Abstract
A mixed variational inequality problem involving generalized Yosida approximation operator is considered and studied in -uniformly smooth Banach space. We have shown that mixed variational inequality problem involving generalized Yosida approximation operator is equivalent to a fixed-point equation. The fixed-point formulation is applied to establish an algorithm to obtain the solution of mixed variational inequality problem involving generalized Yosida approximation operator. Convergence criteria are also discussed. In support of our main result, we provide an example using Matlab program together with a computation table and convergence graphs. To check the validity of mixed variational inequality problem involving generalized Yosida approximation operator and its fixed-point formulation, we construct one more example.
1. Introduction
In 1959, Antonio Signorini posed the Signorini problem in the form of a variational inequality problem and solved by Gaetano Fichera in 1963, which was the first paper related to variational inequalities. After that, in order to study the regularity problem for partial differential equations, Guido Stampacchia proved a theorem called Lax–Milgram theorem and invented the name “variational inequality” for all the problems of the same nature involving the inequalities.
In fact, variational inequality is an inequality which embroils a functional, which is to be solved for all feasible values of a given variable, generally associated with a convex set. At first, this theory was developed to solve some equilibrium problems. In this problem, the functional involved was obtained as the first variation of the involved potential energy. That is why it has a variational origin.
Mixed variational inequality problem was originally studied by Lescarret [1] and Browder [2] due to their various applications. Konnov and Volotskaya [3] considered general equilibrium problems and oligopolistic equilibrium problems which can be formulated as mixed variational inequality problems. Mixed variational inequalities involve a nonlinear term, and due to this nonlinear term, all projection methods fail to solve the mixed variational inequalities. In this case, the technique of resolvent of a maximal monotone operator is applicable.
Many authors have shown that the applications of variational inequalities and its generalized forms can be found in fluid flow through porous media, elasticity, economics, transportation, regional and engineering sciences, odd-order obstacle problems, problems of lubrication, computational procedures, spatial price equilibrium problems, oligopolistic market equilibrium problems, environment network problems, etc., see [4–13] and references therein.
Moreau’s theorem demonstrates that convex functions are well behaved on Hilbert spaces and are differentiable. In this case, derivatives are approximated by Yosida approximation, which is explained with reference to the resolvent operator. Yosida approximation operator are applicable to solve first-order differential evolution inclusions, nonlinear nonhomogeneous evolution inclusions, heat and wave equations, etc.
If we choose Hilbert space structure to solve problems related to variational inequalities, then it is easy for us to use tools of functional analysis. That is, it is easy to use polarization identity, parallelogram law, and many other concepts. In view of applications of several problems, the Hilbert space structure is not much suitable. The Banach space structure is appropriate from the application point of view because of availability of the same geometric structure such as polarization identity and parallelogram law. Moreover, in Banach spaces, we can use the property of uniform convexity or uniform smoothness, duality mapping etc. For more details and applications, see [13–18].
Motivated by the abovementioned facts discussed above, in this paper, we introduce and study a mixed variational inequality problem involving generalized Yosida approximation operator in -uniformly smooth Banach space. An algorithm is suggested which is based on a fixed-point formulation to obtain the solution of our problem. An existence and convergence is proved. Two examples are constructed.
2. Basic Tools and Required Results
Let be a real Banach space with its norm , be its topological dual, be the pairing between and , be the metric induced by the norm, be the family of all nonempty closed and bounded subsets of , and be the Hausdorff metric on defined bywhere and .
The generalized duality mapping is defined bywhere is a constant. For , the generalized duality mapping coincides with normalized duality mapping. It is well known that , for all , and is single-valued if is strictly convex.
The modulus of smoothness of is the function defined by
A Banach space is called uniformly smooth if
is called -uniformly smooth if there exists a constant such that
The following important result of Xu [19] is instrumental to prove our main result.
Theorem 1. Let be a real uniformly smooth Banach space. Then, is -uniformly smooth if and only if there exists a constant such that, for all ,
Definition 1. Let be a functional. A vector is called subgradient of at ifThe subdifferential of denoted by is defined as
Definition 2. Let be a single-valued mapping and be a proper and subdifferential functional. The generalized proximal operator associated with and , for some , is defined as
Definition 3. The generalized Yosida approximation operator associated with proximal operator is defined aswhere is a constant.
Definition 4. Let be a mapping. Then,(i) is said to be Lipschitz continuous if there exists a constant such that(ii) is said to be accretive if(iii) is said to be strongly accretive if there exists a constant such that
Definition 5. A multivalued mapping is said to be -Lipschitz continuous if, for all , there exists a constant , such thatNow, we prove some results which are required in the sequel.
Proposition 1. Let be strongly accretive with constant , and the subdifferential operator of is accretive. Then, the generalized proximal operator is -Lipschitz continuous, that is,
Proof. Let and be any given points in . It follows from Definition 2 thatFrom the above, we haveSince is accretive, we obtainTherefore, we haveSince is strongly accretive with constant , we haveThus,
Proposition 2. Let be Lipschitz continuous with constant and the generalized proximal operator be Lipschitz continuous with constant . Then, the generalized Yosida approximation operator is -Lipschitz continuous, where .
Proof. Using Definition 3, we havewhere .
Proposition 3. Let be strongly accretive with constant and the generalized proximal operator is Lipschitz continuous with constant . Then, the generalized Yosida approximation operator is strongly accretive with constant , where .
Proof. Using Definition 3, we havewhere and .
3. Construction of the Problem and Fixed-Point Formulation
Let and be the single-valued mappings and be the multivalued mappings. Let be a functional on such that , where is the subdifferential of . Let be the generalized Yosida approximation operator. We consider the following problem.
Find , and such that
Problem (24) is called mixed variational inequality problem involving generalized Yosida approximation operator.
It is to be noted that, for appropriate choices of operators, one can obtain many variant forms of problem (24) available in the literature. That is, problems studied by Verma [20], Lee et al.[21], Ding [22], Ahmad et al. [23], Ahmad and Siddiqi [24], etc. can be obtained from problem (24).
The fixed-point formulation of mixed variational inequality problem involving generalized Yosida approximation operator is given below.
Lemma 1. The set of elements , where , and , is a solution of mixed variational inequality problem involving generalized Yosida approximation operator (24) if and only if the following equation is satisfied:
Proof. Let satisfy equation (25); then, we haveApplying the definition of generalized proximal operator , we obtainUsing the definition of subdifferential of , we obtain the required mixed variational inequality problem involving generalized Yosida approximation operator (24). That is,for all .
Based on Lemma 1, we construct the following algorithm for solving mixed variational inequality problem involving generalized Yosida approximation operator (24).
Algorithm 1. For given , , letSince , by Nadler [25], there exist such thatContinuing the above process inductively, we can obtain the sequences , and aswhere .
Theorem 2. Let be a real -uniformly smooth Banach space and , be the single-valued mappings such that is Lipschitz continuous with constant , be Lipschitz continuous with constant , be Lipschitz with constant , be Lipschitz continuous with constant , and be Lipschitz continuous with constant . Suppose is a subdifferentiable, proper functional satisfying , where is the subdifferential of .
Let be the generalized Yosida approximation operator such that is Lipschitz continuous with constant and is strongly accretive with constant . Let , be the multivalued mappings such that , and are -Lipschitz continuous mappings with constants , and , respectively. If the following condition is satisfied,where and is the same as in Theorem 1.
Then, the mixed variational inequality problem involving generalized Yosida approximation operator (24) has a solution.
Proof. Applying (31) of Algorithm 1, we haveUsing Theorem 1, we haveSince is strongly accretive with constant , we haveAs and are Lipschitz continuous mappings with constants and , respectively, we haveThus,Combining (38) and (40) with (37), we obtainwhich gives usUsing the Lipschitz continuity of , and , we obtainApplying Lipschitz continuity of and , -Lipschitz continuity of and using (32), (37), and (38) of Algorithm 1, we obtainCombining (43) and (44), we obtainCombining (42) and (43) with (36), we obtainwhereClearly, , as . It follows from (35) that , and hence, , for sufficiently large. From (46), it is clear that is a Cauchy sequence and .
Furthermore, we will show that , and . It follows from (32), (33), and (34) of Algorithm 1 and -Lipschitz continuity of , and thatFrom (48), (49), and (50), we obtain that , and are all Cauchy sequences in . Therefore, , and , as . We haveHence, . Similarly, we can show that and . This completes the proof.
4. Numerical Example
In support of concepts used in Theorem 2, we provide numerical example 1 using MATLAB R2018a with a computation table and convergence graphs. Thenceforward, in example 2, it is shown that mixed variational inequality problem involving generalized Yosida approximation operator (24) is satisfied together with its fixed-point formulation (25).
Example 1. Let and be the single-valued mappings such that, for all ,Then,(i) is Lipschitz continuous mapping with constant . That is,(ii) is Lipschitz continuous mapping with constant . That is, Also, is strongly accretive mapping with constant . That is,(iii)Suppose that be the operator such that, for all , Then, the subdifferential of , that is, .For , we calculate the generalized proximal operator and generalized Yosida operator associated with and :and It is easy to check that the generalized proximal operator and generalized Yosida operator are and -Lipschitz continuous, respectively, where and .(iv)The generalized Yosida approximation operator is strongly accretive with constant , for all ,(v) is strongly accretive with constant . As,therefore,(vi)Let be the single-valued mappings and be the multivalued mapping such thatAccordingly,Using (31) of Algorithm 1, we havewhich gives usIt is shown in Figures 1, 2, and 3 that, for initial values , the sequence converges to 0.137 997 5, respectively. In Figure 4, we combine all the graphs plotted in Figures 1–3 (Table 1).
Example 2. Let all the mappings be the same as considered in Example 1; then, we check the validity of mixed variational inequality problem involving generalized Yosida approximation operator (24) and its fixed-point formulation (25):We obtained in Example 1 that is one of the solutions of mixed variational inequality problem involving generalized Yosida approximation operator (24). For and , we obtainandFrom the above computation, we haveLet be another solution of mixed variational inequality problem involving generalized Yosida approximation operator (24); then,andThat is,Combining (69) and (72), we haveWe now verify the fixed-point formulation (25):For , we observe that the fixed-point formulation (25) is fulfilled:That is,
5. Conclusion
In this work, we study a new problem, that is, a mixed variational inequality problem involving a generalized Yosida apoproximation operator. A fixed-point formulation is provided in order to define an iterative algorithm.
Finally, an existence and convergence result is obtained for mixed variational inequality problem involving generalized Yosida approximation operator in -uniformly smooth Banach space. A numerical example is provided in support of concepts used in our main result, and another numerical example is furnished to check the validity of mixed variational inequality problem involving generalized Yosida approximation operator (24) and its fixed-point formulation (25).
Furthermore, we remark that our result can be extended in higher dimensional spaces.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.