Abstract
As well known, the existence and nonexistence of solutions for nonlinear algebraic systems are very important since they can provide the necessary information on limiting behaviors of many dynamic systems, such as the discrete reaction-diffusion equations, the coupled map lattices, the compartmental systems, the strongly damped lattice systems, the complex dynamical networks, the discrete-time recurrent neural networks, and the discrete Turing models. In this paper, both the existence of nonzero solution pairs and the nonexistence of nontrivial or nonzero solutions for a nonlinear algebraic system will be considered by using the critical point theory and Lusternik-Schnirelmann category theory. The process of proofs on the obtained results is simple, the conditions of theorems are also easy to be verified, however, some of them improve the known ones even if the system is reduced to the precial cases, in particular, others of them are still new.
Highlights
In this paper, the nonlinear algebraic system, Ax λf x, 1.1 will be considered, where λ > 0 is a parameter, x x1, x2, . . . , xn T, fx f1 x1, f2 x2, . . . , fn xn T1.2 are column vectors with fk is a continuous function defined on R and fk −u −fk u for u ∈ R and k ∈ {1, 2, . . . , n} 1, n, and n is a positive integer
Zhou et al 39 considered the following discrete-time recurrent neural networks, which is thought to describe the dynamical characteristics of transiently chaotic neural network: n vi t 1 kvi t wij uj t ai − wiia0i, i ∈ 1, n, 2.31
For the functional −H x, we ask that −H x ≥ 0 for large x, which implies that Fk xk ≤ 0 for k ∈ 1, n and large |xk|
Summary
The nonlinear algebraic system, Ax λf x , 1.1 will be considered, where λ > 0 is a parameter, x x1, x2, . . . , xn T , fx f1 x1 , f2 x2 , . . . , fn xn T. Negative, and strongly nonzero vector x are denoted by x > 0, x < 0 and x ∦ 0 respectively. If there exists k0 ∈ 1, n such that xk0 / 0, it will be called nontrivial solution of 1.1 In this case, it is denoted by x / 0. The existence of solution pairs for the nonlinear algebraic system 1.1 will be considered by using the critical point theory and Lusternik-Schnirelmann category theory 69 or 70.
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