We consider a class of nonlinear difference equations the fourth order, which extends some in literature. It is shown that solvable closed form explaining theoretically, among other things, solvability previously considered very special cases. also present applications main theorem through two examples, show results literature are not correct.

This work is devoted to study the existence and the regularity of solutions of two nonlinear problems of fourth order governed by p-biharmonic operators in nonresonance cases. In the first problem we establish the nonresonance part of the Fredholm alternative. In the second problem, nonresonance relative to the first eigenvalue is considered for p = 2 at the case where the nonresonance is betwe...

We consider a class of fourth-order nonlinear difference equations. The classification of nonoscillatory solutions is given. Next, we divide the set of solutions of these equations into two types: F+and F−-solutions. Relations between these types of solutions and their nonoscillatory behavior are obtained. Necessary and sufficient conditions are obtained for the difference equation to admit the...

In this paper we study the existence of solutions for the nonlinear elliptic system ⎪⎪⎨ ⎪⎪⎩ Δu−Δu+V1(x)u = fu(x,u,v), Δv−Δv+V2(x)v = fv(x,u,v), u,v ∈ H(R) x ∈ R , where V1(x) and V2(x) are positive continue functions. Under some assumptions on fu(x,u,v) and fv(x,u,v) , we prove the existence of many nontrivial high and small energy solutions by variant Fountain theorems. This generalizes the re...

The existence of multiple solutions for a class of fourth-order elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.

By fixed point theorem of a mixed monotone operator, we study boundary value problems to nonlinear singular fourth-order differential equations, and provide sufficient conditions for the existence and uniqueness of positive solution. The nonlinear term in the differential equation may be singular.