4. Volterra Integro-Dynamic Equations

This chapter is exclusively devoted to the study of Volterra integro-dynamic equations with or without delay. We will display some exotic Lyapunov functionals to obtain stability and instability of the zero solutions and boundedness of solutions. The delay is in the form of shift operators so that all type of general time scales can be considered and without the requirement that they be additive. We prove variant forms of Gronwall’s inequality, so we can determine function bounds for the solutions of the integro-dynamic equations. We will develop in detail the notion of principal matrix solution for Volterra integro-dynamic equations and then build on the same concept to fully develop the existence of resolvent kernel which will be used along with Lyapunov functionals to obtain necessary and sufficient conditions for uniform stability and uniform asymptotic stability of the zero solution. Then we advance to functional delay dynamic equations which we write as Volterra integro-dynamic equations and study their behaviors via Lyapunov functionals. Toward the end of the chapter we classify the positive and negative solutions of nonlinear systems of Volterra integro-dynamic equations by appealing to Schauder’s and Knaster fixed point theorems. As usual, we end the chapter with some interesting and rewarding open problems. Some results are new and the rest can be found in Adıvar (Electron J Qual Theory Differ Equ 2010(7):1–22, 2010; Glasg Math J 53(3), 463–480, 2011), Bohner (Far East J Appl Math 18(3):289–304, 2005), Burton (J Math Anal Appl 28:545–552, 1969; Nonlinear Anal 1(4):331–338, 1976/1977), Eloe et al. (Dynam Syst Appl 9(3):331–344, 2000), Kulik and Tisdell (Int J Differ Equ 3(1):103–133, 2008), Miller (J Differ Equ 10, 485–506, 1971), Raffoul (JIPAM J Inequal Pure Appl Math 10(3):70, 9, 2009), and Wang (J Math Anal Appl 186(3):8350–861, 1994; J Math Anal Appl 298(1):33–44, 2004).