We prove the existence of weak solutions for discrete nonlinear system of Kirchhoff type. We build some Hilbert spaces with suitable norms. We define the notion of weak solution corresponding to the problem (1.1). The proof of the main result is based on a minimization method of an energy functional J. RESUMEN Probamos la existencia de soluciones débiles para sistemas discretos no-lineales de tipo Kirchhoff. Construimos algunos espacios de Hilbert con normas apropiadas. Definimos la noción de solución débil correspondiente al problema (1.1). La demostración del resultado principal se basa en un método de minimización de un funcional de energía J.

"We prove the existence of at least one weak nontrivial solutions for a discrete nonlinear two-point boundary-value problems of Kirchhoff type. The main existence results are obtained by using the technique of geometric mountain pass and the Ekelands variational principle."

We prove the existence of weak solutions for an anisotropic homoclinic discrete nonlinear system. Suitable Hilbert spaces and norms are constructed. The proof of the main result is based on a minimization method. We also extend the problem by using generalized penality and source functions.

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