If the matrix such that, the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities, and is bounded, then the zero solution of (4.1) is stable.
If C is also bounded, then the existence of solutions of the variational inequality is guaranteed by the nonexpansivity of mapping (operatorname{Proj}_{C}^{H} I-rA)).
If the solution of the pendulum-like system (2.1) is bounded, then the functions, where belongs to a solution of (2.1), are uniformly continuous on.
Moreover, if satisfies the Lipschitz condition in and we can find a matrix such that (2.59). is bounded, then this solution is defined on and we have the expression (2.60).
More generally, we know that when all the solutions of an autonomous linear finite dimensional system are bounded, then all these solutions are almost periodic.
If S is bounded, then with the initial data ξ, the solution has the property lim_{trightarrowinfty} biglVert x t,xi bigrVert leq C quad textit {a.s.}, where (C= sup_{xin S}| x|).
As in the proof of Theorem 3.1, we know that all the solutions of are uniformly bounded, then there exists a large enough such that all the solutions of belong to.
If is -subdifferentiable at each, and the ranges and are totally bounded, then there exists which is a solution of GMEPP (2.1).
We shall show that if r ¯ ≠ 0 and f is bounded, then (1.1) has at least one solution; see Theorem 3.1.
We prove that if the mappings F [0,T]×E→E and B [0,T]×E→L(H,E) satisfy suitable Lipschitz conditions and u0 is F0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in Cλ([0,T] D((−A)θ))) provided λ⩾0 and θ⩾0 satisfy λ+θ<12.
