[Paper Review] Existence of solution to a nonlinear first-order dynamic equation on time scales

Benaoumeur Bayour, Ahmed Hammoudi+1 more|arXiv (Cornell University)|Dec 3, 2015

Nonlinear Differential Equations Analysis7 references3 citations

TL;DR

This paper establishes the existence of solutions to a nonlinear first-order nabla dynamic equation on an arbitrary bounded time scale with periodic boundary conditions. Using the novel concept of a 'tube solution'—a generalization of lower and upper solutions—the authors prove existence via Schauder's fixed point theorem applied to a compact operator derived from the dynamic equation, ensuring solutions remain within a prescribed tube of radius M around a central curve v.

ABSTRACT

We prove existence of solution to a nonlinear first-order nabla dynamic equation on an arbitrary bounded time scale with boundary conditions, where the right-hand side of the dynamic equation is a continuous function.

Motivation & Objective

• To establish existence of solutions for a nonlinear first-order nabla dynamic equation on an arbitrary bounded time scale.
• To unify continuous and discrete boundary value problems within a single framework using time scale calculus.
• To extend the concept of lower and upper solutions to 'tube solutions' for more general existence results.
• To apply fixed point theory to prove existence of solutions within a prescribed tube of radius M around a central curve v.

Proposed method

• Introduces the notion of a tube solution (v, M) where v is a central curve and M is a radius function.
• Defines a nonlinear operator T̂p based on the dynamic equation and boundary conditions.
• Proves the operator T̂p is compact using the Arzelà–Ascoli theorem adapted to time scales.
• Applies Schauder's fixed point theorem to guarantee a fixed point of T̂p, which corresponds to a solution.
• Uses nabla differentiation and integration on time scales to analyze the dynamic equation and its solutions.
• Employs the generalized derivative and jump operators (ρ, σ, ν) to handle both continuous and discrete cases uniformly.

Experimental results

Research questions

• RQ1 Under what conditions does a nonlinear first-order nabla dynamic equation on a time scale admit a solution?
• RQ2 How can the concept of lower and upper solutions be generalized to handle more complex nonlinear dynamics?
• RQ3 Can the existence of solutions be guaranteed within a prescribed tube around a central curve v?
• RQ4 What role does the compactness of the solution operator play in proving existence via fixed point theory?
• RQ5 How do the properties of the time scale (e.g., discrete or continuous) affect the existence and structure of solutions?

Key findings

• The existence of a tube solution (v, M) implies the existence of at least one solution x to the dynamic equation (1.1) such that ‖x(t) − v(t)‖ ≤ M(t) for all t ∈ T.
• The solution operator T̂p is compact, ensuring the existence of a fixed point via Schauder's theorem.
• If t is left-dense, the nabla derivative of the norm function is given by the inner product formula involving the derivative of the vector function.
• For t left-scattered, the nabla derivative of the norm is bounded above by a combination of the vector derivative and the radius function.
• The condition r∇(t) < 0 for t in the set where r(t) ≥ 0 ensures that the solution cannot exit the tube, preserving the solution within the prescribed bounds.
• In Example 3.4, the problem x∇(t) = a₁‖x(t)‖²x(t) − a₂x(t) + a₃φ(t) with a₂ ≥ a₁ + a₃ + 1 has a solution satisfying ‖x(t)‖ ≤ 1 for all t ∈ T, as (v, m) ≡ (0, 1) is a tube solution.