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It is well known that the variational inclusions are equivalent to the fixed point problems. A hybrid proximal point algorithm is considered for solving the general variational inclusions. And convergence analysis for this algorithm is explored along with some results on the resolvent operator corresponding to(AA, ηη)-maximal operators. Introduction and preliminaries Variational inequalities and variational inclusions are among the most interesting and important mathematical problems and have been studied intensively in the past years, since they have wide applications in the optimization and control, economics and transportation equilibrium, engineering science. For these reasons, many existence results and iterative algorithms for various variational inclusions have been studied. For details, we refer the reader to [1-10]. Pennanen [11] over-relaxed the Eckstein-Bertsekas proximal point algorithm [12] and has shown that the sequence converges linearly to a solution to the following variational inclusion problem: for finding x ∈ H ,such that 0 ∈ M(u) , where H is Hilbert space,M: H → 2HH is a set-valued mapping. On the basis of this new version of the proximal point algorithm, Pennanen [11] studied a localized version of the maximal monotonicity, and has shown that it ensures the local convergence of the over-relaxed proximal point algorithm. Furthermore, the local convergence of multiplier methods for a general class of problems is established. This, in a way, presents specializations as new convergence results for multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming. Recently Verma [13] develop a hybrid version of the Eckstein-Bertsekas proximal point algorithm based on the notions of A-maximal monotonicity [4] and (AA, ηη)-maximal monotonicity [14] for solving the variational inclusion problem (1). These notions generalize the general class of maximal monotone set-valued mappings, including the notion of H-maximal monotonicity introduced by Fang and Huang [15] in a Hilbert space. Motivated and inspired by the research work going on this field, in this paper, we extend the hybrid proximal point algorithm to solve the general variational inclusion. Let H be a real Hilbert space with the norm ‖. ‖ and the inner product 〈. , . 〉 For given nonlinear operators TT,gg:HH → HH and an (AA, ηη)-maximal monotone mapping M: H → 2HH , we consider the problem of finding uu ∈ HH such that 0 ∈ TTuu + MM(gg(uu)) (1) which is called the general variational inclusion. Variational inclusions have been studied and considered by many authors. For g=I, the identity operator, problem (2) is equivalent to finding u ∈ H such that 0 ∈ TTuu + MM(uu) (2) which is known as the variational inclusion problem. If M(. ) = ∂φ(. ), where φ(. ) is the subdifferential of a proper, convex and semi-continuous function φ: H → R ∪ {+∞} , then problem (2) reduces to finding uu ∈ HH such that 4th International Conference on Mechatronics, Materials, Chemistry and Computer Engineering (ICMMCCE 2015) © 2015. The authors Published by Atlantis Press 1060 0 ∈ TT(uu) + ∂∂∂∂gg(uu) (3) or equivalently, finding uu ∈ HH,gg(uu) ∈ HH, such that 〈TTuu,gg(vv) − gg(uu)〉 + ∂∂gg(vv) − ∂∂gg(uu) ≥ 0,∀gg(vv) ∈ HH (4) Definition 1.1 Let MM:HH → 2HH be a multivalued mapping, and FF:HH → HH be another operator. The map M is said to be : (i)(r)-strongly monotone if there exists a positive constant r such that: ∀uu, vv ∈ HH,uu∗ ∈ MM(uu), vv∗ ∈ MM(vv) 〈uu∗ − vv∗,uu − vv〉 ≥ rr‖uu − vv‖2 (5) (ii)(m)-relaxed monotone if there exists a positive constant m such that:∀uu, vv ∈ HH,uu∗ ∈ MM(uu), vv∗ ∈ MM(vv) 〈uu∗ − vv∗,uu − vv〉 ≥ (−mm)‖uu − vv‖2 (6) (iii)(r)-strongly monotone with respect to F if there exists a positive constant r such that: ∀uu, vv ∈ HH,uu∗ ∈ MM(uu), vv∗ ∈ MM(vv) 〈uu∗ − vv∗,FFuu − FFvv〉 ≥ rr‖uu − vv‖2 (7) Definition 1.2 Let MM:HH → 2HH be a multivalued mapping on H, and let ηη:HH × HH → HH be another mapping. The map M is said to be (rr, ηη)-strongly monotone if there exists a positive constant r such that ∀uu, vv ∈ HH,uu∗ ∈ MM(uu), vv∗ ∈ MM(vv) 〈uu∗ − vv∗, ηη(uu, vv)〉 ≥ rr‖uu − vv‖2 (8) Where ηη is said to be (ττ)-Lipschitz continuous if there is a positive constant ττ such that ηη‖(uu, vv)‖ ≤ ττ‖uu − vv‖. (9) Definition 1.3 ([4]) Let AA:HH → HH be (rr)-strongly monotone. The map MM → 2HH is said to be A-maximal monotone if (i) M is (mm)-relaxed monotone, (ii)RR(AA + ρρMM) = HH, for ρρ > 0 Definition 1.4 Let AA:HH → HH be (rr, ηη)-strongly monotone. The map MM:HH → 2HH is said to be (AA, ηη)-maximal monotone if (i) M is (mm, ηη)-relaxed monotone, (ii) RR(AA + ρρMM) = HH for ρρ > 0 Definition 1.5 Let AA:HH → HH,be an (rr, ηη)-strongly monotone mapping and let MM:HH → 2HH be an (AA, ηη) -maximal monotone mapping. Then the generalized resolvent operator JJρρ,AA :HH → HH is defined by JJρρ,AA (uu) = (AA + ρρMM)−1(uu). (10) Proposition 1.1 ([14]). Let AA:HH → HH be an (rr, ηη)-strongly monotone single-valued mapping and let MM:HH → 2HH be an (AA, ηη)-maximal monotone mapping. Then AA + ρρMM is maximal monotone for ρρ > 0. Proposition 1.2 Let AA:HH → HH be an (rr, ηη)-strongly monotone mapping and let MM:HH → 2HH be an (AA, ηη)-monotone mapping. Then the operator (AA + ρρMM)−1 is single-valued. Hybrid proximal point algorithms This section deals with a hybrid proximal point algorithm for the over-relaxed version of the Eckstein-Bertsekas proximal point algorithm [12] and its application to approximation solvability of the inclusion problem (2) based on the (AA, ηη)-maximal monotonicity. Lemma 2.1 ([14]) Let H be a real Hilbert space. Let AA:HH → HH be an (rr, ηη)-strongly monotone, MM:HH → 2HH be an (AA, ηη)-maximal monotone, and ηη:HH × HH → HH be (ττ)-Lipschitz continuous. Then the generalized resolvent operator associated with M and defined by JJρρ,AA (uu) = (AA + ρρMM)−1(uu),∀uu ∈ HH, (11) is  ττ rr−ρρρρ -Lipschitz continuous. Furthermore, we have 〈JJρρ,AA (uu) − JJρρ,AA (vv),uu − vv〉 ≤ ττ rr−ρρρρ ‖uu − vv‖2. (12) Theorem 2.1 Let H be a real Hilbert space. LetAA:HH → 2HH be(rr, ηη)-strongly monotone, MM:HH →

Near Point Algorithm and Approximate Beam Method for Solving Variational Inclusion Problem