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Functional Difference Equations

In: Qualitative Theory of Volterra Difference Equations

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  • Youssef N. Raffoul

    (University of Dayton, Department of Mathematics)

Abstract

In this chapter we consider functional difference equations that we apply to all types of Volterra Volterra difference equation difference equations. Our general theorems will require the construction of suitable Lyapunov functionals, a task that is difficult but possible. As we have seen in Chapter 1 , the concept of resolvent Resolvent can only apply to linear Volterra Volterra difference equation difference systems. The theorems on functional difference equations will enable us to qualitatively analyze the theory of boundedness, uniform ultimate boundedness, and stability of solutions of vectors and scalar Scalar s Volterra Volterra difference equation difference equations. We extend and prove parallel theorems regarding functional difference equations with finite or infinite delay, Finite delay and provide many applications. In addition, we will point out the need of more research in delay difference equations. In the second part of the chapter, we state and prove theorems that guide us on how to systematically construct suitable Lyapunov functionals for a specific nonlinear Volterra Volterra difference equation difference equation. We end the chapter with open problems. Most of the results of this chapter can be found in [37, 38, 128, 133, 135, 141, 147, 181], and [182].

Suggested Citation

  • Youssef N. Raffoul, 2018. "Functional Difference Equations," Springer Books, in: Qualitative Theory of Volterra Difference Equations, chapter 0, pages 55-92, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-97190-2_2
    DOI: 10.1007/978-3-319-97190-2_2
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