Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous difference equations of monotone type

作者：Rouhani Behind Djafari^*; Khatibzadeh Hadi

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 72(3-4): 1570-1579.

DOI：10.1016/j.na.2009.08.043

• Second order difference equation
• Maximal monotone operator
• Asymptotic behavior
• Ergodic theorem

As a continuation of our previous work in Djafari Rouhani and Khatibzadeh (2008) [1], we investigate the asymptotic behavior of solutions to the following system of second order nonhomogeneous difference equations
{u(n+1) - (1 + theta(n))u(n) + theta(n)u(n-1) is an element of C(n)Au(n) + f(n) n >= 1 u(0) = a is an element of H, sup(n >= 0) vertical bar u(n)vertical bar < + infinity
where A is a maximal monotone operator in a real Hilbert space H, {c(n)} and {theta(n)} are positive real sequences and {f(n)} is a sequence in H. With suitable conditions on A and the sequences {c(n)}, {theta(n)} and {f(n)}. we show the weak or strong convergence of {u(n)} or its weighted average to an element of A(-1) (0), which is also the asymptotic center of the sequence {u(n)}, implying therefore in particular that the existence of a solution {u(n)} implies that A(-1) (0) not equal empty set. Our results extend some previous results by Apreutesei (2007, 2003. 2003) [13,23,24], Morosanu (1988, 1979) [4,20], and Mitidieri and Morosanu (1985/86) [31], whose proofs use the assumption A(-1) (0) not equal empty set, as well as the authors Djafari Rouhani and Khatibzadeh (2008) [1] (as mentioned there in the section on future directions), to the nonhomogeneous case with {theta(n)} not equal 1. We also present some applications of our results to optimization.

• 出版日期2010-2-1