Abstract
In this paper, we present a generalized concept of second-order differentiability for interval-valued functions. The global existence of solutions for interval-valued second-order differential equations (ISDEs) under generalized H-differentiability is proved. We present some examples of linear interval-valued second-order differential equations with initial conditions having four different solutions. Finally, we propose a new algorithm based on the analysis of a crisp solution. Some examples are given by using our proposed algorithm.MSC:34K05, 34K30, 47G20.
Highlights
1 Introduction The set-valued differential and integral equations are an important part of the theory of set-valued analysis
We study four kinds of solutions to interval-valued second-order differential equations (ISDEs)
In Section, we present the global existence and uniqueness theorem of a solution to the interval-valued second-order differential equation under two kinds of the Hukuhara derivative
Summary
The set-valued differential and integral equations are an important part of the theory of set-valued analysis. Note that this definition of derivative is very restrictive; for instance, we can see that if X(t) = A · x(t), where A is an interval-valued constant and x : [a, b] → R+ is a real function with x (t) < , X(t) is not differentiable To avoid this difficulty, Stefanini and Bede [ ] solved the above mentioned approach under strongly generalized differentiability of interval-valued functions. In this case, the derivative exists and the solution of an interval-valued differential equation may have decreasing length of the support, but the uniqueness is lost. In Section , we present the global existence and uniqueness theorem of a solution to the interval-valued second-order differential equation under two kinds of the Hukuhara derivative. In Section , we propose a new algorithm based on the analysis of a crisp solution
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