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**Applications of variational inequalities to nonlinear analysis.**
*(English)*
Zbl 0734.49003

Let K be a nonempty closed convex subset in the real Hilbert space H.

Theorem 1. Let T be a monotone operator from K into H which is continuous on finite dimensional subspaces and has the property that there is \(x_ 0\in K\) such that \(\liminf_{x\to \infty}<x-x_ 0,T(x)>>0\); then there is \(x\in K\) such that \(<u-x,T(x)>\geq 0\) for all \(u\in K.\)

Theorem 2. Let T be a hemicontinuous operator from K into itself and there is a \(k<1\) such that \(<x-y,T(x)-T(y)>\leq k| x-y|^ 2\) for all x,y\(\in K\); then T has a unique fixed point.

Theorem 3. Let T be an operator from K into H, there is a \(k\leq 0\) such that \(<x-y,T(x)-T(y)>\leq k| x-y|^ 2\) for all x,y\(\in K\), there is an \(m\geq 1\) such that \(| T(x)-T(y)| \leq m| x-y|\) for all x,y\(\in K\); then if \(x_ 0\in K\) and \(x_{n+1}=P_ K((1- \lambda)x_ n+\lambda T(x_ n))\) for \(n\in N\), where \(P_ K\) denotes the projection operator on K and \(0<\lambda <2(1-k)/(1-2k+m^ 2)\), the sequence \((x_ n)_{n\in {\mathbb{N}}}\) converges to the unique fixed point of T.

Theorem 1. Let T be a monotone operator from K into H which is continuous on finite dimensional subspaces and has the property that there is \(x_ 0\in K\) such that \(\liminf_{x\to \infty}<x-x_ 0,T(x)>>0\); then there is \(x\in K\) such that \(<u-x,T(x)>\geq 0\) for all \(u\in K.\)

Theorem 2. Let T be a hemicontinuous operator from K into itself and there is a \(k<1\) such that \(<x-y,T(x)-T(y)>\leq k| x-y|^ 2\) for all x,y\(\in K\); then T has a unique fixed point.

Theorem 3. Let T be an operator from K into H, there is a \(k\leq 0\) such that \(<x-y,T(x)-T(y)>\leq k| x-y|^ 2\) for all x,y\(\in K\), there is an \(m\geq 1\) such that \(| T(x)-T(y)| \leq m| x-y|\) for all x,y\(\in K\); then if \(x_ 0\in K\) and \(x_{n+1}=P_ K((1- \lambda)x_ n+\lambda T(x_ n))\) for \(n\in N\), where \(P_ K\) denotes the projection operator on K and \(0<\lambda <2(1-k)/(1-2k+m^ 2)\), the sequence \((x_ n)_{n\in {\mathbb{N}}}\) converges to the unique fixed point of T.

Reviewer: G.Bottaro (Genova)

### MSC:

49J40 | Variational inequalities |

47H05 | Monotone operators and generalizations |

47H10 | Fixed-point theorems |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

### References:

[1] | Stampacchia, G., Variational inequalities, (Ghizzetti, A., Theory and Applications of Monotone Operators (1969), Edizioni Oderisi: Edizioni Oderisi Gubbio, Italy) · Zbl 0152.34601 |

[2] | Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities (1980), Academic Press: Academic Press New York · Zbl 0457.35001 |

[3] | Hartman, G. J.; Stampacchia, G., On some nonlinear elliptic differential functional equations, Acta Math., 115, 271-310 (1966) · Zbl 0142.38102 |

[4] | Allen, G., Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl., 58, 1-10 (1977) · Zbl 0383.49005 |

[5] | Browder, F. E., Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 17, 780-785 (1965) · Zbl 0138.39902 |

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