A VARIATIONAL APPROACH TO THE EXISTENCE OF INFINITELY MANY SOLUTIONS FOR DIFFERENCE EQUATIONS

Subject Areas : Statistics

M. Khaleghi Moghadam ¹ , S. Tersian ² , M. Avci ³

1 - Department of Basic Science, Sari Agricultural Sciences and natural Resources University, 578 Sari, Iran
2 - bDepartment of Mathematics, University of Ruse, Ruse, Bulgaria. Associate at Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113- Sofia, Bulgaria
3 - Department of Science, Grande Prairie Regional College, Grande Prairie, AB T8W 0A8, Canada

Received: 2018-09-04 Accepted : 2019-09-18 Published : 2020-01-21

Keywords: تئوری نقطه بحرانی, بینهایت جواب, مساله مقدار مرزی غیر خطی گسسته, روش تغییراتی,

Abstract :

The existence of infinitely many solutions for an anisotropic discrete non-linear problem with variable exponent according to p(k)–Laplacian operator with Dirichlet boundary value condition, under appropriate behaviors of the non-linear term, is investigated. The technical approach is based on a local minimum theorem for differentiable functionals due to Ricceri. We point out a theorem as a special case of our main results. We define two differentiable functionals and set up the variational framework and present an applied lemma which $lambda$ lying in a well-defined interval. Bearing in mind this fundamental lemma and the local minimum theorem due to Ricceri, we obtain our result which is the existence of a sequence of infinitely many solutions which converges to zero depending on the nonlinear term has suitable behavior at zero. We ensure exact interval of the parameter $lambda$, in which the anisotropic discrete non-linear problem admits infinitely solutions such that their norm converges to zero. Some remarks and corollaries and the proof of especial case theorem are provided. Some examples are inserted to illustrate the importance of the main results.

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