ON THE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS TO A SECOND ORDER DIFFERENCE EQUATION

Behzad Djafari Rouhani
Dept. of Mathematical Sciences, UTEP
behzad@math.utep.edu

We investigate the asymptotic behavior of solutions to the following second order difference equation

 
         +----
         |  
         |  U_(n+1) - 2U_n + U_(n-1) belongs C_n A U_n; n >= 1
         |  
         |  U_0 belongs H, sup_(n >= 0) | U_n | < +infinity
         |  
         +----
where A is a maximal monotone operator in a real Hilbert space H, and {Cn} is a positive real sequence. With suitable conditions on {Cn}, we show the convergence of {U_n} or its weighted average to an element of A^(-1) (0), implying in particular that the existence of a solution to (1) is equivalent to A^(-1) (0) != phi. Our results extend and give simpler proofs to previous results by several authors, whose proofs use the assumption A^(-1) (0) != phi.