Stability of nonlinear Volterra equations

Open access


Using a novel approach, we present some new explicit criteria for global exponential stability of the zero solution of general nonlinear time-varying Volterra difference equations. Furthermore, an explicit stability bound for equations subject to nonlinear time-varying perturbations is given. Finally, the obtained results are used to study uniform attraction of equilibrium of discrete-time bidirectional associative memory (BAM) neural networks. Some illustrative examples are given.

[1] J.A.D. Appleby, I. Gyori, and D.W. Reynolds, “On exact convergence rates for solutions of linear systems of Volterra difference equations”, J. Difference Equ. Appl. 12, 1257‒1275 (2006).

[2] H. Brunner, and P.J. Houwen, The Numerical Solution of Volterra Equations, CWI. Monographs, North-Holland, Amsterdam, 1986.

[3] L. Burlando, “Continuity of spectrum and spectral radius in algebras of operators”, Ann. Fac. Sci. Toulouse Math. 9, 5‒54 (1988).

[4] M.R. Crisci, V.B. Kolmanovskii, E. Russo, and A. Vecchio, “Stability of difference Volterra equations: direct Liapunov method and numerical procedure”, Comput. Math. Appl. 36, 77‒97 (1998).

[5] M.R. Crisci, V.B. Kolmanovskii, E. Russo, and A. Vecchio, “On the exponential stability of discrete Volterra equations”, J. Difference Equ. Appl. 6, 667‒680 (2000).

[6] C. Cuevas, F. Dantas, M. Choquehuanca, and H. Soto, “Boundedness properties for Volterra difference equations”, Appl. Math. Comput. 219 , 6986‒6999 (2013).

[7] S. Elaydi, An Introduction to Difference Equations, Springer Verlag, 2005.

[8] S. Elaydi, and S. Murakami, “Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type”, J. Difference Equ. Appl. 2, 401‒410 (1996).

[9] S. Elaydi, and V. Kocic, “Global stability of a nonlinear Volterra difference equations”, Diff. Eqns. Dyn. Sys. 2, 337‒345 (1994).

[10] S. Elaydi, “Stability and asymptoticity of Volterra difference equations: A progress report”, J. Comput. Appl. Math. 228, 504‒513 (2009).

[11] P.W. Eloe, M.N. Islam, and Y.N. Raffoul, “Uniform asymptotic stability in nonlinear Volterra discrete equations”, Comput. Math. Appl. 45, 1033‒1039 (2003).

[12] G. Gripenberg., S.O. Londen, and O. Staffans, Volterra integral and functional equations, Cambridge University Press, Vol. 34, 1990.

[13] I. Gyori, and D.W. Reynolds, “On admissibility of the resolvent of discrete Volterra equations”, J. Difference Equ. Appl. 16, 1393‒1412 (2010).

[14] D. Hinrichsen, and N.K. Son, “Stability radii of positive discrete- time equations under affine parameter perturbations”, Internat. J. Robust Nonlinear Control 8, 1169‒1188 (1988).

[15] V.B. Kolmanovskii, E. Castellanos-Velasco, and J. A. Torres-Munoz, “A survey: stability and boundedness of Volterra difference equations”, Nonlinear Anal. 53, 861‒928 (2003).

[16] J.J. Levin, and J.A. Nohel, “The integrodifferential equations of a class of nuclear reactors with delayed neutrons“, Arch. Ration. Mech. Anal. 31, 151‒172, (1968).

[17] N. Levinson, “A nonlinear Volterra equation arising in the theory of superfluidity”, J. Math. Anal. Appl. 1, 1‒11 (1960).

[18] W. Li, L. Panga, H. Sua, and K.Wang, “Global stability for discrete Cohen-Grossberg neural networks with finite and infinite delays”, Appl. Math. Lett. 25, 2246‒225 (2012).

[19] W.R. Mann, and F. Wolf, “Heat transfer between solids and gases under nonlinear boundary conditions“, Quart. Appl. Math. 9, 163‒184 (1951).

[20] P.H.A. Ngoc, and L.T. Hieu, “New criteria for exponential stability of nonlinear difference systems with time-varying delay”, Internat. J. Control 86, No. 9, 1646‒1651 (2013).

[21] P.H.A. Ngoc, T. Naito, J.S. Shin, and S. Murakami, “Stability and robust stability of positive linear Volterra difference equations”, Internat. J. Robust Nonlinear Control 19, 552‒568 (2008).

[22] Y.N. Raffoul, and Y.M. Dib, “Boundedness and stability in nonlinear discrete dystems with nonlinear perturbation”, J. Difference Equ. Appl. 9, 853‒862 (2003).

[23] Y. Song, and C.T.H. Baker, “Perturbation of Volterra difference equations”, J. Difference Equ. Appl. 10, 379‒397 (2004).

[24] T. Zhou, Y. Liu, X. Li, and Y. Liu, “A new criterion to global exponential periodicity for discrete-time BAM neural network with infinite delays”, Chaos Solitons Fractals 39, 332‒341 (2009).

Bulletin of the Polish Academy of Sciences Technical Sciences

The Journal of Polish Academy of Sciences

Journal Information

IMPACT FACTOR 2016: 1.156
5-year IMPACT FACTOR: 1.238

CiteScore 2016: 1.50

SCImago Journal Rank (SJR) 2016: 0.457
Source Normalized Impact per Paper (SNIP) 2016: 1.239


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 179 143 8
PDF Downloads 88 82 6