[Paper Review] Lyapunov-Type Inequalities for Discrete Riemann-Liouville Fractional Boundary Value Problems

Jagan Mohan Jonnalagadda|arXiv (Cornell University)|Jul 31, 2018

Nonlinear Differential Equations Analysis37 references4 citations

TL;DR

This paper establishes Lyapunov-type inequalities for two-point discrete Riemann-Liouville fractional boundary value problems using nabla calculus. It derives a lower bound for the sum of absolute values of the potential function, proving that nontrivial solutions exist only if this sum exceeds a threshold involving the gamma function and generalized rising factorial. The key contribution is a new Lyapunov inequality for nabla fractional differences, which enables nonexistence criteria for solutions, eigenvalue bounds, and zero-free conditions for discrete Mittag-Leffler functions.

ABSTRACT

In this article we establish a few Lyapunov-type inequalities for two-point discrete fractional boundary value problems involving Riemann-Liouville type backward differences. To illustrate the applicability of established results, we obtain criteria for the nonexistence of nontrivial solutions and estimate lower bounds for eigenvalues of the corresponding eigenvalue problems. We also apply these inequalities to deduce criteria for the nonexistence of real zeros of certain discrete Mittag-Leffler functions.

Motivation & Objective

• To derive new Lyapunov-type inequalities for discrete fractional boundary value problems involving nabla Riemann-Liouville differences.
• To provide sufficient conditions for the nonexistence of nontrivial solutions in discrete fractional boundary value problems.
• To estimate lower bounds for eigenvalues of corresponding eigenvalue problems.
• To deduce criteria for the nonexistence of real zeros of certain discrete Mittag-Leffler functions.

Proposed method

• Derives the Green's function for a right focal discrete fractional boundary value problem using nabla fractional calculus.
• Uses the Green's function to establish pointwise bounds and maximum estimates over the domain.
• Applies the maximum of the Green's function to derive a Lyapunov-type inequality involving the gamma function and generalized rising factorial.
• Utilizes the inequality to prove nonexistence of nontrivial solutions under a threshold condition on the potential function.
• Applies the inequality to eigenvalue problems to derive a lower bound proportional to $ \Gamma(\alpha) / (b-a-1)^{\overline{\alpha-1}} $.
• Connects the inequality to the nonexistence of real zeros of discrete Mittag-Leffler functions via eigenvalue analysis.

Experimental results

Research questions

• RQ1 What is the sharp lower bound on $ \sum_{s=a+2}^{b} |q(s)| $ that guarantees the nonexistence of nontrivial solutions for a discrete fractional boundary value problem with nabla Riemann-Liouville differences?
• RQ2 How can Lyapunov-type inequalities be extended to discrete fractional problems with right focal boundary conditions?
• RQ3 What lower bound can be established for the eigenvalues of the corresponding discrete fractional eigenvalue problem?
• RQ4 Under what condition does the discrete Mittag-Leffler function $ E_{-\lambda,\alpha,0}(n,0) + \lambda E_{-\lambda,\alpha,\alpha-1}(n,0) $ have no real zeros?

Key findings

• The Lyapunov-type inequality $ \sum_{s=a+2}^{b} |q(s)| \geq \frac{\Gamma(\alpha)}{(b-a-1)^{\overline{\alpha-1}}} $ holds for nontrivial solutions of the discrete fractional boundary value problem.
• If $ \sum_{s=a+2}^{b} |q(s)| < \frac{\Gamma(\alpha)}{(b-a-1)^{\overline{\alpha-1}}} $, then no nontrivial solution exists.
• For the eigenvalue problem $ \nabla^\alpha_0 u(t) + \lambda u(t) = 0 $, the eigenvalue satisfies $ |\lambda| \geq \frac{\Gamma(\alpha)}{(n-1)(n-1)^{\overline{\alpha-1}}} $.
• The discrete Mittag-Leffler function $ E_{-\lambda,\alpha,0}(n,0) + \lambda E_{-\lambda,\alpha,\alpha-1}(n,0) $ has no real zeros when $ |\lambda| < \frac{\Gamma(\alpha)}{(n-1)(n-1)^{\overline{\alpha-1}}} $.
• The maximum of the Green's function $ G_r(t,s) $ over the domain is $ \frac{(b-a-1)^{\overline{\alpha-1}}}{\Gamma(\alpha)} $, which is critical for deriving the inequality.
• The proof relies on bounding the sum of the Green's function over the domain and using properties of the generalized rising factorial and gamma function.