On stability and boundedness of solutions of certain non autonomous fourth-order delay differential equations (Q2800733)

The authors consider a nonlinear differential equation of fourth order with constant delay NEWLINE\[NEWLINE\left(g(x)x''\right)''+a(t)\left(p(x)x''\right)'+b(t)\left(q(x)x'\right)'+c(t)f(x)x'+d(t)h(x(t-r))=0,\tag{1}NEWLINE\]NEWLINE where \(r\) is a positive constant, the functions \(a\), \(b\), \(c\) and \(d\) are continuously differentiable, the functions \(f\), \(g\), \(h\), \(p\) and \(q\) are continuous. It is also assume that the derivatives \(g'(x)\), \(p'(x)\), \(q'(x)\), \(f'(x)\) and \(h'(x)\) exist and are continuous.NEWLINENEWLINEBy defining a new Lyapunov functional, the authors establish new sufficient conditions to guarantee the asymptotic stability of the zero solution of equation (1). They also give an example to illustrate the obtained result.