## 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 $$Q$$.
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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### References:

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