Indian J. Pure Appl. Math., Ahead of print c  Indian National Science Academy DOI: 10.1007/s13226-015-0162-3 INERTIAL PROXIMAL ALGORITHM FOR DIFFERENCE OF TWO MAXIMAL MONOTONE OPERATORS M. Alimohammady and M. Ramazannejad Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran, 47416-1468 e-mails: {m.alimohammady, m.ramzannezhad}@gmail.com (Received 30 November 2013; after final revision 24 February 2015; accepted 6 July 2015) In this note, a new algorithm is presented for finding a zero of difference of two maximal mono- tone operators T and S, i.e., T − S in finite dimensional real Hilbert space H in which operator S has local boundedness property. This condition is weaker than Moudafi’s condition on operator S in [13]. Moreover, applying some conditions on inertia term in new algorithm, one can improve speed of convergence of sequence. Key words : Maximal monotone operator; proximal point algorithm. 1. PRELIMINARIES Let H be a Hilbert space. The notation 〈., .〉 will be used for inner product in H × H and ‖.‖ for the corresponding norm. A set valued operator T : H → 2 H is said to be monotone if 〈x ∗ − y ∗ ,x − y〉≥ 0, ∀(x, x ∗ ), (y,y ∗ ) ∈ G(T ), wherein G(T ) := {(x, y) ∈ H × H ; y ∈ Tx} is graph of T . The domain of T is D(T ) := {x ∈ H ; T (x) = ∅}. A monotone operator T is called maximal monotone if its graph is maximal in the sense of inclu- sion. Associated with a given monotone operator T , the resolvent operator for T and parameter λ> 0 is J T λ := (I + λT ) −1 . The resolvent J T λ of a monotone operator T is a single valued nonexpansive map from Im(I + λT ) to H [5, Proposition 3.5.3]. Moreover, the resolvent has full domain precisely