On Solvability Of A Fourth Order Nonlinear Partial Difference Equation

This paper investigate the following fourth order nonlinear partial difference equation△2n(am·n△2m(xm·n+bm·nxm-t0·n-σ0))+f(m,n,xm-t1m·n-σ1,n,...,xm-tk,m·n-σk,n)=cm.n,m≥m0,n≥n0. where m0,n0∈N0,k,τ0,σ0∈N,{am.n｝(m.n)∈Nm(1.m),｛bm.n｝(m.n)∈m(1.m),｛cm.n｝(m.n)∈Nm(1.m)are real sequences with am.n≠0，bm.n≠±1for(m,n)∈Nm0.n0,f:Nm0.n0×Rk�'Rand｛τ1.m,σ1.n: m,n∈N0,ι∈｛1,2,...,｝｝∈Z with limm�'∞(m-τ1.m)=limn�'∞(n-σ1.n)=+∞,1∈｛1,2,...,｝.The first chapter states the importance of the study for the above equation.When nonlinear term take special linear term,the above equation grows many cases of the study in [1-28].The second chapter gives some related definitions and lemmas.The third chapter with the aid of the Banach fixed point theorem and some natural modification of the technique in the literature,this paper establishes the existence results of uncountably many bounded positive solutions for the above difference equation,put forward a few Mann type iterative approximation schemes and get several error estimates between the iterative approximations and the bounded positive solutions.The fourth chapter constructs four nontrivial examples to illustrate the main results.