Homoclinic solutions for a second order difference equation with p-Laplacian

Homoclinic solutions for a higher order difference equation withp-Laplacian

“…10 However, condition (AR) is not needed in this paper and there are many functions which do not 11 satisfy such a condition. 12 Below, we present an example of the function F which does dot satisfy condition (AR) but 13 does satisfy all the related conditions in this paper.…”

“…When n = 1, Corollary 3.1 reduces to[12, Theorem 3.1].13 4. Proofs of the main results14In this section, we prove our results.…”

Sequences of positive homoclinic solutions to difference equations with variable exponent

“…To study such problems on unbounded intervals directly by variational methods, [10] and [16] introduced coercive weight functions which allow to preserve of certain compactness properties on l p -type spaces. That method was used in the following papers [9,12,21,22,23,24,25].…”

“…The goal of the present paper is to establish sufficient conditions for the existence of a sequence of positive solutions to problem (1.1) which norm tend to infinity or zero. Infinitely many solutions for a constant {p k } were obtained in [25] by employing Nehari manifold methods, in [12,24] by applying a variant of the fountain theorem, in [21,22,23] by use of the Ricceri's theorem (see [3,20]) and in [22,23] by applying a direct argumentation. The existence of a nontrivial or infinitely many solutions to problem (1.1) with variable exponent were proved in [1,7,11] by the mountain pass theorem.…”

On the periodicity of a max-type rational difference equation

“…Although difference equations which appear naturally as discrete analogues in the numerical solutions of differential and delay differential equations have been predominantly studied so far (see, for example, [11,16,21] and the references therein), the study of nonlinear difference equations which are not discrete analogues of differential equations has been also of a great interest recently (see, for example, [7,9,13] and the references therein). Recently, so called, max-type difference equations have attracted some attention (see, for example, [6,17,20] and the references therein) because the max operator have great importance in automatic control models (see, [15,19]) and have wide applications in biology (see, [8]), ecology (see, [12,18]), and physics (see, [4,10]).…”