Abstract

First we prove existence of a fixed point for mappings defined on a complete modular space satisfying a general contractive inequality of integral type. Then we generalize fixed-point theorem for a quasicontraction mapping given by Khamsi (2008) and Ciric (1974).

1. Introduction

In [1], Branciari established that a function defined on a complete metric space satisfying a contraction condition of the form has a unique attractive fixed point where is a Lebesgue-integrable mapping and .

In [2], Rhoades extended this result to a quasicontraction function . The purpose of this paper is to extend these theorems in modular space.

First, we introduce the notion of modular space.

Definition 1.1. Let be an arbitrary vector space over or . A functional is called modular if(1) if and only if ; (2) for with ,for all ;(3) if , , for all.

If (2.14) in Definition 1.1 is replaced by for , with an , then the modular is called an -convex modular; and if , is called a convex modular.

Definition 1.2. A modular defines a corresponding modular space, that is, the space is given by

Definition 1.3. Let be a modular space. (1)A sequence in is said to be (a)-convergent to if as , (b)-Cauchy if as . (2) is -complete if any -Cauchy sequence is -convergent. (3)A subset is said to be -closed if for any sequence with then . denotes the closure of in the sense of . (4)A subset is called -bounded if where is called the -diameter of . (5)We say that has Fatou property if whenever (6) is said to satisfy the -condition if: as whenever as .

Remark 1.4. Note that since does not satisfy a priori the triangle inequality, we cannot expect that if and are -convergent, respectively, to and then is -convergent to , neither that a -convergent sequence is -Cauchy.

2. Main Result

Theorem 2.1. Let be a complete modular space, where satisfies the -condition. Assume that is an increasing and upper semicontinuous function satisfying Let be a nonnegative Lebesgue-integrable mapping which is summable on each compact subset of and such that for , and let be a mapping such that there are where , for each . Then has a unique fixed point in .

Proof. First, we show that for , the sequence converges to 0. For , we have Consequently, is decreasing and bounded from below. Therefore converges to a nonnegative point .
Now, if , then which is a contradiction, so and
This concludes . Suppose that then there exist a and a sequence such that then we get the following contradiction:
Now, we prove for each the sequence is a -Cauchy sequence.
Assume that there is an such that for each there exist that ,
Then we choose the sequence and such that for each , is minimal in the sense that But for each .
Now, let be such that , then we have
Thus, as , by -condition, . Therefore
Now, If we get which is a contradiction for . Therefore is a -Cauchy sequence and by -condition is -Cauchy. By the fact that is -complete, there is a such that as . Furthermore, is the fixed point for . In fact then and .
Now, assume that we have more than one fixed point for . Let and be two distinct fixed points, then which is a contradiction. So and the proof is complete.

Corollary 2.2 (see [1]). Let be a complete modular space, where satisfies the -condition. Let be a mapping such that there exists an and where and for each , then has a unique fixed point.

Corollary 2.3 (see [3]). Let be a complete modular space, where satisfies the -condition. Assume that is an increasing and upper semicontinuous function satisfying Let be a -closed subset of and be a mapping such that there exist with , for all . Then has a fixed point.

In the next theorem we use the following notation:

Theorem 2.4. Let be a -complete modular space that satisfies the -condition and let be a mapping such that for each , where and are as in Theorem 2.1. Then has a unique fixed point.

Proof. Let , we will show that is a Cauchy sequence. First, we prove that converges to 0. From (2.23),
By the definition of ,
Hence, and therefore, This means that is decreasing and since it is bounded from below, it is a convergent sequence. Similarly to Theorem 2.1, it is easy to show that
Now, we show that is Cauchy. If not, then there exist an and subsequences and such that with
From (2.22),
By using (2.28), we get
On the other hand, thus by the -condition,
For the last term in by the fact that is an increasing function of we have
Hence, from (2.28) we get
Therefore from (2.31), (2.33), and (2.35) it can be concluded that which is a contradiction, when is large enough. Therefore, is Cauchy and since is -complete there is an that . Now, we should prove that is the fixed point for . In fact, by the definition of . It follows that .
Let be another fixed point of . Then, That is because thus .