Potentiality conditions for difference-differential equations. (English. Russian original) Zbl 0947.34067

Popov, A. M.

Differ. Equations 34, No. 3, 423-426 (1998); translation from Differ. Uravn. 34, No. 3, 422-424 (1998).

The short paper is devoted to studying the potentiality conditions for the second-order differential-difference equation of neutral type \[ N(u)\equiv F(x,u_{-\tau},u,u_{+\tau},u_{-\tau}', u',u_{+ \tau}', u_{-\tau}'',u'', u_{+\tau}'')=0, \] with \(u^{(k)}_{\pm\tau} =u^{(k)}(x \pm\tau)\), \(k=0,1,2\), \(\tau>0\) is a constant, and \(F\) is a thrice continuously differentiable function with respect to all arguments in some domain \(W\in\mathbb{R}^{10}\). The operator \(N\) is said to be potential with respect to the given bilinear form \(\langle v,g \rangle =\int_{x_1}^{x_2} v(x)g(x)\) if there exists a functional \(f_N:D(N) \to \mathbb{R}\) such that its Gateaux derivative is equal to \(\langle N(u), h\rangle\), where \(D(N)\) is the set of twice piecewise differentiable functions on \([x_1-\tau,x_2 +\tau]\). The author obtains necessary and sufficient Helmholtz conditions for potentiality of the operator \(N\). These results are applied to a linear differential equation with constant delay.

Reviewer: V.Chernyatin (Szczecin)

MSC:

34K40	Neutral functional-differential equations
34K06	Linear functional-differential equations

Keywords:

bilinear form; Gateaux derivative; constant delay; Helmholtz conditions; second-order differential-difference equation of the neutral type