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**Fixed point theory for weakly contractive maps with applications to operator inclusions in Banach spaces relative to the weak topology.**
*(English)*
Zbl 0911.47057

The author presents a new fixed point theory for weakly contractive multivalued maps between Banach spaces.

Theorem 2.2. Let \(Q\) be a non-empty, bounded, convex, closed set in a Banach space \(E\). Assume \(F:Q\to C(Q)\) is weakly sequentially upper semicontinuous and \(\alpha\) \(W\)-contractive. Then \(F\) has a fixed point; here \(C(Q)\) denotes the family of non-empty, closed, convex subsets of \(Q\).

Theorem 2.3. Let \(Q\) and \(C\) be closed bounded, convex subsets of a Banach space \(E\) and \(Q\subset C\). In addition, let \(U\) be a weakly open subset of \(Q\) with \(0\in U\), \(U^w\) a weakly compact subset of \(Q\) and \(F: U^w\to CK(Q)\) a weakly sequentially upper semicontinuous, \(\alpha\) \(w\)-contractive (here \(0\leq\alpha< 1\)) map, here \(CK(C)\) denotes the family of non-empty, convex, weakly compact subsets of \(C\) and \(\overline{U^w}\) the weak closure of \(U\) in \(E\). Then either:

\((\text{A}_1)\) \(F\) has a fixed point; or

\((\text{A}_2)\) there is a point \(u\in\partial_QU\) (the weak boundary of \(U\) in \(Q\)) and \(\lambda\in(0,1)\) with \(u\in\lambda Fu\).

Theorem 2.4. Let \(E=(E,\|\cdot\|)\) be a separable and reflexive Banach space, \(C\) and \(Q\) are closed, bounded, convex subsets of \(E\) with \(Q\subset C\) and \(0\in Q\). Also, assume \(F: Q\to CK(Q)\) is a weakly sequentially upper semicontinuous (and weakly compact) map. In addition suppose the following:

For any \(\Omega_\varepsilon= \{x\in E: d(x,Q)\leq \varepsilon\}\) \((\varepsilon>0)\), if \(\{(x_j,\lambda_j)\}^\infty_{j= 1}\) is a sequence in \(Q\times[0, 1)\) with \(x_j\rightharpoonup x\in\partial\Omega_\varepsilon Q\) and \(\lambda_j\to\lambda\) and if \(x\in\lambda F(x)\), \(0\leq\lambda< 1\), then \(\{\lambda_jF(x_j)\}\subset Q\) for \(j\) sufficiently large (\(\partial\Omega_\varepsilon Q\) is the weak boundary of \(Q\) relative to \(\Omega_\varepsilon\), \(d(x,y)=| x-y|\) and \(\rightharpoonup\) denotes weak convergence). Then \(F\) has a fixed point in \(Q\).

The author uses Theorems 2.2 and 2.4 to establish some general existence principles for nonlinear abstract operator inclusion \[ y(t)\in Fy(t)\quad\text{on}\quad [0,T].\tag{1} \] Theorem 2.5. Let \(E_1\) be a Banach space and let \(E\) be either \(C([0,T],E_1)\) or \(L^p([0,T],E_1)\), \(1\leq p<\infty\). Let \(Q\) be a non-empty, bounded, convex, closed subset of \(E\) and assume \(F: Q\to C(Q)\) is a weakly sequentially upper semicontinuous and \(\alpha\) \(w\)-contractive \((0\leq \alpha<1)\) map. Then (1) has a solution in \(Q\).

Theorem 2.6. Let \(E_1\) be a separable and reflexive Banach space and let \(Q\) and \(C\) be closed, bounded, convex subsets of \(L^p([0,T],E_1)\), \(1\leq p<\infty\), with \(Q\subset C\) and \(0\in Q\). Assume \(F:Q\to CK(Q)\) is a weakly sequentially upper semicontinuous map such that (2.4) holds. Then (1) has a solution in \(Q\).

Theorem 2.9. Let \(E\) be a Banach space with \(Q\) a non-empty, closed, convex subset of

\(C([0,T], E_w)\). Also, assume \(Q\) is a closed, bounded, subset of \(C([0,T], E)\), \(F: Q\to C_c(Q)\) is \(w\)-upper semicontinuous, and there exists \(\alpha\), \(0\leq\alpha< 1\), with \(w(F(X))\leq \alpha w(X)\) for all bounded subsets \(X\subset Q\). In addition, suppose the family \(F(Q)\) is weakly equicontinuous. Then (1) has a solution in \(Q\), here \(C_c(Q)\) denotes the family of non-empty, convex (subset of \(C([0,T]),E)\)), closed (in \(C([0,T],E)\)) subsets of \(Q\).

Theorem 2.2. Let \(Q\) be a non-empty, bounded, convex, closed set in a Banach space \(E\). Assume \(F:Q\to C(Q)\) is weakly sequentially upper semicontinuous and \(\alpha\) \(W\)-contractive. Then \(F\) has a fixed point; here \(C(Q)\) denotes the family of non-empty, closed, convex subsets of \(Q\).

Theorem 2.3. Let \(Q\) and \(C\) be closed bounded, convex subsets of a Banach space \(E\) and \(Q\subset C\). In addition, let \(U\) be a weakly open subset of \(Q\) with \(0\in U\), \(U^w\) a weakly compact subset of \(Q\) and \(F: U^w\to CK(Q)\) a weakly sequentially upper semicontinuous, \(\alpha\) \(w\)-contractive (here \(0\leq\alpha< 1\)) map, here \(CK(C)\) denotes the family of non-empty, convex, weakly compact subsets of \(C\) and \(\overline{U^w}\) the weak closure of \(U\) in \(E\). Then either:

\((\text{A}_1)\) \(F\) has a fixed point; or

\((\text{A}_2)\) there is a point \(u\in\partial_QU\) (the weak boundary of \(U\) in \(Q\)) and \(\lambda\in(0,1)\) with \(u\in\lambda Fu\).

Theorem 2.4. Let \(E=(E,\|\cdot\|)\) be a separable and reflexive Banach space, \(C\) and \(Q\) are closed, bounded, convex subsets of \(E\) with \(Q\subset C\) and \(0\in Q\). Also, assume \(F: Q\to CK(Q)\) is a weakly sequentially upper semicontinuous (and weakly compact) map. In addition suppose the following:

For any \(\Omega_\varepsilon= \{x\in E: d(x,Q)\leq \varepsilon\}\) \((\varepsilon>0)\), if \(\{(x_j,\lambda_j)\}^\infty_{j= 1}\) is a sequence in \(Q\times[0, 1)\) with \(x_j\rightharpoonup x\in\partial\Omega_\varepsilon Q\) and \(\lambda_j\to\lambda\) and if \(x\in\lambda F(x)\), \(0\leq\lambda< 1\), then \(\{\lambda_jF(x_j)\}\subset Q\) for \(j\) sufficiently large (\(\partial\Omega_\varepsilon Q\) is the weak boundary of \(Q\) relative to \(\Omega_\varepsilon\), \(d(x,y)=| x-y|\) and \(\rightharpoonup\) denotes weak convergence). Then \(F\) has a fixed point in \(Q\).

The author uses Theorems 2.2 and 2.4 to establish some general existence principles for nonlinear abstract operator inclusion \[ y(t)\in Fy(t)\quad\text{on}\quad [0,T].\tag{1} \] Theorem 2.5. Let \(E_1\) be a Banach space and let \(E\) be either \(C([0,T],E_1)\) or \(L^p([0,T],E_1)\), \(1\leq p<\infty\). Let \(Q\) be a non-empty, bounded, convex, closed subset of \(E\) and assume \(F: Q\to C(Q)\) is a weakly sequentially upper semicontinuous and \(\alpha\) \(w\)-contractive \((0\leq \alpha<1)\) map. Then (1) has a solution in \(Q\).

Theorem 2.6. Let \(E_1\) be a separable and reflexive Banach space and let \(Q\) and \(C\) be closed, bounded, convex subsets of \(L^p([0,T],E_1)\), \(1\leq p<\infty\), with \(Q\subset C\) and \(0\in Q\). Assume \(F:Q\to CK(Q)\) is a weakly sequentially upper semicontinuous map such that (2.4) holds. Then (1) has a solution in \(Q\).

Theorem 2.9. Let \(E\) be a Banach space with \(Q\) a non-empty, closed, convex subset of

\(C([0,T], E_w)\). Also, assume \(Q\) is a closed, bounded, subset of \(C([0,T], E)\), \(F: Q\to C_c(Q)\) is \(w\)-upper semicontinuous, and there exists \(\alpha\), \(0\leq\alpha< 1\), with \(w(F(X))\leq \alpha w(X)\) for all bounded subsets \(X\subset Q\). In addition, suppose the family \(F(Q)\) is weakly equicontinuous. Then (1) has a solution in \(Q\), here \(C_c(Q)\) denotes the family of non-empty, convex (subset of \(C([0,T]),E)\)), closed (in \(C([0,T],E)\)) subsets of \(Q\).

Reviewer: V.Popa (Bacau)

### MSC:

47H10 | Fixed-point theorems |

47J05 | Equations involving nonlinear operators (general) |

54C60 | Set-valued maps in general topology |

47H04 | Set-valued operators |

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