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Khatibzadeh, Hadi

On the convergence of solutions to a second order difference equation with monotone operator. (English) Zbl 1241.39002

Numer. Funct. Anal. Optim. 32, No. 12, 1271-1282 (2011).
The author investigates the asymptotic behavior of solutions to the following second order difference equation \(u_{i+1}-2u_{i}+u_{i-1}\in c_{i}Au_{i}\), \(i\geq 1\), \(u_{0}=x\), \(\sup_{i\geq 0}|u_i| <+\infty \), where \(A\) is a maximal monotone operator in a real Hilbert space \(H\) and \(\{ c_i\}\) is a positive real sequence. The author shows weak and strong convergence of the solutions to an element of \(A^{-1}(0)\), under appropriate assumptions on \(A\) and the sequence \(\{c_i\}\). The author’s results extend and improve previous results by G. Morosanu [Numer. Funct. Anal. Optimization 1, 441–450 (1979; Zbl 0441.39005)] and E. Mitidieri and G. Morosanu [ibid. 8(1985/86), 419–434 (1986; Zbl 0628.39004)] (see also [G. Morosanu, Nonlinear evolution equations and applications. Transl. from the Romanian by Gheorge Morosanu. Mathematics and Its Applications: East European Series, 26. Dordrecht etc.: D. Reidel Publishing Company; Bucuresti: Editura Academiei (1988; Zbl 0696.35078)]) under more general assumptions on \(c_i\).
Reviewer: Bilender P. Allahverdiev (Isparta)

Cited in 6 Documents

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
47H05 Monotone operators and generalizations

Keywords:

asymptotic behavior; maximal monotone operator; second order difference equation; strong convergence; subdifferential; weak convergence; Hilbert space

Citations:

Zbl 0441.39005; Zbl 0628.39004; Zbl 0696.35078
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Full Text: DOI

References:

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