New technique in asymptotic stability for second order nonlinear delay integro differential equations
Abstract
The second order nonlinear integro-differential equation
x(t)+f(t,x(t),x(t))x(t)+∑_{j=1}^{N}∫_{t-τ_{j}(t)}^{t}a_{j}(t,s)g_{j}(s,x(s))ds=0,
with variable delays τ_{j}(t)≥0, 1≤j≤N, is investigated with low restrictions on the delays. Omitting assumptions such as differentiability on τ_{j} or invisibility of functions t-τ_{j}(t), makes the variation of parameters method difficult to apply to the equation. To circumvent the difficulties we choose conditions for f, a_{j}, g_{j} and we, carefully, amplify space of functions so that the equation takes a suitable form that facilitates the inversion of the equation into an equivalent one from which we derive a fixed point mapping. The end result is not only conditions for existence and uniqueness of solutions of the equation, but also for boundedness and stability of the zero solution of that equation. We also provide conditions that make zero solution asymptotically stable. The technique we use here avoids many difficulties which we often encounter in studying any class of second order nonlinear equations with variables delays and offers, what we hope, a new way to investigate the stability by fixed point theory. Our work extends and improves previous results in the literature such as, D. Pi pi1.
x(t)+f(t,x(t),x(t))x(t)+∑_{j=1}^{N}∫_{t-τ_{j}(t)}^{t}a_{j}(t,s)g_{j}(s,x(s))ds=0,
with variable delays τ_{j}(t)≥0, 1≤j≤N, is investigated with low restrictions on the delays. Omitting assumptions such as differentiability on τ_{j} or invisibility of functions t-τ_{j}(t), makes the variation of parameters method difficult to apply to the equation. To circumvent the difficulties we choose conditions for f, a_{j}, g_{j} and we, carefully, amplify space of functions so that the equation takes a suitable form that facilitates the inversion of the equation into an equivalent one from which we derive a fixed point mapping. The end result is not only conditions for existence and uniqueness of solutions of the equation, but also for boundedness and stability of the zero solution of that equation. We also provide conditions that make zero solution asymptotically stable. The technique we use here avoids many difficulties which we often encounter in studying any class of second order nonlinear equations with variables delays and offers, what we hope, a new way to investigate the stability by fixed point theory. Our work extends and improves previous results in the literature such as, D. Pi pi1.
DOI: http://dx.doi.org/10.14510%2Flm-ns.v0i0.1419