Another flux vector becomes a system of two waves and one, two or three stationary discontinuities depending on the dimension of the Euler equations.
The classical Ważewski theorem claims that the condition p ij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalities x ′ ( t ) + ∑ j = 1 n p i j ( t ) x ( t ) ≥ 0, x i (0) ≥ 0, i =1,..., n.
Then the q columns of C and the column of D corresponding to the vector (u_{n}) form a system of (q+1) linearly independent vectors of dimension q, which is impossible.
Ansari et al. [18] introduced and studied a system of vector equilibrium problems and a system of vector variational inequalities using a fixed point theorem.
A system of vector semiparametric nonlinear time series models is studied with possible dependence structures and nonstationarities in the parametric and nonparametric components.
Ansari et al. [2] considered a system of vector variational inequalities and obtained its existence results.
This concept of a system of vector quasi-equilibrium problems was first introduced and studied by Ansari et al. [1].
The Grushin space ({R}^{h+1}={(x,y):xin{R}^{h}, yin{R}}) is a Carnot-Carathéodory space with a system of vector fields X_{i}=frac{partial}{partial x_{i}},quad i=1,ldots,hquad mbox{and}quad Y= vert x vert ^{alpha}frac{partial}{partial y}, where (alpha>0) is a given real number and (vert x vert ) is the standard Euclidean norm of x.
A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if B is countable).
Our method slightly differs from theirs by relying on the Banach Grothendieck theorem and introducing tools from pseudodifferential operators, useful in our local setting of a system of complex vector fields with variable coefficients.
