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

O’Regan, D.

Z. Anal. Anwend. 17, No. 2, 281-296 (1998).

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 \(Q\).

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

Keywords:

fixed point; weakly contractive multivalued maps; nonlinear abstract operator inclusion

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