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Volterra Integral Dynamic Equations

In: Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales

Author

Listed:
  • Murat Adıvar

    (Fayetteville State University, Broadwell College of Business and Economics)

  • Youssef N. Raffoul

    (University of Dayton, Department of Mathematics)

Abstract

Summary In this chapter, we apply the concept of resolvent that we developed in Sect. 1.4.1 for vector Volterra integral dynamic equations and show the boundedness of solutions. The resolvent is an abstract term which makes it difficult, if not impossible, to make efficient use of it. However, by the help of Lyapunov functionals and variation of parameters, we will be able to verify all the conditions that are related to the resolvent. In Sect. 6.1.1 we make use of the resolvent along with fixed point theory Fixed point and show the existence and uniqueness of solutions of nonlinear Volterra dynamic equations. Later in the chapter, we consider nonlinear Volterra integral dynamic equations and construct suitable Lyapunov functional to obtain results regarding boundedness and stability of the zero solution. Contents of this chapter are totally new and not published anywhere else except those of Sect. 6.3 that can be found in Adıvar and Raffoul (Appl Math Comput 273:258–266, 2016).

Suggested Citation

  • Murat Adıvar & Youssef N. Raffoul, 2020. "Volterra Integral Dynamic Equations," Springer Books, in: Stability, Periodicity and Boundedness in Functional Dynamical Systems on Time Scales, chapter 0, pages 237-263, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-42117-5_6
    DOI: 10.1007/978-3-030-42117-5_6
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