Asymptotics of a Difference Equation and Zeroes of Monotone Operators
Behzad Djafari Rouhani, Hadi Khatibzadeh · Numerical Functional Analysis and Optimization · 2014
We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation: where A is a maximal monotone operator in a real Hilbert space H, and {c n } is a positive sequence of real numbers. We do not assume that A −1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being nonempty. These theorems provide new approximation results for zeros of monotone operators, as well as unify and extend previously known results in [2 N. C. Apreutesei ( 2003 ). On a class of difference equations of monotone type . J. Math. Anal. Appl. 288 : 833 – 851 .[Crossref], [Web of Science ®] , [Google Scholar], 3 N. C. Apreutesei ( 2003 ). Existence and asymptotic behavior for a class of second order difference equations . J. Difference Equ. Appl. 9 : 751 – 763 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 7 R. E. Bruck ( 1978 ). On the strong convergence of an averaging iteration for the solution of operator equations involving monotone operators in Hilbert space . J. Math. Anal. Appl. 64 : 319 – 327 .[Crossref], [Web of Science ®] , [Google Scholar], 10 B. Djafari Rouhani and H. Khatibzadeh ( 2008 ). A note on the asymptotic behavior of solutions to a second order difference equation . J. Difference Equ. Appl. 14 : 429 – 432 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 18 H. Khatibzadeh ( 2011 ). On the convergence of solutions to a second order difference equation with monotone operator . Numer. Funct. Anal. Optim. 32 : 1271 – 1282 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 22 E. Mitidieri and G. Morosanu ( 1985–1986 ). Asymptotic behavior of the solutions of second-order difference equations associated to monotone operators . Numer. Funct. Anal. Optim. 8 : 419 – 434 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 23 G. Morosanu ( 1988 ). Nonlinear Evolution Equations and Applications . Editura Academiei Romane (and D. Reidel Publishing) , Bucharest . [Google Scholar], 25 G. Morosanu ( 1979 ). Second order difference equations of monotone type . Numer. Funct. Anal. Optim. 1 : 441 – 450 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] by considering much weaker conditions on the coefficients {c n }. In particular, our new strong ergodic theorem extends the results of [7 R. E. Bruck ( 1978 ). On the strong convergence of an averaging iteration for the solution of operator equations involving monotone operators in Hilbert space . J. Math. Anal. Appl. 64 : 319 – 327 .[Crossref], [Web of Science ®] , [Google Scholar]] and [23 G. Morosanu ( 1988 ). Nonlinear Evolution Equations and Applications . Editura Academiei Romane (and D. Reidel Publishing) , Bucharest . [Google Scholar], Theorem 3.3] for first-order difference equations, to the case of second order difference equations, and implies also a new strong convergence theorem.