GENERALIZED MONOTONE MAPPING AND RESOLVENT EQUATION TECHNIQUE WITH AN APPLICATION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The objective of this paper is to study generalized monotone mapping, which is the addition of cocoercive mapping and monotone mapping. First resolvent operator is obtained and discussion of its few properties. Then we give the resolvent equation associated with the resolvent operator and find a solution to a variational-like inclusion problem.

Recently, Sahu et al. [26] proved the existence of solutions for a class of nonlinear implicit variational inclusion problems in semi-inner product spaces, which is more general than the results studied in [27]. Very recently Luo and Huang [23], introduced and studied (H, ϕ)-ηmonotone mapping in Banch spaces. Bhat and Zahoor [4,5] introduced and studied (H, φ )-ηmonotone mapping in semi-inner product space. For the applications point of view we refer to see [7]- [10], [17,22,23], [25]- [29], [31,32]. The proposed work is impelled by the noble research works mentioned above. First we study the generalized monotone mapping which is the addition of cocoercive mapping and monotone mapping and call it H(., ., .)-ϕ-η-cocoercive mapping in semi-inner product spaces. Then, resolvent operator and its resolvent equation are obtained and discuss its few properties. In last existence and convergence results are obtained for a variational inclusion problem in 2-uniformly smooth Banach spaces. Our work is extension and refinement of some result. For details, see [7]- [10], [12]- [18], [22,23], [25]- [29], [31,32]. Definition 1.1. [24,26] Let us consider the vector space Y over the field F of real or complex The pair (Y, [., .]) is called a semi-inner product space.
"We observed that u 1 = [u 1 , u 1 ] 1/2 is a norm and we can say a semi-inner product space is a normed linear space with the norm. Every normed linear space can be made into a semi-inner product space in infinitely many different ways. Giles [11] had shown that if the underlying space Y is a uniformly convex smooth Banach space then it is possible to define a semi-inner product uniquely. For a detailed study and fundamental results on semi-inner product spaces, one may refer to Lumer [24], Giles [11] and Koehler [21]," [4]. Definition 1.2. [26,30] The Y be a Banach space, then (ii) There is a constant k > 0 such that for every v 1 , w 1 ∈ Y , the following inequality holds "Every normed linear space Y is a semi-inner product space (see [24]). Infact, by Hahn-Banach theorem, for each v 1 ∈ Y , there exists at least one functional Given any such mapping f : Y → Y * , we can verify that [w 1 , v 1 ] = w 1 , f v 1 defines a semi-inner product. Hence we can write the inequality (2.1) as The constant s is known as constant of smoothness of Y , is chosen with best possible minimum value", [26].

PRELIMINARIES
Let Y be a 2-uniformly smooth Banach space. Its norm and topological dual space is given by . and Y * , respectively. The semi-inner product [., .] signify the dual pair among Y and Y * .
In order to proceed the next, we recall some basic concepts, which will be needed in the subsequent sections.
Definition 2.2. [17] Let us consider the single-valued mappings Q, R, S : Y → Y , mapping η : Similarly we can define the Lipschitz continuity for H(., ., .) in regards second and third component.
The Range of (M) is given by The inverse of (M) is given by For any two set-valued mappings N and M, and any real number β , we define For a mapping A and a set-valued map M : where d(., .) is the induced metric on Y and CB(Y ) denotes the family of all nonempty closed and bounded subsets of X. Since assumption M 1 , M 2 hold, we have Since µ > γ, α > β , δ > 0, it follows that y − z ≤ 0. We get y = z, therefore (H(Q, R, S) + λ ϕoM) −1 is single-valued.
Proof. Let any given points y, z ∈ Y . From Since ϕoM is (m, η)-relaxed monotone in the first arguments, we have

Now, we have
Since assumption M 1 -M 3 hold and η is τ-Lipschitz continuous Hence, we get the required result.

FORMULATION OF THE PROBLEM AND EXISTENCE OF SOLUTION
Now we make an attempt to show that generalized H(., ., .)-ϕ-η-cocorecive operator under acceptable assumptions can be used as a powerful tool to solve variational inclusion problems.
It is given that Θ < 1, then {t k } is a Cauchy sequence in Banach space Y , then t k → t as k → ∞.
From (4.17), {u k } is also Cauchy sequence in Banach space Y , then there exist u such that u k → u.
From equation (4.5)-(4.7) and Algorithm 4.3, the sequences {v k } and {w k } are also Cauchy sequences in Y . Thus, there exist v and w such that v k → v and w k → w as k → ∞. Next we will prove that v ∈ V (u). Since v k ∈ V (u), then which gives d(v, V (u)) = 0. Due to V (u) ∈ CB(Y ), we have v ∈ V (u). In the same manner, we easily show that w ∈ W (u).