We introduce a class of mixed general bivariational inequalities in a real Hilbert space and show that several known models, including mixed variational inequalities, bivariational inequalities, general variational inequalities, and complementarity problems, arise as special cases. An auxiliary-principle framework is then used to derive predictor–corrector type iterative schemes. A basic descent estimate is established under a g-partially relaxed monotonicity assumption, and a convergence theorem is obtained under natural continuity and uniqueness hypotheses. The presentation has been streamlined to make the algorithmic steps explicit, and a scalar example is included to illustrate the resolvent formulation.
Variational inequality theory provides a unified framework for equilibrium, obstacle, optimization, and complementarity problems. Since the foundational works of Stampacchia [1] and Lions and Stampacchia [2], the subject has been extended in several directions to accommodate nonlinear operators, nonsmooth terms, and generalized constraints. For broad background on theory, applications, and numerical methods, we refer to [1– 17].
A useful extension of the classical variational inequality is the mixed variational inequality, where a nondifferentiable term is incorporated into the model. Such problems arise naturally in equilibrium theory, optimization, mechanics, and engineering applications. Because of the nonsmooth term, standard projection- and descent-type arguments are not always directly applicable. One fruitful alternative is the auxiliary principle technique introduced by Trémolieres, Lions, & Glowinski [6]; see also Noor [8, 12] and Zhu and Marcotte [17]. This technique has repeatedly proved useful for deriving iterative schemes with transparent convergence mechanisms.
In this paper we study a mixed general bivariational inequality involving a bifunction \(N(\cdot,\cdot)\), operators \(A\), \(T\), and \(g\), and a lower-semicontinuous function \(\varphi\). The formulation unifies several classes of variational inequalities and complementarity problems. We derive auxiliary-principle-based predictor–corrector methods and establish a Fejér-type descent estimate under a \(g\)-partially relaxed monotonicity condition. To keep the convergence analysis rigorous, the main convergence statement is formulated under explicit continuity, cluster-point, and uniqueness assumptions. We also include a scalar example that clarifies the resolvent interpretation of the proposed algorithm.
Let \(H\) be a real Hilbert space with inner product \(\left\langle \cdot,\cdot \right\rangle\) and norm \(\|\cdot\|\). Let \(K\) be a nonempty closed convex subset of \(H\), and let \(\varphi:H\to \mathbb{R}\cup\{+\infty\}\) be a proper function.
For a bifunction \(N:H\times H\to H\) and continuous operators \(A,T,g:H\to H\), we consider the problem of finding \(u\in H\) such that \[\label{eq:mainVI} \left\langle N(A(u),T(u)),g(v)-g(u)\right\rangle+\varphi(g(v))-\varphi(g(u))\ge 0,\qquad \forall v\in H. \tag{1}\]
We call (1) the mixed general bivariational inequality. When \(\varphi\) is proper, convex, and lower semicontinuous, (1) is equivalent to the inclusion \[\label{eq:mainIncl} 0\in N(A(u),T(u))+\partial\varphi(g(u)), \tag{2}\] where \(\partial\varphi\) denotes the subdifferential of \(\varphi\).
If \(A=I\) and \(T=I\), then (1) becomes \[\label{eq:sp1} \left\langle N(u,u),g(v)-g(u)\right\rangle+\varphi(g(v))-\varphi(g(u))\ge 0,\qquad \forall v\in H, \tag{3}\] which is a mixed general variational inequality [11].
If \(g=I\), then (1) reduces to \[\label{eq:sp2} \left\langle N(A(u),T(u)),v-u\right\rangle+\varphi(v)-\varphi(u)\ge 0,\qquad \forall v\in H, \tag{4}\] which is a mixed bivariational inequality.
If \(\varphi=I_K\) is the indicator function of a closed convex set \(K\), then (1) is equivalent to \[\label{eq:sp3} \left\langle N(A(u),T(u)),g(v)-g(u)\right\rangle\ge 0,\qquad \forall v\in K, \tag{5}\] which is a general bivariational inequality.
If, in addition, \(N(A(u),T(u))\equiv T(u)\), then (5) becomes \[\label{eq:sp4} \left\langle T(u),g(v)-g(u)\right\rangle\ge 0,\qquad \forall v\in K, \tag{6}\] which is the general variational inequality introduced in [7].
If \(K\) is a convex cone and \(K^{*}=\{w\in H:\left\langle w,v\right\rangle\ge 0\ \text{for all } v\in K\}\) is its polar cone, then (5) is equivalent to finding \(u\in H\) such that \[\label{eq:sp5} g(u)\in K,\qquad N(A(u),T(u))\in K^{*},\qquad \left\langle N(A(u),T(u)),g(u)\right\rangle=0, \tag{7}\] which is a general bicomplementarity problem [13].
If \(g=I\) in (7), then we obtain the bicomplementarity problem: find \(u\in K\) such that \[\label{eq:sp6} u\in K,\qquad N(A(u),T(u))\in K^{*},\qquad \left\langle N(A(u),T(u)),u\right\rangle=0. \tag{8}\]
If \(g=I\) in (5), then we recover the bivariational inequality \[\label{eq:sp7} \left\langle N(A(u),T(u)),v-u\right\rangle\ge 0,\qquad \forall v\in K. \tag{9}\]
If \(N(A(u),T(u))\equiv T(u)\) in (9), then we obtain the classical variational inequality \[\label{eq:sp8} \left\langle T(u),v-u\right\rangle\ge 0,\qquad \forall v\in K, \tag{10}\] introduced by Lions and Stampacchia [2].
Thus, for suitable choices of \(N\), \(A\), \(T\), \(g\), and \(\varphi\), the mixed general bivariational inequality (1) provides a unifying model for a broad family of problems.
We also recall a simple Hilbert space identity.
Lemma 1. For all \(u,v\in H\), \[\label{eq:identity} 2\left\langle u,v\right\rangle=\|u+v\|^{2}-\|u\|^{2}-\|v\|^{2}. \tag{11}\]
Definition 1. For all \(u,v,z\in H\), the bifunction \(N(\cdot,\cdot):H\times H\to H\) involving the operators \(A\) and \(T\) with respect to \(g\) is said to be:
\(g\)-partially relaxed strongly monotone if there exists \(\alpha>0\) such that \[\left\langle N(A(u),T(u))-N(A(v),T(v)),g(z)-g(v)\right\rangle\ge -\alpha\|g(u)-g(z)\|^{2};\]
\(g\)-partially relaxed monotone if \[\left\langle N(A(u),T(u))-N(A(v),T(v)),g(z)-g(v)\right\rangle\ge 0;\]
Lipschitz continuous if there exists \(\beta>0\) such that \[\|N(A(u),T(u))-N(A(v),T(v))\|\le \beta\|u-v\|.\]
When \(z=u\), Definition 1(i) reduces to the usual \(g\)-monotonicity of the composite mapping \(u\mapsto N(A(u),T(u))\). In particular, the partially relaxed formulation is weaker than the coercive assumptions commonly used in proximal or descent analyses; compare [9, 11].
Definition 2([4]).If \(B\) is a maximal monotone operator on \(H\), then its resolvent is defined by \[J_{B}(u)=(I+\lambda B)^{-1}(u),\qquad \forall u\in H,\] where \(\lambda>0\) is fixed.
It is well known that the resolvent of a maximal monotone operator is single-valued and nonexpansive. Since \(\partial\varphi\) is maximal monotone whenever \(\varphi\) is proper, convex, and lower semicontinuous, we write \[\label{eq:Jphi} J_{\varphi}(u):=(I+\lambda\partial\varphi)^{-1}(u). \tag{12}\]
Lemma 2. Let \(\varphi\) be proper, convex, and lower semicontinuous. For given \(u,z\in H\), the following are equivalent:
\[\label{eq:reschar} \left\langle u-z,v-u\right\rangle+\varphi(v)-\varphi(u)\ge 0,\qquad \forall v\in H; \tag{13}\]
\(u=J_{\varphi}(z)\).
Let \(u\in H\) satisfy (1). Consider the auxiliary problem of finding \(w\in H\) such that \[\label{eq:auxiliary} \left\langle \rho N(A(u),T(u))+g(w)-g(u),g(v)-g(w)\right\rangle+\rho\varphi(g(v))-\rho\varphi(g(w))\ge 0, \qquad \forall v\in H, \tag{14}\] where \(\rho>0\) is fixed. If \(w=u\), then (14) reduces to (1). This observation leads to the following predictor–corrector scheme.
Algorithm 1. Given \(u_{0}\in H\), generate sequences \(\{u_n\}\) and \(\{w_n\}\) by solving \[\label{eq:alg1} \left\langle \rho N(A(w_n),T(w_n))+g(u_{n+1})-g(u_n),g(v)-g(u_{n+1})\right\rangle +\rho\varphi(g(v))-\rho\varphi(g(u_{n+1}))\ge 0, \quad \forall v\in H, \tag{15}\] and \[\label{eq:alg2} \left\langle \beta N(A(u_n),T(u_n))+g(w_n)-g(u_n),g(v)-g(w_n)\right\rangle +\beta\varphi(g(v))-\beta\varphi(g(w_n))\ge 0, \quad \forall v\in H, \tag{16}\] where \(\rho>0\) and \(\beta>0\) are constants.
Remark 1. If \(g=I\), then Algorithm 1 takes the form \[\left\langle \rho N(A(w_n),T(w_n))+u_{n+1}-u_n,v-u_{n+1}\right\rangle+\rho\varphi(v)-\rho\varphi(u_{n+1})\ge 0,\] \[\left\langle \beta N(A(u_n),T(u_n))+w_n-u_n,v-w_n\right\rangle+\beta\varphi(v)-\beta\varphi(w_n)\ge 0,\] for all \(v\in H\).
Remark 2. If \(\varphi\) is proper, convex, and lower semicontinuous, then Lemma 2 yields the equivalent resolvent formulation \[\label{eq:resolventAlg} g(u_{n+1})=J_{\varphi}[g(w_n)-\rho N(A(w_n),T(w_n))], \qquad g(w_n)=J_{\varphi}[g(u_n)-\beta N(A(u_n),T(u_n))]. \tag{17}\]
If, moreover, \(\varphi=I_K\), then Algorithm 1 reduces to the corresponding projection-type method on \(K\).
The first result gives the basic descent inequalities needed in the convergence analysis.
Theorem 1. Let \(\bar u\in H\) be a solution of (1), and let \(\{u_n\}\) and \(\{w_n\}\) be generated by Algorithm 1. If \(N(\cdot,\cdot)\) is \(g\)-partially relaxed monotone, then for every \(n\ge 0\), \[ \|g(u_{n+1})-g(\bar u)\|^{2} \le \|g(u_n)-g(\bar u)\|^{2}-\|g(u_{n+1})-g(u_n)\|^{2}, \label{eq:descent1}\ \tag{18}\] \[\|g(w_n)-g(\bar u)\|^{2} \le \|g(u_n)-g(\bar u)\|^{2}-\|g(w_n)-g(u_n)\|^{2}. \label{eq:descent2} \tag{19}\]
Proof. Because \(\bar u\) solves (1), we have \[ \left\langle \rho N(A(\bar u),T(\bar u)),g(v)-g(\bar u)\right\rangle +\rho\varphi(g(v))-\rho\varphi(g(\bar u)) \ge 0, \qquad \forall v\in H, \label{eq:sol1}\ \tag{20}\] \[\left\langle \beta N(A(\bar u),T(\bar u)),g(v)-g(\bar u)\right\rangle +\beta\varphi(g(v))-\beta\varphi(g(\bar u)) \ge 0, \qquad \forall v\in H. \label{eq:sol2} \tag{21}\]
Taking \(v=u_{n+1}\) in (20) and \(v=\bar u\) in (15), and then adding the resulting inequalities, we obtain \[\label{eq:key1} \left\langle g(u_{n+1})-g(u_n),g(\bar u)-g(u_{n+1})\right\rangle \ge \rho\left\langle N(A(w_n),T(w_n))-N(A(\bar u),T(\bar u)),g(u_{n+1})-g(\bar u)\right\rangle. \tag{22}\]
By \(g\)-partial relaxed monotonicity, the right-hand side is nonnegative, and therefore \[\label{eq:key2} \left\langle g(u_{n+1})-g(u_n),g(\bar u)-g(u_{n+1})\right\rangle\ge 0. \tag{23}\]
Applying Lemma 1 with \[\xi=g(\bar u)-g(u_{n+1}),\qquad \eta=g(u_{n+1})-g(u_n),\] yields \[2\left\langle g(u_{n+1})-g(u_n),g(\bar u)-g(u_{n+1})\right\rangle =\|g(\bar u)-g(u_n)\|^{2}-\|g(\bar u)-g(u_{n+1})\|^{2}-\|g(u_{n+1})-g(u_n)\|^{2}.\]
Combining this identity with (23) gives (18).
Next, take \(v=\bar u\) in (16) and \(v=w_n\) in (21). Adding the two relations, we obtain \[\label{eq:key3} \left\langle g(w_n)-g(u_n),g(\bar u)-g(w_n)\right\rangle \ge \beta\left\langle N(A(u_n),T(u_n))-N(A(\bar u),T(\bar u)),g(w_n)-g(\bar u)\right\rangle. \tag{24}\]
Again, the right-hand side is nonnegative by Definition 1(ii), so \[\label{eq:key4} \left\langle g(w_n)-g(u_n),g(\bar u)-g(w_n)\right\rangle\ge 0. \tag{25}\]
Applying Lemma 1 with \[\xi=g(\bar u)-g(w_n),\qquad \eta=g(w_n)-g(u_n),\] we obtain (19). ◻
Corollary 1. Under the assumptions of Theorem 1, the sequence \(\{\|g(u_n)-g(\bar u)\|\}\) is nonincreasing and \[\label{eq:areg1} \sum_{n=0}^{\infty}\|g(u_{n+1})-g(u_n)\|^{2}<\infty. \tag{26}\]
Consequently, \[\label{eq:areg2} \lim_{n\to\infty}\|g(u_{n+1})-g(u_n)\|=0. \tag{27}\]
Moreover, (19) implies \[\label{eq:areg3} \|g(w_n)-g(u_n)\|^{2}\le \|g(u_n)-g(\bar u)\|^{2}-\|g(w_n)-g(\bar u)\|^{2}, \tag{28}\] so in particular \(\|g(w_n)-g(u_n)\|\to 0\) whenever \(\|g(u_n)-g(\bar u)\|\) and \(\|g(w_n)-g(\bar u)\|\) have the same limit.
The next theorem gives a rigorous convergence statement. It is formulated with an explicit cluster-point assumption, which is automatic, for example, when \(H\) is finite dimensional and \(\{u_n\}\) is bounded.
Theorem 2. Assume that the solution of (1) is unique and denote it by \(\bar u\). Suppose that the mapping \[F(u):=N(A(u),T(u)),\] is continuous, that \(g:H\to H\) is invertible with continuous inverse, and that the sequence \(\{u_n\}\) generated by Algorithm 1 has a cluster point. Then \[u_n\to \bar u\qquad\text{and}\qquad w_n\to \bar u.\]
Proof. Let \(\hat u\) be a cluster point of \(\{u_n\}\), and let \(u_{n_j}\to \hat u\) along a subsequence. By (27) and continuity of \(g^{-1}\), we also have \[u_{n_j+1}\to \hat u.\]
From (19), the nonnegative quantity \[\|g(w_n)-g(u_n)\|^{2}\le \|g(u_n)-g(\bar u)\|^{2}-\|g(w_n)-g(\bar u)\|^{2},\] vanishes along the same subsequence, hence \(g(w_{n_j})-g(u_{n_j})\to 0\). Since \(g^{-1}\) is continuous, \[w_{n_j}\to \hat u.\]
Fix \(v\in H\). From (15), \[\begin{aligned} \rho\varphi(g(u_{n_j+1})) &\le \rho\varphi(g(v)) +\left\langle \rho F(w_{n_j})+g(u_{n_j+1})-g(u_{n_j}),g(v)-g(u_{n_j+1})\right\rangle. \end{aligned}\]
Passing to the limit superior and using the continuity of \(F\), together with \(u_{n_j+1}\to\hat u\), \(w_{n_j}\to\hat u\), and \(g(u_{n_j+1})-g(u_{n_j})\to 0\), we obtain \[\label{eq:limsupineq} \limsup_{j\to\infty}\varphi(g(u_{n_j+1})) \le \varphi(g(v))+\left\langle F(\hat u),g(v)-g(\hat u)\right\rangle. \tag{29}\]
Since \(\varphi\) is lower semicontinuous, \[\varphi(g(\hat u))\le \liminf_{j\to\infty}\varphi(g(u_{n_j+1})).\]
Combining this with (29) gives \[\left\langle F(\hat u),g(v)-g(\hat u)\right\rangle+\varphi(g(v))-\varphi(g(\hat u))\ge 0, \qquad \forall v\in H.\]
Thus \(\hat u\) solves (1). By uniqueness, \(\hat u=\bar u\).
Now Theorem 1 shows that the sequence \(d_n:=\|g(u_n)-g(\bar u)\|\) is nonincreasing. Because \(u_{n_j}\to\bar u\), we have \(d_{n_j}\to 0\), hence \(d_n\to 0\). Since \(g^{-1}\) is continuous, it follows that \(u_n\to\bar u\). Finally, (19) yields \(\|g(w_n)-g(u_n)\|\to 0\), and therefore \(w_n\to\bar u\) as well. ◻
Remark 3. Theorem 2 is intentionally stated under explicit continuity and cluster-point hypotheses. These assumptions make the limit passage transparent and avoid hidden compactness assumptions in infinite-dimensional Hilbert spaces.
We include a simple scalar example to make the resolvent formulation concrete.
Example 1. Let \(H=\mathbb{R}\), let \(A=T=g=I\), define \[N(A(u),T(u))\equiv 0, \qquad \varphi(x)=\mu|x|,\quad \mu>0.\]
Then (1) becomes \[\label{eq:scalarvi} \mu|v|-\mu|u|\ge 0,\qquad \forall v\in\mathbb{R}. \tag{30}\]
Hence the unique solution is \(u=0\). In this case the composite mapping \(u\mapsto N(A(u),T(u))\) is constant, so the \(g\)-partial relaxed monotonicity condition is satisfied trivially.
Since \(\varphi\) is proper, convex, and lower semicontinuous, Algorithm 1 reduces to the resolvent iteration \[w_n=J_{\varphi}(u_n), \qquad u_{n+1}=J_{\varphi}(w_n).\]
For \(\varphi(x)=\mu|x|\), the resolvent \(J_{\varphi}\) is the soft-thresholding operator \[S_{\lambda\mu}(x)=\operatorname{sgn}(x)\max\{|x|-\lambda\mu,0\}.\]
Therefore, \[w_n=S_{\lambda\mu}(u_n), \qquad u_{n+1}=S_{\lambda\mu}(w_n).\]
Starting from any \(u_0\in\mathbb{R}\), the iterates move monotonically toward \(0\) and reach the exact solution in finitely many steps whenever the accumulated threshold dominates \(|u_0|\). This example illustrates the proximal structure behind the general algorithm.
We have studied a mixed general bivariational inequality that unifies several well-known models in variational inequality and complementarity theory. The auxiliary principle technique leads naturally to a two-step predictor–corrector method and to its resolvent representation. The main analytical outcome is a pair of descent inequalities under a \(g\)-partially relaxed monotonicity condition. These estimates yield asymptotic regularity of the iterates and, under continuity, cluster-point, and uniqueness hypotheses, convergence to the exact solution.
The analysis in this paper is theoretical. Further work may address existence theory under problem-specific hypotheses, sharper convergence rates, and computational studies for concrete application classes.
Both authors contributed to the conception of the study, the mathematical analysis, and the preparation of the manuscript.
The authors declare that they have no conflict of interest.
No data sets were generated or analyzed in this study.
We wish to express our sincere gratitude to our respected professors, teachers, students, colleagues, collaborators, reviewers, editors and friends, who have direct or indirect contributions in the process of this paper. The authors are grateful to the Vice Chancellor, University of Wah, Wah Cantt, Pakistan for providing the excellent research facilities and support in our research endeavors. The authors would like to thank the referee and managing editor, for their many helpful constructive comments and suggestions towards the improvement of this paper.
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