus. Given ∅ ≠ 𝕋 ⊆ ℝ, if 𝕋 is closed and equipped with the subspace topology inherited from the Euclidean topology on ℝ, then 𝕋 is called a measure chain; Hilger [74]. Within the context of measure chains, such a subset 𝕋 of ℝ came to be called a time scale [6]. Now, almost exclusively, if 𝕋 is a nonempty, closed subset of ℝ and is endowed with the subspace Euclidean topology, then 𝕋 is called a time scale…

Many of the methods employed in this book for uniqueness and existence of boundary value problems for ordinary differential equations have also been applied to boundary value problems that are described as being “between conjugate and right focal” boundary value problems for ordinary differential equations; namely, for an ordinary differential equation, if solutions of the right focal boundary value problems are unique, when solutions exist, then by applications of Rolle's Theorem, solutions of the “between” boundary value problems are unique, when solutions exist, whereas uniqueness of solutions of the “between” boundary value problems implies, by applications of Rolle's Theorem, that solutions of the conjugate boundary value problems are unique, when solutions exist…

The following sections are included:

• Bibliography
• Index