VARIATIONAL-LIKE INCLUSION INVOLVING INFINITE FAMILY OF SET-VALUED MAPPINGS GOVERNED BY RESOLVENT EQUATIONS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The purpose of this work is to investigate generalized H(., ., .)-φ-η-cocoercive operator and use its application via resolvent equation approach to solve the variational-like inclusion involving infinite family of set-valued mappings in semi-inner product spaces. We aim to establish an equivalence between the set-valued variational-like inclusion problem and fixed point problem. A relationship also obtain between the set-valued variational-like inclusion problem and the resolvent equation problem. This equivalent formulation suggests an idea to construct an iterative algorithm to find a solution of the resolvent equation problem.


INTRODUCTION
Variational Inequality theory is very important due to its large application in various problem e.g. partial differential equation and optimization problems, see [3]. Therefore it have been developed and generalized in numerous directions. Variational inclusions is a natural generalization of variational inequalities. Monotonicity have a very crucial role in the study of variational inclusions. Therefore researchers introduced and studied many types of monotonicity e.g. maximal monotone mapping, relaxed monotone mapping, H-monotone mapping, A-monotone mapping etc., and discussed the solvability of different variational inclusion problems with the help of underlying different monotone mappings, see [4,5], [7]- [9], [19,20], [22]- [24], [25,26].
The resolvent operator technique which is the generalized form of projection technique, is very efficient tool to solve variational inclusions and their generalizations. The resolvent equation is also a very significant approach. The resolvent operator equations technique is utilized to expand significant and feasible numerical approaches to find a the solution of many variational inequalities (inclusions) and linked optimization problems, see [1,2].
"Recently, Sahu et al. [23] 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 [24]. Moreover, they constructed an iterative algorithm for approximating the solution for the class of implicit variational inclusion problems involving A-monotone and Hmonotone operators by using the generalized resolvent operator technique. It is remarked that they discussed the existence and convergence analysis by relaxing the condition of monotonicity on the set-valued map considered", [4].
Very recently Luo and Huang [20], introduced and studied (H, ϕ)-η-monotone mapping in Banach spaces which provides a unifying framework for various classes of monotone mapping.
The considered work is motivated by the noble research works discussed above. First, we investigate the notion generalized H(., ., .)-ϕ-η-cocoercive operator which is the generalization of H(., ., .)-η-cocoercive operator [15,16]. Then we consider the variational inclusion involving infinite family of set-valued mappings. First, we obtain a relation between the variational-like inclusion and fixed point problem and also obtain a equivalance between the variational-like inclusion amd the resolvent operator equation involving generalized H(., ., .)-ϕ-η-cocoercive operator. These equivalant fixed point problem and the resolvent equation formulation suggest us an idea to develop an iterative algorithm. As an application of resolvent equation approach, we will solve the considered variational-like inclusion problem. The obtained results are quite similar to above discussed research work but we utilize distinguished notion and approach to solve variational inclusion problems in 2-uniformly smooth Banach space. Our work is the extension and refinement of the existing results, see [1,2,4,5,14,18,20,30].
"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 [10] 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" [4]. Remark 1.2. "This unique semi-inner product has the following nice properties: (iii) The semi-inner product is continuous, that is for each [4].
The real sequence space l p f or 1 < p < 1 is a semi-inner product space with the semi-inner product defined by Definition 1.4. [10,23] The real Banach space L p (Y, µ) for 1 < p < 1 is a semi-inner product space with the semi-inner product defined by Definition 1.5. [23,27] The Y be a Banach space, then (iv) Y be 2-uniformly smooth if there exists c > 0 such that ρ Y (s) ≤ cs 2 .
Lemma 1.6. [23,27] Let p > 1 be a real number and Y be a smooth Banach space. Then the following statements are equivalent: (ii) There is a constant k > 0 such that for every v 1 , w 1 ∈ Y , the following inequality holds Remark 1.7. "Every normed linear space Y is a semi-inner product space (see [21]). Infact, by 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", [23].
The function space L p is 2-uniformly smooth for p ≥ 2 and it is p-uniformly where the constant of smoothness is p − 1", [23].

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 * .
Definition 2.2. [15,16] Let us consider the single-valued mappings Q, R, S : Y → Y , mapping Similarly we can define the Lipschitz continuity for H(., ., .) in regards second and third com- 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 : Since assumption M 1 holds, we have Since assumption M 2 holds, 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. .
Since ϕoM is (m, η)-relaxed monotone in the first arguments, we have Since assumptions M 1 , M 2 , M 3 hold and η is τ-Lipschitz continuous, we have Hence, Hence, we get the required result.

APPLICATION
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.
Varaitionl inclusion problem type of (4.1), studied by Ahmad and Dilshad [1] and Wang [29] in the setting of real Banach space, .  a solution of problem (4.1) if and only if (v, a, (v 1 , v 2 , ...)) satisfies the following relation: The resolvent equation corresponding to generalized set-valued variational-like inclusion problem (4.1).
Next, we find the convergence of the iterative algorithm for the resolvent equation problem Then there exist q, v ∈ Y , a ∈ V (v) and v i ∈ W i (v) that satisfy the resolvent equation problem Proof. Using Algorithms 4.3 and λ V , β i -D Lipschitz continuity of V,W i , we have where k = 1, 2, .....
It is given that Θ < 1, then {q k } is a Cauchy sequence in Banach space Y , then q k → q as k → ∞. From (4.15), {v k } is also Cauchy sequence in Banach space Y , then there exist v such that v k → v.
From equation (4.5)-(4.6) and Algorithm 4.3, the sequences {v k i } and {a k } are also Cauchy sequences in Y . Thus, there exist v i and a such that v k i → v i and a k → a as k → ∞. Next we will prove that v i ∈ W i (v). Since v k i ∈ W i (v), then which gives d(v i , W i (v)) = 0. Due to W i (v) ∈ CB(Y ), we have v i ∈ W i (v), i = 1, 2, .... In the same manner, we easily show that a ∈ V (v).