Since is reflexive and convex, -type spaces have many useful applications, and it is natural to consider a geometric structure of these spaces. From this point of view, we generalized the space by using the Zweier operator and then obtained the equality , that is, it was seen that the structure of the space was preserved. In this section, our goal is to investigate a geometric structure of the modular space related to the fixed point theory. For this, we will examine property and the uniform Opial property for . Finally, we will give some fixed point results. To do this, we need some results which are important in our opinion.
Lemma 3.1 [2]
If , then for any and , there exists such that
where with and .
Lemma 3.2 [2]
If , convergence in norm and in modular are equivalent in .
Lemma 3.3 [2]
If , then for any , there exists such that implies .
Now we give the following two lemmas without proof.
Lemma 3.4 If for any , then .
Lemma 3.5 For any , if and only if .
Lemma 3.6 If , then for any , there exist and such that
for all with , where
and .
Proof Let be fixed. Then there exists such that . Let γ be a real number , and so there exists such that for all . Choose such that . Therefore, we have
for each and . □
Lemma 3.7 If , then for any , there exists such that implies .
Proof Suppose that lemma does not hold. So, there exist and such that and . Take , and so as . Let . There exists such that
for every with , since . We have
for all by (3.1). Therefore, and from Lemma 3.5 we have
This is a contradiction. So, the proof is complete. □
Theorem 3.8 The space has property .
Proof Let and with and . For each , we can find such that . Let
Since for each , is bounded, by using the diagonal method, we can find a subsequence of such that converges for each with . Therefore, there exists an increasing sequence of positive integers such that . Thus, there exists a sequence of positive integers with such that for all . Since , there is such that
(3.4)
from Lemma 3.3. However, there exist and such that
(3.5)
for all and by Lemma 3.6. There exists such that
(3.6)
for any by Lemma 3.7.
By Lemma 3.1, there exists such that
(3.7)
where and . Hence, we get that since . Then there exists such that . Let and . Then
We obtain from (3.3) and (3.5) that
(3.8)
Choose . By the inequalities (3.2), (3.3), (3.6) and the convexity of the function , we have
So, the inequality (3.4) implies that . Consequently, the space possesses property . □
Since property implies NUC, NUC implies property and property implies reflexivity, we can give the following result from Theorem 3.8.
Corollary 3.9 The space is nearly uniform convex, reflexive and also it has property .
Theorem 3.10 The space has the uniform Opial property.
Proof Let and be such that and be a weakly null sequence in . By , there exists independent of x such that by Lemma 3.2. Also since , by Lemma 3.1, there is such that
(3.10)
whenever and . Take such that
(3.11)
Hence, we have
(3.12)
and this implies that
(3.13)
Since , by the inequality (3.10), there exists such that
(3.14)
Again, by , there is such that for all
(3.15)
where for all . Therefore, we obtain that
(3.16)
by the triangle inequality of the norm. It follows from the definition of the Luxemburg norm that
(3.17)
and this implies that
(3.18)
By (3.7), (3.8), (3.11), (3.15) and since (coordinatewise), we have for any that
Since , it follows from Lemma 3.3 that there is τ depending on ζ only such that . □
Corollary 3.11 The space has property and the fixed point property.