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Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph
Fixed Point Theory and Applications volume 2014, Article number: 220 (2014)
Abstract
Jachymski (Proc. Am. Math. Soc. 136:1359-1373, 2008) gave a modified version of a Banach fixed point theorem on a metric space endowed with a graph. The aim of this paper is to present fixed point results of mappings satisfying integral type contractive conditions in the framework of modular spaces endowed with a graph. Some examples are presented to support the results proved herein. Our results generalize and extend various comparable results in the existing literature.
MSC:47H10, 54H25, 54E50.
1 Introduction
Fixed point theory for nonlinear mappings is an important subject of nonlinear functional analysis. One of the basic and the most widely applied fixed point theorem in all of analysis is the ‘Banach (or Banach-Caccioppoli) contraction principle’ due to Banach [1]. This Banach contraction principle [1] is a simple and powerful result with a wide range of applications, including iterative methods for solving linear, nonlinear, differential, integral, and difference equations. Due to its applications in mathematics and other related disciplines, the Banach contraction principle has been generalized in many directions.
The existence of fixed points in ordered metric spaces has been discussed by Ran and Reurings [2]. Recently, many researchers have obtained fixed point and common fixed point results for single valued maps defined on partially ordered metric spaces (see, e.g., [3, 4]). Jachymski [5] investigated a new approach in metric fixed point theory by replacing an order structure with graph structure on a metric space. In this way, the results proved in ordered metric spaces are generalized (see for details [5] and the references therein). For further work in this direction, we refer to, e.g., [6–8].
In 1968, Kannan [9] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point. This paper led to the genesis for a multitude of fixed point papers over the next two decades. Since then, there have been many theorems dealing with mappings satisfying various types of contractive inequalities involving linear and nonlinear expressions. For a thorough survey, we refer to [10] and the references therein. On the other hand, Branciari [11] obtained a fixed point theorem for a single valued mapping satisfying an analog of Banach’s contraction principle for an integral type inequality. Recently, Akram et al. [12] introduced a new class of contraction maps, called A-contractions, which is a proper generalization of Kannan’s mappings [9], Bianchini’s mappings [13], and Reich type mappings [14].
The theory of modular spaces was initiated by Nakano [15] in connection with the theory of ordered spaces which was further generalized by Musielak and Orlicz [16] (see also [17–21]). The study of fixed point theory in the context of modular function spaces was initiated by Khamsi et al. [22] (see also [22–26]). Also, some fixed point theorems have been proved for mappings satisfying contractive conditions of integral type in modular space [27, 28].
In this paper, we introduce three new classes of mappings satisfying integral type contractive conditions in the setup of modular space endowed with graphs. We study the existence, uniqueness, and iterative approximations of fixed points for such mappings. Our results extend, unify, and generalize the comparable results in [5, 11, 12].
2 Preliminaries
A mapping T from a metric space into is called a Picard operator (PO) if T has a unique fixed point and for all .
Define Φ = {: φ is a Lebesgue integral mapping which is summable, nonnegative and satisfies , for each }.
Let A = {: α is continuous and for some whenever or or for all a, b}.
Let be a nondecreasing mapping which satisfies the following conditions:
() if and only if ;
() for a sequence in , we have if and only if as ;
() for every , we have .
The collection of all such mappings will be denoted by Ψ.
Define
Theorem 2.1 [11]
Let be a complete metric space, , and a mapping. Suppose that
is satisfied for every , where . Then T has a unique fixed point and for each , we have .
Lemma 2.2 [29]
Let be a metric space, , and a nonnegative sequence. Then
-
(a)
implies that ;
-
(b)
if and only if .
Definition 2.3 [12]
A selfmap T on a metric space X is called an A-contraction if for any and for some , the following condition holds:
Now, we recall some basic facts and notations as regards modular spaces. For more details the reader may consult [16].
Definition 2.4 Let X be an arbitrary vector space. A functional is called modular if for any arbitrary x, y in X:
(m1) if and only if ;
(m2) for every scalar α with ;
(m3) if , , .
If (m3) is replaced by if , where , , then we say that ρ is s-convex modular. If , then ρ is called convex modular.
defined by is a simple example of a modular functional.
The vector space given by
is called a modular space. In general the modular ρ is not sub-additive and therefore does not behave as a norm or a distance. One can associate to a modular an x-norm.
Remark 2.5 [28]
The following are immediate consequences of condition (m3):
(r1) if (set of all real numbers) with , then for all ;
(r2) if are nonnegative real numbers with , then we have
Define the ρ-ball, , centered at with radius r as
A point is called a fixed point of if .
A function modular is said to satisfy the -type condition if there exists such that for any we have . A modular ρ is said to satisfy the -condition if as , whenever as .
Definition 2.6 Let be a modular space. The sequence is said to be:
(t1) ρ-convergent to if as ;
(t2) ρ-Cauchy if as n and .
is ρ-complete if any ρ-Cauchy sequence is ρ-convergent. Note that ρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangle inequality. In fact, one can show that this will happen if and only if ρ satisfies the -condition.
Proposition 2.7 [30]
Let be a modular space. If with , then .
Proposition 2.8 [30]
Suppose that is a modular space, ρ satisfies the -condition and is a sequence in . If , then , as , where with and . Now we give some basic definitions from graph theory needed in the sequel.
Throughout this paper, denotes the diagonal of , where X is any nonempty set. Let G be a directed graph such that the set of its vertices coincides with X and be the set of edges of the graph such that . Further assume that G has no parallel edge and G is a weighted graph in the sense that each edge is assigned a distance between their vertices x and y and each vertex x is assigned a weight . The graph G is identified by the pair .
If x and y are vertices of G, then a path in G from x to y of length is a finite sequence , of vertices such that and for .
Recall that a graph G is connected if there is a path between any two vertices and it is weakly connected if is connected, where denotes the undirected graph obtained from G by ignoring the direction of edges. Denote by the graph obtained from G by reversing the direction of the edges. Thus
Since it is more convenient to treat as a directed graph for which the set of its edges is symmetric, under this convention we have
Let be the component of G consisting of all the edges and vertices which are contained in some path in G beginning at x. If G is such that is symmetric, then for , the equivalence class defined on by the rule R ( if there is a path in G from x to y) is such that .
Definition 2.9 [[5], Definition 2.1]
A mapping is called a Banach G-contraction if and only if:
-
(a)
for each x, y in X with , we have , that is, T preserves edges of G;
-
(b)
there exists α in such that for each with implies
(3)
That is, T decreases weights of edges of G.
For any , such that , , is called a subgraph of G.
3 Main results
In this section, we obtain several fixed point results in the setup of a modular space endowed with a graph. We start with the following definitions. Let be a modular space endowed with a graph G and let be a mapping. Denote
Definition 3.1 Let be a sequence, there exists such that
and
Then a graph G is called a -graph if there exists a subsequence of such that for .
Definition 3.2 A mapping is called orbitally -continuous for all and any sequence of positive integers, if there exists such that
as .
Definition 3.3 A mapping T is called a -contraction if it satisfies the following conditions:
(A1) T preserves edges of G;
(A2) there exist nonnegative numbers l, c with such that
holds for each , and some and .
Remark 3.4 Let be a -contraction. If there exists such that , then
-
(i)
T is both a -contraction and a -contraction,
-
(ii)
is T-invariant and is a -contraction.
Lemma 3.5 Let be a -contraction. If , then there exists such that
holds for all , where .
Proof Let , that is, . Then by induction, we have for all . Now, we have
By the definition of α, we obtain
for some . Thus we have
That is, for all , where . □
Theorem 3.6 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be a -contraction. If the set is nonempty, the graph G is weakly connected and a -graph, then T is a PO.
Proof If , then and for all . Let with . Note that
Using Lemma 3.5, we have
It follows that is a ρ-Cauchy sequence in . Since is ρ-complete, there exists a point such that . Consequently, .
Now we show that is a fixed point of T. As , for all and G is a -graph, there exists a subsequence of such that for each . Since and T is a -contraction, it follows that
which on taking the limit as gives
By the definition of function α, we have
From Lemma 2.2, it follows that and .
Next, we prove that is a unique fixed point. Suppose that T has another fixed point . Since G is a -graph, there exists a subsequence of such that and for each . Furthermore, G is weakly connected, , and we have
By the definition of α and Lemma 2.2, we have , , and . □
In Theorem 3.6, if we replace the condition that G is a -graph with orbitally -continuity of T, then we have the following theorem.
Theorem 3.7 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be a -contraction and orbitally -continuous. If the set is nonempty and the graph G is weakly connected, then T is a PO.
Proof If , then Theorem 3.6 implies that is a ρ-Cauchy sequence in . Owing to ρ-completeness of , there exists such that . As for all and T is orbitally -continuous, we have , as . That is, . Assume that is another fixed point of T. Following arguments similar to those in the proof of Theorem 3.6, we obtain . □
Corollary 3.8 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be edge-preserving, the set nonempty and graph G be weakly connected and a -graph. If there exist nonnegative numbers l, c with such that
holds for all and some , then T is a PO.
Now, we introduce Hardy-Rogers type -contraction and obtain related fixed point results.
Definition 3.9 Let be a modular space. A mapping is called a Hardy-Rogers type -contraction if the following conditions hold:
(H1) T preserves edges of G;
(H2) there exist nonnegative numbers , c with for such that
holds for each with nonnegative numbers η, β, γ such that and .
Remark 3.10 Let be a modular space endowed with a graph G and let be a Hardy-Rogers type -contraction. If there exists such that , then
-
(i)
T is both a Hardy-Rogers type -contraction and a Hardy-Rogers type -contraction,
-
(ii)
is T-invariant and is a Hardy-Rogers type -contraction.
Theorem 3.11 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be a Hardy-Rogers type -contraction. Assume that the set is nonempty and the -graph G is weakly connected. Then T is a PO.
Proof If , then and for all . Note that
It follows that
where . Also,
Taking the limit as , and using Lemma 2.2, we get
which implies that .
Let with . By (4) and Remark 2.5, we get
Thus, is a ρ-Cauchy sequence in . Since is ρ-complete, there exists a point such that . As G is a -graph, there exists a subsequence such that for all . Also, for all . Now we have
Taking the limit as , we have
As , so and .
Next, we prove that is a unique fixed point. Suppose that T has another fixed point . Since G is a -graph, there exists a subsequence of such that and for each . As G is weakly connected, we have , and
which further implies that
Since , . The result follows. □
In Theorem 3.11, if we replace the condition that G is a -graph with orbitally -continuity of T, then we have the following theorem.
Theorem 3.12 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be a Hardy-Rogers type -contraction, which is orbitally -continuous. Assume that the set is nonempty and the graph G is weakly connected. Then T is a PO.
In the following suppose that is a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and is edge-preserving such that the set is nonempty.
Corollary 3.13 Assume
-
(i)
the -graph G is weakly connected and
-
(ii)
there exist nonnegative numbers l, c with such that
holds for each with and . Then T is a PO.
Corollary 3.14 Assume
-
(i)
the -graph G is weakly connected and
-
(ii)
there exist nonnegative numbers , , c with such that
holds for each with and . Then T is a PO.
Corollary 3.15 Assume
-
(i)
the -graph G is weakly connected and
-
(ii)
there exist nonnegative numbers , , c with such that
for each with and . Then T is a PO.
Now we introduce the -contraction and obtain some fixed point results.
Definition 3.16 A mapping is called a -contraction if the following conditions hold:
(Q1) T preserves edges of G;
(Q2) there exist nonnegative numbers l, c with such that
holds for each , where , , and .
Remark 3.17 Let be a -contraction. If there exists such that , then
-
(i)
T is both a -contraction and a -contraction,
-
(ii)
is T-invariant and is a -contraction.
Theorem 3.18 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be a -contraction. If is nonempty and -graph G is weakly connected, then T is a PO.
Proof If , then for all . First, we show that the sequence converges to 0. From Definition 3.16, we have
Thus,
is decreasing and bounded from below and so
converges to a nonnegative number L. If , we obtain
that is, , a contradiction. Hence as . It follows that . Suppose that
Then there exists a and a sequence such that
Hence we get the following:
Assume that there is an and there exist such that for each and
Then we choose the sequence and such that for each , is minimal in the sense that but , for all . Now, let be such that , then we have
Thus, taking the limit as , and Proposition 2.8, we have
Therefore,
Now,
If , then we have
a contradiction for . Hence, is a ρ-Cauchy sequence. By ρ-completeness of , there exists such that as and for all and G is a -graph, then there exists a subsequence such that for each . Also, for each . From Remark 2.5 and (), it follows that
Taking the limit as , we have
Since (), we obtain
which implies that , as . Thus,
Hence and .
Finally, we prove that is a unique fixed point. Suppose that T has another fixed point . Since G is a -graph, there exists a subsequence of such that and for each . Furthermore, as G is weakly connected, . We have
a contradiction. Hence, . □
In Theorem 3.18, if we replace the condition that G is a -graph with orbitally -continuity of T, then we have the following theorem.
Theorem 3.19 Let be a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be a -contraction, which is orbitally -continuous. Assume that the set is nonempty and the graph G is weakly connected. Then T is a PO.
In the following corollaries, suppose that is a ρ-complete modular space endowed with a graph G, where ρ satisfies the -condition and let be edge-preserving and the set be nonempty.
Corollary 3.20 Assume
-
(i)
the -graph G is weakly connected and
-
(ii)
there exist nonnegative numbers l, c with such that
hold for each with and . Then T is a PO.
Corollary 3.21 Assume
-
(i)
the -graph G is weakly connected and
-
(ii)
there exist nonnegative numbers l, c with such that
for each with and . Then T is a PO.
Now we provide examples in support of our results.
Example 3.22 Let and , for all . Consider
and , . Then G is weakly connected and -graph, is nonempty and T is a -contraction where , , . Hence, T is a PO.
Example 3.23 Let and , for all . Consider
Define as follows:
Then G is weakly connected and -graph, is nonempty, and T is a Hardy-Rogers type -contraction where , , , , , , and . Moreover, 0 is a unique fixed point of T.
Example 3.24 Let and , for all . Consider
and , . Then G is weakly connected and a -graph, is nonempty and T is a -contraction where , , , , and . Thus all conditions of Theorem 3.18 are satisfied. Moreover, T is a PO.
Remark 3.25 In the above examples, if we use , the conclusions remain the same.
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Ran ACM, Reuring MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4
Abbas M, Khamsi MA, Khan AR: Common fixed point and invariant approximation in hyperbolic ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 25 10.1186/1687-1812-2011-25
Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060
Jachymski J: The contraction principle for mappings on a metric space endowed with a graph. Proc. Am. Math. Soc. 2008, 136: 1359–1373.
Abbas M, Nazir T: Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph. Fixed Point Theory Appl. 2013., 2013: Article ID 20
Öztürk M, Girgin E: On some fixed-point theorems for ψ -contraction on metric space involving a graph. J. Inequal. Appl. 2014., 2014: Article ID 39 10.1186/1029-242X-2014-39
Samreen M, Kamran T: Fixed point theorems for integral G -contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 149
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60: 71–76.
Rhaodes BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 26: 257–290.
Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2002, 29: 531–536. 10.1155/S0161171202007524
Akram M, Zafar AA, Siddiqui AA: A general class of contractions: A -contractions. Novi Sad J. Math. 2002, 38(1):25–33.
Bianchini R: Su un problema di S. Reich riguardante la teoria dei punti fissi. Boll. Unione Mat. Ital. 1972, 5: 103–108.
Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 5: 1–11.
Nakano H Tokyo Math. Book Ser. 1. In Modulared Semi-Ordered Linear Spaces. Maruzen, Tokyo; 1950.
Musielak T, Orlicz W: On modular spaces. Stud. Math. 1959, 18: 49–65.
Koshi S, Shimogaki T: On F -norms of quasi-modular spaces. J. Fac. Sci. Hokkaido Univ., Ser. I 1961, 15(3–4):202–218.
Mazur S, Orlicz W: On some classes of linear spaces. Stud. Math. 1958, 17: 97–119. (Reprinted in Wladyslaw Orlicz Collected Papers, pp. 981–1003, PWN, Warszawa (1988))
Yamamuro S: On conjugate spaces of Nakano spaces. Trans. Am. Math. Soc. 1959, 90: 291–311. 10.1090/S0002-9947-1959-0132378-1
Turpin PH: Fubini inequalities and bounded multiplier property in generalized modular spaces. Comment. Math. 1978, 1: 331–353. (Tomus specialis in honorem Ladislai Orlicz)
Luxemburg, WAJ: Banach function spaces. Thesis, Delft Inst. of Techn. Asser., The Netherlands (1955)
Khamsi MA, Kozolowski WK, Reich S: Fixed point theory in modular function spaces. Nonlinear Anal. 1990, 14: 935–953. 10.1016/0362-546X(90)90111-S
Benavides TD, Khamsi MA, Samadi S: Asymptotically non-expansive mappings in modular function spaces. J. Math. Anal. Appl. 2002, 265: 249–263. 10.1006/jmaa.2000.7275
Benavides TD, Khamsi MA, Samadi S: Asymptotically regular mappings in modular function spaces. Sci. Math. Jpn. 2001, 53: 295–304.
Benavides TD, Khamsi MA, Samadi S: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Anal. 2001, 46: 267–278. 10.1016/S0362-546X(00)00117-6
Khamsi MA: A convexity property in modular function spaces. Math. Jpn. 1996, 44: 269–279.
Mongkolkeha C, Kumam P: Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 705943
Beygmohammadi M, Razani A: Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space. Int. J. Math. Math. Sci. 2010., 2010: Article ID 317107
Liu Z, Li X, Kang SM, Cho SY: Fixed point theorems for mappings satisfying contractive conditions of integral type and applications. Fixed Point Theory Appl. 2011., 2011: Article ID 64
Mongkolkeha C, Kumam P: Some fixed point results for generalized weak contraction mappings in modular spaces. Int. J. Anal. 2013., 2013: Article ID 247378
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Öztürk, M., Abbas, M. & Girgin, E. Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph. Fixed Point Theory Appl 2014, 220 (2014). https://doi.org/10.1186/1687-1812-2014-220
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DOI: https://doi.org/10.1186/1687-1812-2014-220