Let (f:Hto{mathbf{R}}) be a continuous differentiable function.
Let f : H 1 → R be a continuous differentiable function.
Let (g C rightarrow mathbb{R}) be a continuous differentiable function.
Let (u(t),a(t),b(t),h(t)in C mathbf{R}_,mathbf{R}_)), (alpha(t)) be a continuous, differentiable and increasing function on ({mathbf{R}_) with (alpha(t)le t), (alpha(0)= 0).
Let (a(t), f_{1} (t),f_{2}(t), f_{3}(t)in C mathbf{R}_,mathbf{R}_)), and (a(t)) is a nondecreasing function, and let (alpha(t)) be a continuous, differentiable and increasing function on ({mathbf{R}_) with (alpha(t)le t), (alpha(0)= 0).
We conclude the main result in the following: Assume that (H1) holds and the electric potential V is a continuous differentiable function.
f ( y i ( k ) ) = [ f ( y i 1 ( k ) ), f ( y i 2 ( k ) ), …, f ( y i n ( k ) ) ] ∈ R n is a continuous differentiable vector function.
where x ∗ ∈ R n is the unique equilibrium point and T : R n → R n is a continuous differentiable nonlinear operator in R n, and lim ∥ x ∥ → ∞ sup ∥ T ′ ( x ) ∥ = λ < 1 holds for a positive constant λ (e.g., 0 < λ < 1 ).
Let (varepsilon in R), and define a family of curves x t)=x^{(t)}+varepsiloneta(t), (3.6) where (eta(t)) is a continuous differentiable function for all given, which satisfy the boundary conditions, i.e., eta(t_{0}) = 0. (3.7) Due to the changing terminal time (t_{f}), each has its own trajectory terminal point (t_{f}).
Suppose that (V t)in R^{1}) is a continuous differentiable and nonnegative function, which satisfies textstylebegin{cases} D^{alpha}V t)leq-a V t)+b V t-tau),& 0< alpha< 1 V t-taurphi(t)geq0,& tin[-tau, 0<, end{calpha<(6) where (tin[0, +infty)). If (a>b>0 ) for all (varphi(t)geq0, tau >0), then (lim_{trightarrow+infty}V(t)=0). ([34]).
We consider the variational inequality problem (VI for abbreviation), which is to find a vector (x^inOmega) such that text{VI}(Omega,F quad bigl x-x^ bigl x-x^p}F bigr(x^{top})geq0, quadForall xinOmega, (1) where Ω is a nonempty, closed and convex subigl of ({mathcal {R}}^{n}) and F is a continuous differentiable mapping from ({mathcal {R}}^{n}) into ({mathcal {R}}^{n}).
