Separation for the biharmonic differential operator in the Hilbert space associated with the existence and uniqueness theorem

AbstractIn this paper, we have studied the separation for the following biharmonic differential operator:Au=ΔΔu+V(x)u(x),x∈Rn, in the Hilbert space H=L2(Rn,H1) with the operator potential V(x)∈C1(Rn,L(H1)), where L(H1) is the space of all bounded linear operators on the Hilbert space H1 and ΔΔu is the biharmonic differential operator, while Δu=∑i=1n∂2u∂xi2 is the Laplace operator in Rn. Moreover, we have studied the existence and uniqueness of the solution of the biharmonic differential equationAu=ΔΔu+V(x)u(x)=f(x) in the Hilbert space H, where f(x)∈H

On the dynamics of the nonlinear rational difference equation xn+1 = Axn + Bxn−k + Cxn−l + bxn−k/dxn−k − exn−l

In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1=Axn+Bxn-k+Cxn-l+bxn-kdxn-k-exn-l,n=0,1,2,… where the coefficients A,B,C,b,d,e∈(0,∞), while k and l are positive integers. The initial conditions x-l,…,x-k,…,x-1,x0 are arbitrary positive real numbers such that k<l. Some numerical examples will be given to illustrate our results