Solving System of Mixed Variational Inclusions Involving Generalized Cayley Operator and Generalized Yosida Approximation Operator with Error Terms in q -Uniformly Smooth Space

: In this paper, we solve a system of mixed variational inclusions involving a generalized Cayley operator and the generalized Yosida approximation operator. An iterative algorithm is suggested to discuss the convergence analysis. We have shown that our system admits a unique solution by using the properties of q -uniformly smooth Banach space, and we discuss the convergence criteria for sequences generated by iterative algorithm. Two examples are constructed, and an application is provided.


Introduction
By using the clever transformation, Baiocchi [1] investigated the fact that the free boundary value problem associated with seepage through an earth dam is equivalent to a class of variational inequality.Variational inequalities have a tremendous impact in this field, and consequently variational inequalities are applied rather than other methods.
The study of variational inequality theory is twofold.On the one hand, it reveals the basic facts regarding the qualitative behaviour of solutions related to many nonlinear boundary value problems.On the other hand, it produces effective numerical methods to solve free and moving boundary value problems (see for more details [2][3][4][5]).
It is well known that projection methods are not applicable to solve variational inclusion problems and such an issue was solved by Hassouni and Moudafi [6] by using the resolvent operator technique.Generalized resolvent operators were introduced by several authors by using accretive operators, H-accretive operators, m-accretive operators, etc. (see [7][8][9]).The system of variational inclusions can be regarded as a natural extension of the system of variational inequalities.Many problems related to mathematical and convex analysis, biological sciences, image recovery processes, biomedical sciences, elasticity, data compression, mechanics, computer programming, and mathematical physics, etc., can be worked out by using the framework of system of variational inclusions.Due to their applications, system of variational inclusions (inequalities) were considered and studied by many authors, that is, by Pang [10], Cohen and Chaplais [11], Bianchi [12], Ansari and Yao [13], Yan et al. [14], etc.For more details on variational inclusions and their systems, we refer to [15][16][17][18][19][20][21][22][23][24][25][26][27] and references therein.
It is well-known that monotone operators can be structured into single-valued Lipschitzian monotone operators through a process known as the Yosida approximation process.The applications of the Yosida approximation operator can be found while dealing with wave equations, heat equations, linearized equations of coupled sound, and heat flow, etc.
Compared with other normed spaces, Banach spaces have the advantage that it is easy to obtain the convergence of a sequence of vectors.That is why we choose a q-uniformly smooth Banach space in order to achieve better results.
Conjoining the above facts, in this paper, we study a system of mixed variational inclusions which involve a generalized Cayley operator and a generalized Yosida approximation operator in a q-uniformly smooth Banach space.To obtain convergence result, we define an algorithm with error terms to take into account inexact computation.We prove that our system admits a unique solution, and we discuss convergence criteria for sequences generated by algorithm.
In support of our system, we provide an example.Moreover, another example is constructed to show that the generalized Cayley operator is Lipschitz-continuous and the generalized Yosida approximation operator is Lipschitz-continuous, as well as strongly accretive.Lipschitz continuity of both the operators is shown in Figures in Section 5, respectively.An application is also given.

Fundamental Concepts
Let Ẽ be real Banach space with its topological dual Ẽ * .The norm on Ẽ is denoted by • , duality pairing between Ẽ and Ẽ * by •, • and the class of subsets of Ẽ by 2 Ẽ.
It is well known that generalized duality mapping J q : Ẽ → 2 Ẽ * , is defined by For q = 2, generalized duality mapping reduces to normalized duality mapping.Note that if Ẽ is uniformly smooth then J q is single-valued.
A Banach space Ẽ is called q-uniformly smooth if where q > 1, k > 0 are constants and ρ Ẽ(t) is the modulus of smoothness.
The following result of Xu [35] is important to prove the main result.

Lemma 1.
Let Ẽ be a real, uniformly smooth Banach space.Then Ẽ is q-uniformly smooth if and only if there exists a constant C q such that for all x, y ∈ Ẽ, x + y q ≤ x q + q y, J q (x) + C q y q .
Now, we prove that the generalized Cayley operator is Lipschitz-continuous and the generalized Yosida approximation operator is strongly accretive as well as Lipschitzcontinuous.
Proof.For any x, y ∈ Ẽ and using the Lipschitz continuity of R M P,λ and P, we evaluate Thus, That is, the generalized Cayley operator is λ C -Lipschitz-continuous.Proof.(i) For any x, y ∈ Ẽ, λ > 0 and using the Lipschitz continuity of P and R M P,λ , we have that is, Thus, the generalized Yosida approximation operator Y M P,λ is λ Y -Lipschitz-continuous.(ii) For any x, y ∈ Ẽ and using the Lipschitz continuity of R M P,λ , we have Thus, the generalized Yosida approximation operator Y M P,λ is δ Y -strongly accretive.
Find (x, y) For suitable choices of operators involved in system (1), one can obtain many previously studied systems of variational inclusions and variational inequalities.
The following Lemma ensures the equivalence between system of mixed variational inclusions involving generalized Cayley operator and generalized Yosida approximation operator (1) and a system of equations.Lemma 3. The system of mixed variational inclusions involving generalized Cayley operator and the generalized Yosida approximation operator (1) admits a solution (x, y) ∈ Ẽ × Ẽ if and only if the following equations are satisfied: Proof.From Equation (2), we have By using the definition of generalized resolvent operator R M P 1 ,λ , we obtain ).In a similar way, by using Equation (3) and definition of generalized resolvent operator R N P 2 ,ρ , we obtain 0 ∈ Y M P 2 ,ρ (B(y)) + N(g(x)).
Theorem 2. Let Ẽ be a q-uniformly smooth Banach space.Let A, B, g, P 1 , P 2 : Ẽ → Ẽ be singlevalued mappings such that A is λ A -Lipschitz-continuous, B is λ B -Lipschitz-continuous and ξ B -expansive, g is λ g -Lipschitz-continuous, P 1 is λ P 1 -Lipschitz-continuous and r 1 -strongly accretive, P 2 is λ P 2 -Lipschitz-continuous and r 2 -strongly accretive.Suppose that M, N : Ẽ → 2 Ẽ be multi-valued mappings and generalized resolvent operators R M P 1 ,λ and R N P 2 ,ρ are 1 r 1 and 1 r 2 -Lipschitz-continuous, respectively.Suppose that generalized Cayley operator C M P 1 ,λ is λ C -Lipschitzcontinuous, generalized Yosida approximation operator Y N P 2 ,ρ is λ Y -Lipschitz-continuous and δ Ystrongly accretive with respect to B. Suppose that the following conditions are satisfied: where λr .Then, the system of mixed variational inclusions involving generalized Cayley operator and the generalized Yosida approximation operator (1) admits a unique solution.

That is,
By Lemma 3, we conclude that (x, y) is the unique solution of system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator (1).

Algorithm and Convergence Result
An algorithm is designed to establish convergence result for system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator (1).
By using Lemma 3, we suggest the following iterative algorithm for solving system (1).Now, we prove convergence of sequences {x n } and {y n } generated by Algorithm 1.
For initial points x 0 , y 0 ∈ Ẽ, let For next iterative points x 1 , Continuing in the same manner, compute sequences {x n } and {y n } by the scheme: where λ, ρ > 0 are constants, {e n } ∈ Ẽ and {r n } ∈ Ẽ are sequences introduced as error terms for inexact computation such that lim n→∞ e n = 0 = lim n→∞ r n and {α n } is a sequence such that Theorem 3. Let all the conditions of Theorem 2 be satisfied and additionally if the following conditions are satisfied: (iv) g is δ g -strongly accretive, then sequences {x n } and {y n } defined by Algorithm 1 converge strongly to x and y, respectively, where (x, y) is the unique solution of system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator (1).
Proof.It follows from Theorem 2 that the system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator (1) has a unique solution (x, y) ∈ Ẽ × Ẽ.By Lemma 3, we have and By using ( 14) of Algorithm 1, ( 16) and Lipschitz continuity of g, R M P 1 ,λ , P 1 , A, C M P 1 ,λ , we evaluate By using (15) of Algorithm 1, ( 17) and Lipschitz continuity of g, R N P 2 ,ρ , P 2 , B, we evaluate By using the same arguments as for (7), we have By combining ( 19) and (20), we have By accretiveness of g with constant δ g , we have which implies that Similarly, From ( 18) and ( 22), we have From ( 21) and ( 23), we have + 1 Adding ( 24) and ( 25), we have where θ = Because λ g ≤ δ g and δ g ≥ 1, (26) becomes where Applying condition (i), we have . Thus, all the conditions of Lemma 2 are satisfied.

Numerical Example
We construct following example in support of system (1).It is shown that (0, 0) is the unique solution of system (1).
For λ = 1 and ρ = 1, we evaluate the generalized resolvent operators x, for all x ∈ R.
Consequently, the generalized Cayley operators and the generalized Yosida approximation operators are calculated.
and Y N P 2 ,ρ (x) = x, for all x ∈ R.
In continuation of Example 1, we construct another example showing that the generalized Cayley operator is Lipschitz continuous and generalized Yosida approximation operator is Lipschitz-continuous as well as strongly accretive.Lipschitz continuity for both operators is also shown by graphs.
Example 2. Let E = R and all the mappings remain same as in Example 1.That is, x, Because P 1 (x) = It is a well-known fact that for a Lipschitz-continuous function, there exists a double cone whose origin can be moved along the graph so that the whole graph always stays outside the double cone.The following figures (Figures 1 and 2) demonstrate the Lipschitz continuity of generalized Cayley operator and generalized Yosida approximation operator calculated above, respectively.

Application
A dynamical system is a system that changes over time according to a set of fixed rules and determine how one state of the system moves to another state.On the other hand, a dynamical system describes the disequilibrium adjustment processes which may produce important transient phenomenon prior to the achievement of steady state.Dynamical system is a generalization of classical mechanics where the equation of motion postulated directly and are not constrained to be Euler-Lagrange equations of a least action principle.
Dynamical system theory has been applied in the field of neuroscience, cognitive development, equation of motion, electronic circuits, chaotic system (double pendulum), etc.
As an application of system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator (1), we mention a system of resolvent dynamical systems.
It can be shown easily that by using the techniques of Noor [38], the Gronwall lemma and Lyapunov function, which are the trajectory of the solution of the system of resolvent dynamical systems (30), converge globally exponentially to the unique solution of system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator (1).

Conclusions
It is well known that the Cayley operator, the Yosida approximation operator, and a system of variational inclusions are application oriented.This paper is focused on the study of a system of mixed variational inclusions involving the generalized Cayley operator and the generalized Yosida approximation operator in q-uniformly smooth Banach space.We obtain the unique solution of our system, and we discuss the convergence criteria by suggesting an iterative algorithm.Two examples are provided with an application.
Our results can be extended further and may be useful for other scientists.