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Chirasak Mongkolkeha, Poom Kumam, "Some Geometric Properties of Lacunary Sequence Spaces Related to Fixed Point Property", Abstract and Applied Analysis, vol. 2011, Article ID 903736, 13 pages, 2011. https://doi.org/10.1155/2011/903736
Some Geometric Properties of Lacunary Sequence Spaces Related to Fixed Point Property
The main purpose of this paper is considering the lacunary sequence spaces defined by Karakaya (2007), by proving the property () and Uniform Opial property.
Let be a real Banach space and let (resp., ) be a closed unit ball (resp., the unit sphere) of . For any subset of , we denote by the convex hull of . The Banach space is uniformly convex , if for each there exists such that for the inequality implies (see ). A Banach space has the property if for each there exists such that implies , where denotes the Kuratowski measure noncompactness of a subset of defined as the infimum of all such that can be covered by a finite union of sets of diameter less than . The following characterization of the property is very useful (see ): A Banach space has the property if and only if for each there exists such that for each element and each sequence in with there is an index for which where . A Banach space is nearly uniformly convex if for each and every sequence in with , there exists such that . Define for any the drop determined by by . A Banach space has the drop property (write ) if for every closed set disjoint with there exists an element such that . A point is an of if for any sequence in such that as , the week convergence of to implies that as . If every point in is an -point of , then is said to have the property . A Banach space is said to have the uniform Kadec-Klee property (abbreviated as (UKK)) if for every there exists such that for every sequence in with and as , we have . Every (UKK) Banach space has -property (see ). The following implications are true in any Banach spaces,(1.1)
where denotes the property of reflexivity (see [3–6]). A Banach space is said to have the Opial property (see ) if every sequence weakly convergent to satisfies for every . Opial proved in  that the sequence space have this property but do not have it. A Banach space is said to have the uniform Opial property (see ), if for each there exists such that for any weakly null sequence in and with there holds For example, the space in [9, 10] has the uniform Opial property. The Opial property is important because Banach spaces with this property have the weak fixed point property (see ) and the geometric property involving fixed point theory can be found, for example, in [9, 12–14].
For a bounded subset , the set measure of noncompactness was defined in  by The ball measure of noncompactness was defined in [16, 17] by The functions and are called the Kuratowski measure of noncompactness and the Hausdorff measure of noncompactness in , respectively. We can associate these functions with the notions of the set-contraction and ball contraction (see ). These notions are very useful tools to study nonlinear operator propblems (see [8, 18]). For each define that is closed convex subset of with . The function is called the modulus of noncompact convexity (see ). A Banach space is said to have property if . It has been proved in  that property is a useful tool in the fixed point theory and that a Banach space has property if and only if it is reflexive and has the uniform Opial property.
For a real vector space , a function is called a modular if it satisfies the following conditions:(i) if and only if ;(ii) for all scalar with ;(iii), for all and all with ; the modular is called convex if(iv), for all and all with .
For modular on , the space is called the modular space.
A sequence in is called modular convergent to if there exists a such that as .
A modular is said to satisfy the if for any there exist constants and such that for all with .
If satisfies the for any with dependent on , we say that satisfies the strong .
By a lacunary sequence , where , we will mean an increasing sequence of nonnegative integers with as . The intervals determined by will be denote by . We write and the ratio , will be denoted by . The space of lacunary strongly convergent sequence was defined by Freedman et al.  as It is well known that there is very closed connection between the space of lacunary strongly convergent sequence and the space of strongly Cesaro summability sequences. This connection can be found in [18–23], because a lot of these connection, a lot of geometric property of Cesaro sequence spaces can generalize the lacunary sequence spaces.
Let be the space of all real sequences. Let be a bounded sequence of the positive real numbers. In 2007, Karakaya  introduced the new sequence spaces involving lacunary sequence as follows: and paranorm on is given by where . If for all , we will use the notation in place of . The norm on is given by By using the properties of lacunary sequence in the space , we get the following sequences. If , then . If and for all , then . For defined the modular on by It is easy to see that if then . The Luxembourg norm on is defined by The Luxembourg norm on can be reduced to a usual norm on , that is,
Throughout this paper, we assume that and and for , , we denote The following results are very important for our consideration.
Lemma 1.1 (see [25, Lemma 2.1]). If , then for any and , there exists such that whenever with , and .
Lemma 1.2 (see [25, Lemma 2.3]). Convergence in norm and in modular are equivalent in if .
Lemma 1.3 (see [25, Corollary 2.2, Lemma 2.3]). If , then for any sequence in , if and only if as .
Lemma 1.4 (see [25, Lemma 2.4]). If , then for any there exists such that whenever .
Lemma 1.5 (see [24, Lemma 2.3]). The functional is a convex modular on .
Lemma 1.6 (see [24, Lemma 2.5]).
(i) For any , if , then .
(ii) For any , if and only if .
2. The Main Results
In this section, we prove the property and uniform Opial property in lacunary sequence and connect to the fixed point property. First we shall give some results which are very important for our consideration.
Lemma 2.1. For any , there exists and such that for all with , where
Proof. Let be fixed. So there exist such that . Let be a real number such that , then there exists such that for all . Choose as a real such that . Then for each and , we have
Lemma 2.2. For any and there exists such that implies .
Proof. Suppose that the lemma does not hold, then there exist and such that and . Let . Then as . Let . Since there exists such that for every with . By (2.3), we have for all . Hence . By Lemmas 1.5 and 1.6(ii), we have which is a contradiction.
Theorem 2.3. The space is Banach spaces with respect to the Luxemburg norm.
Proof. Let be a Cauchy sequence in and . Thus there exists such that for all . By Lemma 1.6 (i), we have That is, For fixed , we get that Thus let be a Cauchy sequence in for all . Since is complete, then there exists such that as for all . Thus for fixed , we have This implies that, for all , This means that, for all , as . By (2.6), we have This implies that as . So we have . By the linearity of the sequence space , we have . Therefore the sequence space is Banach space, with respect to the Luxemburg norm and the proof is complete.
Theorem 2.4. The space has property .
Proof. Let and with . For each , there exist such that is a minimal element in . Let Since for each , is bounded. By using the diagonal method, we have that for each we can find subsequence of such that converges for each . Therefore, for any there exists an increasing sequence such that . Hence for each there exists sequence of positive integers with such that , and since , by Lemma 1.3 we may assume that there exists such that for all , that is, for all . On the other hand by Lemma 2.1, there exist and such that for all and . From Lemma 2.2, there exist such that for any , Since again , by Lemma 1.1, there exists such that whenever and . Since , we have that . Then there exits such that . We put and , From (2.14) and (2.16), we have By (2.13), (2.16), (2.18), and convexity of function , for all , we have So it follows from (2.15) that Therefore, the space has property .
By the facts presented in the introduction, following results are obtained directly from Theorem 2.4.
Corollary 2.5. The space has property .
Corollary 2.6. The space is nearly uniform convexity, has drop property, and is reflexive.
Corollary 2.7. The space has property (UKK).
Corollary 2.8 (see [24, Theorem 2.9]). The space has property .
Corollary 2.9. The space is nearly uniform convexity, has drop property, and is reflexive.
Corollary 2.10. The space has property (UKK) and .
Theorem 2.11. The space has uniform Opial property.
Proof. Take any and with . Let be weakly null sequence in . By , that is, , hence by Lemma 1.2 there exists independent of such that . Also, by , Lemma 1.1 asserts that there exists such that whenever and . Choose such that So, we have which implies that Since , then there exists such that for all , since weak convergence implies coordinatewise convergence. Again, by , then there exists such that for all where is a minimal element in and with for all . Hence, by the triangle inequality of the norm, we get It follows by the definition of that we have Which implies that for all . By inequality (2.21), (2.25), and (2.29), it yields for any that Since and by Lemma 1.4 there exists depending on only such that , which implies that , hence the prove is complete.
By the facts presented in the introduction and the reflexivity of , we get the following results.
Corollary 2.12. The space has uniform Opial property.
Corollary 2.13. The space has property and the fixed point property.
Corollary 2.14. The space has property and the fixed point property.
The authors are grateful to the referees for their valuable comments. They also wish to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission for financial support (under the Project no. 54000267). Mr. Chirasak Mongkolkeha was supported by the Thailand Research Fund through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for Ph.D. program at KMUTT, Thailand. The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkuts University of Technology Thonburi (Grant no. MRG5380044).
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