Applications of general variational inequalities to coincidence point results (Q2875451)

Variational inequality theory has various applications both inside and outside of mathematics, and has its origins in works by \textit{G. Stampacchia} [C. R. Acad. Sci., Paris 258, 4413--4416 (1964; Zbl 0124.06401)] and \textit{G. Fichera} [Atti Accad. Naz. Lincei, Mem., Cl. Sci. Fis. Mat. Nat., Sez. I, VIII. Ser. 7, 91--140 (1964; Zbl 0146.21204)].NEWLINENEWLINELet \(X\) be a real Banach space and \(X'\) its topological dual. One of the main results is the following.NEWLINENEWLINELet \( K \subseteq X\) be a non-void subset and let \(A:K\rightarrow X'\) and \(a : K\rightarrow X\) be mappings. Assume that \(a(K)\) is weakly compact and convex and that, for every sequence \((x_n)\) in \(K\), the following condition is satisfied: If the sequence \((a(x_n))\) converges weakly to \(a(x) \in a(K)\), then the sequence \((A(x_n))\) is norm convergent to \(A(x)\) in \(X'\). Then the general variational inequality problem, \(VI_S(A,a,K)\), admits solutions, that is to say, there is an element \(x \in K\) such that NEWLINE\[NEWLINE \langle A(x), a(y)-a(x) \rangle \geq 0 NEWLINE\]NEWLINE for all \(y \in K\).NEWLINENEWLINEThis is proved by repatriation of the special case \(a=\text{id}_K\) established by \textit{S. László} [J. Optim. Theory Appl. 150, No. 3, 425--443 (2011; Zbl 1228.49013)].NEWLINENEWLINE Similar results concern monotone mappings \(A\) from \(X\) into \(X'\) relative to \(a\), where \(X\) is reflexive. Moreover, in Section 3, the authors derive a theorem showing the existence of a coincidence point \(x \in K\) to a pair \(f,g\) of functions on \(K\), i.e., \(f(x)=g(x)\) with some corollaries. The special case \(a=\text{id}_K\) yields fixed point theorems. The paper is easy to read and all proofs are included and complete.