Rate of decay of solutions of Volterra equations

Qualitative properties in nonlinear Volterra integro-differential equations with delay

“…(1) and 3, when τ = 0. Our results will also be different from those obtained in the literature (see Adıvar and Raffoul [1], Becker [2], Burton [3][4][5][6], Burton and Haddock [7], Burton et al [8], Burton and Mahfoud [9,10]), Corduneanu [11], Graef and Tunc [13], Gripenberg et al [14], Miller [15], Raffoul [16], Raffoul and Unal [17], Staffans [18], Tunc [19], Vanualailai and Nakagiri [21] and the references thereof). In this way, we mean that the Volterra integro-differential equations discussed and the assumptions to be established here are different from those in the abovementioned papers.…”

“…It should be noted that an important ingredient in the qualitative theory for ordinary and functional differential equations and integro-differential equations is Lyapunov's second method. In particular, Burton et al [8] developed a Lyapunov theory that primarily seems to apply to Volterra integro-differential equations. They use Lyapunov functionals, which are (most of the time) non-increasing or strictly decreasing along solutions.…”

New results on exponential stability of nonlinear Volterra integro-differential equations with constant time-lag

“…That is, the trivial solution of (VIDE) (1.9) is (UEAS [3], [4]), Burton ([5], [6], [7]), Burton et al [8], Burton and Mahfoud ([9], [10]), Eloe et al [13], Furumochi and Matsuoka [14], Graef and Tunç [15], Graef et al [16], Grimmer and Seifert [17], Gripenberg et al [18], Hara et al [19], Islam and Raffoul [20], Jordan [21], Levin [22], Mahfoud [23], Miller [24], Raffoul ([25], [26], [27], [28]), Raffoul and Ren [29], Raffoul and Unal [29], Rama Mohana Rao and Raghavendra [30], Rama Mohana Rao and Srinivas [31], Staffans [32], Tunç ([33], [34], [35]) Tunç and Ayhan [38], Vanualailai and Nakagiri [39], Wang [40], Wazwaz [41], Zhang [42], Da Zhang [43].…”

Asymptotic stability of nonlinear integrodifferential equation

“…Over the years, Lyapunov method for the stability of integro-differential equation have been proposed by different researcher ( [8], [13], [14], [15], [16], [17], [19]). In particular, [7] developed a Lyapunov theory and used nonincreasing or strictly decreasing Lyapunov functionals along solutions that primarily seems to apply to Volterra integro-differential equations. Theoretically, this method is very appealing and there are numerous applications in which it is used in nature.…”