Cayley Approximation Operator with an Application to a System of Set-valued Cayley Type Inclusions

abstract: In this paper, we introduce and study a system of set-valued Cayley type inclusions involving Cayley operator and (H,ψ)-monotone operator in real Banach spaces. We show that Cayley operator associated with the (H,ψ)-monotone operator is Lipschitz type continuous. Using the proximal point operator technique, we establish a fixed point formulation for the system of set-valued Cayley type inclusions. Further, the existence and uniqueness of the approximate solution is proved. Moreover, we suggest an iterative algorithm for the system of set-valued Cayley type inclusions and discuss the strong convergence of the sequences generated by the proposed algorithm. Some examples are constructed to illustrate some concepts used in this paper.


Introduction
It is well known that variational inequalities, complementarity problems and equilibrium problems are among most important and interesting problems in mathematical analysis. Inclusion problems were introduced and studied as a generalization of equilibrium problems. Many nonlinear problems arising in applied sciences such as signal processing, image recovery and machine learning, etc., can be modelled as an inclusion problem. In recent past, variational inclusion problems have been studied extensively by number of researchers due to their wide ranging applications to convex analysis, partial differential equations, optimization, game theory, industry, transportation, mathematical finance, nonlinear programming, economics, ecology, engineering sciences, etc., see; for example, [4,5,6,8,10,11,12,14,19,20,21,22] and references cited therein. Recently, Luo and Huang [16] and Kim et al. [15] introduced a new class of (H, φ) − η and (H, φ, ψ)-η-monotone operators, respectively in Banach spaces. These operators provide a unified framework for class of maximal monotone operators, maximal η-monotone operators, H-monotone operators and (H, η)-monotone operators. Using proximal point operator technique, they studied the convergence analysis of the iterative algorithms for some classes of variational inclusions. Very recently, Ali et al. [2] studied a Cayley inclusion problem involving XOR-operation. They defined a Cayley operator associated with a resolvent operator of a rectangular multi-valued mapping and studied convergence analysis of Cayley inclusion problem.
On the other hand, iterative computation of zeros or fixed points of nonlinear operators have been studied extensively in the literature, see; for example, [1,3,7,13,23,24,25,27]. Zhang et al. [26] introduced an iterative procedure for approaching a solution of the inclusion problem and a fixed point of a non expansive mapping in Hilbert spaces. Peng et al. [18] presented a viscosity algorithm for finding a solution of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem and a fixed point of a non expansive mapping. Motivated by the facts mentioned above, in this paper, we introduce and study an interesting class of inclusions, namely, system of set-valued Cayley type inclusions involving Cayley operator and (H, ψ)monotone operator in real Banach spaces. We show that Cayley operator associated with the (H, ψ)monotone operator is Lipschitz type continuous. Using proximal point operator technique, we establish a fixed point formulation for the system of set-valued Cayley type inclusions. Further, existence and uniqueness of the approximate solution is proved. Moreover, an iterative algorithm for the system of setvalued Cayley type inclusions is suggested to discuss the strong convergence of the sequences generated by the proposed algorithm.

Preliminaries
Now, we mention some definitions, notations and conclusions which are needed in the sequel. Let E be a real Banach space, E * be the topological dual of E, with its norm · and d be a metric induced by the norm · . Let ·, · be the dual pair between E and E * and CB(E)(respectively, 2 E ) be the family of all nonempty closed and bounded subsets (respectively, all nonempty subsets) of E and D(·, ·) be the Hausdorff metric on CB(E) defined by The normalized duality mapping J 2 : E → 2 E * is defined by If E ≡ H, a real Hilbert space, then J 2 becomes the identity mapping on E.
It is known that uniformly convex Banach spaces are reflexive and strictly convex.
is called the modulus of smoothness of E and defined by Lemma 2.1. [9] Let E be a uniformly smooth Banach space and J : E → 2 E * be a normalized duality mapping. Then Lemma 2.2. [17] Let E be a complete metric space with metric d and T : E → CB(E) be a multi-valued mapping. Then for any ǫ > 0 and for any x, y ∈ E, u ∈ T (x); there exists v ∈ T (y) such that and the equality holds if and only if x = y; (iii) δ g -strongly monotone, if there exists a constant δ g > 0 such that (iv) Lipschitz continuous, if there exists a constant λ g > 0 such that (v) k-strongly accretive, if there exists a constant k > 0 such that (ii) M is said to be H-monotone, if M is monotone and (iii) N is said to be Lipschitz continuous in the first argument, if there exists a constant α 1 > 0 such that N (x, ·) − N (y, ·) ≤ α 1 x − y , ∀x, y ∈ E; (iv) N is said to be Lipschitz continuous in the second argument, if there exists a constant α 2 > 0 such that Proof. For any given x * ∈ E * , let u, v ∈ (H + λψ • M ) −1 (x * ). Then, we have It follows from monotonicity of (ψ • M ) that Since H is strictly monotone, we have It follows from ( is called proximal point operator associated with (H, ψ)-monotone mapping, where λ > 0 is a constant.
Proof. Let x * , y * ∈ E * , then it follows from (2.3) that which implies that This completes the proof.
Thus, ψ • M is a monotone mapping. It is easy to see that Thus, the proximal point operator R H,λ M,ψ is 1 n -Lipschitz continuous, for n = 1, 2, 3.

Formulation of the System of Set-valued Cayley Type Inclusions and Convergence Result
This section begins with the formulation of a system of set-valued Cayley type inclusions and we discuss the existence of unique solution.
Let E be a uniformly smooth Banach space with its dual E * , for each be the set-valued mappings. Let H : E → E * be a strongly monotone mapping and M : E ⇒ 2 E * be an (H, ψ)-monotone mapping. We consider the following system of set-valued inclusions. Find (3.1) We call the system (3.1), the system of set-valued Cayley type inclusions.  Let H : E → E * be a strongly monotone mapping and M : E ⇒ 2 E * be an (H, ψ)-monotone mapping.

Then the system of set-valued Cayley type inclusions (3.1) has a solution
if and only if it satisfies following fixed point problem: Similarly, we can prove that This completes the proof. x n+1

Algorithm 2.
For any arbitrary x 0 ∈ E, u 0 ∈ P (x 0 ), v 0 ∈ Q(x 0 ), w 0 ∈ T (x 0 ) and z 0 ∈ G(x 0 ), compute the sequences {x n }, {u n }, {v n }, {w n }, {z n } by the following iterative scheme: and Assume that there exist constants λ 1 , λ 2 > 0 satisfying where, In addition the following condition holds: Then Proof. It follows from Algorithm 1, Theorem 2.8 and Condition (3.7) that Since g is k-strongly accretive and Lipschitz continuous with constant µ, then from Lemma 2.1, we have Cayley Approximation Operator with an Application 9 which implies that Using the Lipschitz continuities of H and g, we get (3.10) Since ψ 1 is ζ 1 -Lipschitz continuous with respect to N 1 (·, ·) and N 1 (·, ·) is (α 1 , β 1 )-Lipschitz continuous with respect to first and second argument, respectively, P 1 and Q 1 are D-Lipschitz continuous with constants δ P1 and δ Q1 , respectively, then we have Also, T 1 is D-Lipschitz continuous with constant δ T1 , then we have Thus, from (3.8)-(3.13), we get (3.14) Again, using the fact that g is k-strongly accretive and Lipschitz continuous with constant µ, we have (3.15) Since ψ 2 is Lipschitz continuous with constant ζ 2 with respect to N 2 (·, ·) and N 2 (·, ·) is (α 2 , β 2 )-Lipschitz continuous with respect to first and second argument, respectively, P 2 and Q 2 are D-Lipschitz continuous with constant δ P2 and δ Q2 , respectively, then we have Since H and g are Lipschitz continuities with constant s and µ, respectively, then we have Again, it follows from the fact that C H,λ2 M,ψ 2 is 2 + γ γ -Lipschitz continuous and G 2 is D-Lipschitz continuous with constant δ G2 , then we have Also, T 2 is D-Lipschitz continuous with constant δ T2 , then we have Thus, from (3.15)-(3.19), we have It follows from (3.14) and (3.20) that where, We see that Θ(P n ) → Θ(P ) as n → ∞, where Θ(P ) = ∆+ l 1 + l 2 + ξ 2 k δ T2 and ∆ = 1 − 2k + 64cδ 2 + It follows from the condition (3.6) that 0 < Θ(P ) < 1, and consequently by (3.22), {x n 1 } is a Cauchy sequence in E. Similarly by (3.20) and (3.22), it follows that {x n 2 } is also a Cauchy sequence in E. Since E is complete, then there exist x 1 , x 2 ∈ E such that x n 1 → x 1 and x n 2 → x 2 as n → ∞. It follows from Algorithm 1 that is a solution of the system of set-valued Cayley type inclusions (3.1). Next, we show that which shows that d(u i , P i (x 1 )) = 0, and hence u i ∈ P i (x 1 ). Similarly, one can show that v i ∈ Q i (x 1 ), w i ∈ T i (x 1 ) and z i ∈ G i (x 1 ), respectively. Now, we prove the uniqueness of the solution ( ) be another solution of the system of set-valued Cayley type inclusions (3.1), then it follows from Theorem 3.2 that Now following the same arguments as mentioned from (3.8)-(3.22), we have (3.29) Since 0 < Θ(P ) < 1, thus we have is unique solution of the system of set-valued Cayley type inclusions (3.1).
Thus, the proximal point operator R H,λ M,ψ is 1 n -Lipschitz continuous, for n ≤ 120.

Concluding Remarks
In this paper, we considered and studied a system of set-valued Cayley type inclusions involving Cayley operator and (H, ψ)-monotone operator in real Banach spaces, which includes many inclusion problems studied in the literature as special cases. We proved that Cayley operator associated with the (H, ψ)-monotone operator is Lipschitz type continuous. Existence and uniqueness of the approximate solution is proved. Moreover, we suggested an iterative algorithm for the system of set-valued Cayley type inclusions and the strong convergence of the sequences generated by the proposed algorithm is discussed.