Fixed Point and Common Fixed Point Theorems for Generalized Weak Contraction Mappings of Integral Type in Modular Spaces

(1)

Volume 2011, Article ID 705943,12pages doi:10.1155/2011/705943

Research Article

Fixed Point and Common Fixed Point Theorems

for Generalized Weak Contraction Mappings of

Integral Type in Modular Spaces

Chirasak Mongkolkeha and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,[email protected]

Received 11 January 2011; Accepted 19 April 2011

Academic Editor: S. M. Gusein-Zade

Copyrightq2011 C. Mongkolkeha and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove new fixed point and common fixed point theorems for generalized weak contractive mappings of integral type in modular spaces. Our results extend and generalize the results of A. Razani and R. Moradi2009and M. Beygmohammadi and A. Razani2010.

1. Introduction

LetX, dbe a metric space. A mappingT :X → Xis a contraction if

dTx, Ty≤kdx, y, 1.1

where 0 < k < 1. The Banach Contraction Mapping Principle appeared in explicit form in Banach’s thesis in 19221. For its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many diﬀerent directions; see2–6. In 1997 Alber and Guerre-Delabriere7introduced the concept of weak contraction in Hilbert spaces, and Rhoades8

has showed that the result by Akber et al. is also valid in complete metric spaces A mapping T :X → Xis said to be weakly contractive if

(2)

whereφ : 0,∞ → 0,∞is continuous and nondecreasing function such thatφt 0 if and only ift 0. If one takesφt 1−ktwhere 0 < k < 1, then1.2reduces to1.1. In 2002, Branciari9gave a fixed point result for a single mapping an analogue of Banach’s contraction principle for an integral-type inequality, which is stated as follow.

Theorem 1.1. LetX, dbe a complete metric space,α∈0,1,f :X → Xa mapping such that for eachx, y∈X,

_dfx,fy

0

ϕtdt≤α

_dx,y

0

ϕtdt, 1.3

whereϕ : → is a Lebesgue integrable which is summable, nonnegative, and for allε > 0,

_ε

0ϕtdt >0. Then, f has a unique fixed pointz∈Xsuch that for eachx∈X,limn→ ∞fnxz.

Afterward, many authors extended this work to more general contractive conditions. The works noted in10–12are some examples from this line of research.

The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano 13 and redefined by Musielak and Orlicz 14. A lot of mathematicians are interested, fixed points of Modular spaces, for example15–22. In 2009, Razani and Moradi

23studied fixed point theorems forρ-compatible maps of integral type in modular spaces. Recently, Beygmohammadi and Razani24proved the existence for mapping defined on a complete modular space satisfying contractive inequality of integral type.

In this paper, we study the existence of fixed point and common fixed point theorems forρ-compatible mapping satisfying a generalize weak contraction of integral type in mod-ular spaces.

First, we start with a brief recollection of basic concepts and facts in modular spaces.

Definition 1.2. Let X be a vector space over or . A functional ρ : X → 0,∞is called

a modular if for arbitrary f and g, elements of X satisfy the following conditions:

1ρf 0 if and only iff0;

2ραf ρffor all scalarαwith|α|1;

3ραf βg≤ρf ρg, wheneverα, β≥0 andαβ1.If we replace3by

4ραf βg≤ αs_ρ_f _βs_ρ_g_{, for}_{α, β} _≥ _0,_αs_βs _{1 with an s} _∈₀_,₁_{, then the}

modularρis called s-convex modular, and ifs1,ρis called convex modular.

Ifρis modular inX, then the set defined by

Xρ

x∈X:ρλx−→0 asλ−→0 1.4

is called a modular space.Xρis a vector subspace ofX.

Definition 1.3. A modular ρ is said to satisfy theΔ2-condition ifρ2fn → 0 as n → ∞,

wheneverρfn → 0 asn → ∞.

Definition 1.4. Let Xρbe a modular space. Then,

1the sequencefn_n∈inXρis said to beρ-convergent tof∈Xρifρfn−f → 0, as

(3)

2the sequencefn_n∈inXρis said to beρ-Cauchy ifρfn−fm → 0, asn, m → ∞,

3a subsetCofXρis said to beρ-closed if theρ-limit of aρ-convergent sequence ofC

always belong toC,

4a subset C of Xρ is said to be ρ-complete if any ρ-Cauchy sequence in C is ρ

-convergent sequence and its is inC,

5a subsetCofXρis said to beρ-bounded ifδρC sup{ρf−g;f, g∈C}<∞.

Definition 1.5. Let C be a subset of Xρ and T : C → Can arbitrary mapping.T is called

aρ-contraction if for eachf, g∈Xρthere existsk <1 such that

ρTf−Tg≤kρf−g. 1.5

Definition 1.6. LetXρbe a modular space, whereρsatisfies theΔ2-condition. Two

self-map-pingsT and f ofXρare calledρ-compatible ifρTfxn −fTxn → 0 asn → ∞, whenever

{xn}n∈is a sequence inXρsuch thatfxn → zandTxn → zfor some pointz∈Xρ.

2. A Common Fixed Point Theorem for

ρ

-Compatible Generalized

Weak Contraction Maps of Integral Type

Theorem 2.1. LetXρbe aρ-completemodular space, whereρsatisfies theΔ2-condition. Letc, l∈ _,_{c > l}_and_{T, f}_:_X

ρ → Xρare twoρ-compatiblemappings such thatTXρ⊆fXρand

_{ρcT x−T y}

0

ϕtdt≤

_{ρlf x−f y}

0

ϕtdt−φ

_{ρlf x−f y}

0

ϕtdt , 2.1

for all x, y ∈ Xρ, where ϕ : 0,∞ → 0,∞ is a Lebesgue integrable which is summable,

nonnegative, and for allε > 0, ₀εϕtdt > 0 andφ : 0,∞ → 0,∞is lower semicontinuous function withφt> 0 for allt > 0 andφt 0 if and only ift0. If one ofT orfis continuous, then there exists a unique common fixed point ofT andf.

Proof. Letx ∈ Xρand generate inductively the sequence{Txn}n∈ as follow:Txn fxn1.

First, we prove that the sequence{ρcTxn−Txn−1}converges to 0. Since,

_{ρcT xn−T xn−1}

0

ϕtdt≤

_{ρlf xn−f xn−1}

0

ϕtdt−φ

0

ϕtdt

≤

0

ϕtdt

≤

_{ρlT xn−1−T xn−2}

0

ϕtdt

<

_{ρcT xn−1−T xn−2}

0

ϕtdt.

(4)

This means that the sequence {₀ρcT xn−T xn−1} is decreasing and bounded below. Hence, there existsr ≥0 such that

lim

n→ ∞

0

ϕtdtr. 2.3

Ifr >0, then limn→ ∞

0 ϕtdtr >0. Takingn → ∞in the inequality2.2which

is a contradiction, thusr0. This implies that

ρcTxn−Txn−1−→0 asn−→ ∞. 2.4

Next, we prove that the sequence{Txn}n∈isρ-Cauchy. Suppose{cTxn}n∈is notρ

-Cauchy, then there existsε > 0 and sequence of integers{mk},{nk}withmk > nk ≥ksuch

that

ρcTxmk−Txnk≥ε fork1,2,3, . . . 2.5

We can assume that

ρcTxmk−1−Txnk< ε. 2.6

Letmkbe the smallest number exceedingnkfor which2.5holds, and

θk

m∈ | ∃nk∈; ρcTxm−Txnk≥ε, m > nk≥k

. 2.7

Sinceθk ⊂ and clearlyθk/∅, by well ordering principle, the minimum element ofθk is

denoted bymkand obviously2.6holds. Now, letα∈ be such thatl/c1/α1, then we

get

ε

0

ϕtdt≤

_{ρcT x}_mk_{−T x}_nk

0

ϕtdt

≤

_{ρlf x}

mk−f xnk

0

ϕtdt−φ

ρlf x_mk−f x_nk

0

ϕtdt

≤

_{ρlf x}_mk_{−f x}_nk

0

ϕtdt

≤

_{ρlT x}

mk−1−T xnk−1

0

ϕtdt,

(5)

ρlTxmk−1−Txnk−1 ρlTxmk−1−TxnkTxnk −Txnk−1

ρ

l

ccTxmk−1−Txnk 1

ααlTxnk −Txnk−1

≤ρcTxmk−1−Txnk ραlTxnk −Txnk−1

< εραlTxnk−Txnk−1.

2.9

Using theΔ2-conditionand2.4, we obtain

lim

n→ ∞ραlTxnk −Txnk−1 0. 2.10

It follows that

lim

k→ ∞

_{ρlT x}

mk−1−T xnk−1

0

ϕtdt <

_ε

0

ϕtdt. 2.11

From2.8and2.11, we also have

_ε

0

ϕtdt≤

_{ρlT x}

mk−1−T xnk−1

0

ϕtdt

<

ε

0

ϕtdt,

2.12

which is a contradiction. Hence,{cTxn}n∈isρ-Cauchyand by theΔ2-condition,{Txn}n∈

isρ-Cauchy. SinceXρisρ-complete, there exists a pointu∈Xρsuch thatρTxn−u → 0 as

n → ∞. IfT is continuous, thenT2_x

n → TuandTfxn → Tuasn → ∞. SinceρcfTxn−

Tfxn → 0 asn → ∞, byρ-compatible,fTxn → Tuasn → ∞. Next, we prove thatuis

a unique fixed point ofT. Indeed,

_ρcT2_{xn−T xn}

0

ϕtdt

_{ρcTT xn−T xn}

0

ϕtdt

≤

_{ρlf T xn−f xn}

0

ϕtdt−φ

ρlf T xn−f xn

0

ϕtdt

≤

_{ρlf T xn−f xn}

0

ϕtdt.

2.13

Takingn → ∞in the inequality2.13, we have

_{ρcT u−u}

0

ϕtdt≤

_{ρlT u−u}

0

(6)

which implies thatρcTu−u 0 andTu u. SinceTXρ ⊆fXρ, there existsu1 such

thatuTufu1. The inequality,

_ρcT2_{xn−T u1}

0

ϕtdt≤

_{ρlf T xn−f u1}

0

ϕtdt−φ

0

ϕtdt

≤

0

ϕtdt

2.15

asn → ∞, yields

_{ρcT u−T u1}

0

ϕtdt≤

_{ρlT u−f u1}

0

ϕtdt 2.16

and, thus,

_{ρcu−T u1}

0

ϕtdt≤

_{ρlu−f u1}

0

ϕtdt

≤

_ρlu−u

0

ϕtdt

0,

2.17

which implies that,uTu1 fu1and alsofu fTu1 Tfu1 Tu usee25. Hence,

fuTu u. Suppose that there existsw ∈Xρsuch thatw Tw fwandw /u, we have

_ρcw−u

0 ϕtdt >0 and

_ρcw−u

0

ϕtdt

_{ρcT w−T u}

0

ϕtdt

≤

_{ρlf w−f u}

0

ϕtdt−φ

ρlf w−f u

0

ϕtdt

<

_{ρlf w−f u}

0

ϕtdt

<

_ρcw−u

0

ϕtdt,

2.18

which is a contradiction. Hence,uwand the proof is complete.

(7)

Corollary 2.2see23. LetXρbe aρ-completemodular space, whereρsatisfies theΔ2-condition.

Supposec, l ∈ ,c > landT, h :Xρ → Xρare twoρ-compatiblemappings such thatTXρ ⊆

hXρand

_{ρcT x−T y}

0

ϕtdt≤k

_ρlhx−hy

0

ϕtdt, 2.19

for somek∈0,1, whereϕ: → is a Lebesgue integrable which is summable, nonnegative, and for allε >0,₀εϕtdt >0. If one ofhorTis continuous, then there exists a unique common fixed point ofhandT.

Corollary 2.3see23. LetXρbe aρ-completemodular space, whereρsatisfies theΔ2-condition.

Supposec, l ∈ ,c > landT, h :Xρ → Xρare twoρ-compatiblemappings such thatTXρ ⊆

hXρand

_{ρcT x−T y}

0

ϕtdt≤ψ

_ρlhx−hy

0

ϕtdt , 2.20

whereϕ : → is a Lebesgue integrable which is summable, nonnegative, and for allε > 0,

ε

0ϕtdt >0 andψ : → is a nondecreasing and right continuous function withψt< tfor

allt >0. If one ofhorTis continuous, then there exists a unique common fixed point ofhandT.

3. A Fixed Point Theorem for Generalized Weak Contraction

Mapping of Integral Type

Theorem 3.1. LetXρbe aρ-completemodular space, whereρsatisfies theΔ2-condition. Letc, l∈ _,_{c > l}_and_T _:_X

ρ → Xρbe a mapping such that for eachx, y∈Xρ,

_{ρcT x−T y}

0

ϕtdt≤

_ρlx−y

0

ϕtdt−φ

ρlx−y

0

ϕtdt , 3.1

whereϕ:0,∞ → 0,∞is a Lebesgue integrable which is summable, nonnegative, and for allε >0,

_ε

0ϕtdt >0 andφ:0,∞ → 0,∞is lower semicontinuous function withφt>0 for allt >0

andφt 0 if and only ift0. Then,T has a unique fixed point.

Proof. First, we prove that the sequence{ρcTn_x₋_Tn−1_x_}_{converges to 0. Since,}

_ρcTn_x−Tn−1_x

0

ϕtdt≤

_ρlTn−1_x−Tn−2_x

0

ϕtdt−φ

0

ϕtdt

≤

0

ϕtdt

<

_ρcTn−1_x−Tn−2_x

0

ϕtdt,

(8)

it follows that the sequence{₀ρcTnx−Tn−1x}is decreasing and bounded below. Hence, there existsr ≥0 such that

lim

n→ ∞

0

ϕtdtr. 3.3

Ifr >0, then limn→ ∞

0 ϕtdtr >0, takingn → ∞in the inequality3.2which

is a contradiction, thusr0. So, we have

ρcTnx−Tn−1x−→0 as n−→ ∞. 3.4

Next, we prove that the sequence {Tn_x_}

n∈ isρ-Cauchy. Suppose {cT

n_x_}

n∈ is notρ

-Cauchy, there existsε >0 and sequence of integers{mk},{nk}withmk> nk≥ksuch that

ρcTmkx−Tnkx≥ε fork1,2,3, . . . . 3.5

We can assume that

ρcTmk−1x−Tnkx< ε. 3.6

Letmkbe the smallest number exceedingnkfor which3.5holds, and

θk

m∈ | ∃nk∈; ρcT

m_x₋_Tnk_x_≥_{ε, m > n}

k≥k

. 3.7

Sinceθk ⊂ and clearlyθk/∅, by well ordering principle, the minimum element ofθk is

denoted bymkand obviously3.6holds. Now, letα∈ be such thatl/c1/α1, then we

get

_ε

0

ϕtdt≤

_ρcTmk_x−Tnk_x

0

ϕtdt

≤

_ρlTmk−1_x−Tnk−1_x

0

ϕtdt−φ

_ρlTmk−1_x−Tnk−1_x

0

ϕtdt

≤

_ρlTmk−1_x−Tnk−1_x

0

ϕtdt,

, 3.8

ρlTmk−1x−Tnk−1xρlTmk−1x−TnkxTnkx−Tnk−1x

ρ

l cc

Tmk−1x−Tnkx 1 ααl

Tnkx−Tnk−1x

≤ρcTmk−1_x₋_Tnk_x_ρ_αl_Tnk_x₋_Tnk−1_x

< εραlTnkx−Tnk−1x.

(9)

Using theΔ2-conditionand3.4, we obtain

lim

k→ ∞ρ

αlTnkx−Tnk−1x0, _3.10

lim

k→ ∞

0 ϕtdt <

ε

0ϕtdt. 3.11

From3.8and3.11, we have

ε

0

ϕtdt≤

0

ϕtdt

<

_ε

0

ϕtdt,

3.12

which is a contradiction. Hence,{cTnx}_n∈ isρ-Cauchy and again by theΔ2-condition,

{Tn_x_}

n∈isρ-Cauchy. SinceXρisρ-complete, there exists a pointu∈Xρsuch thatρT

n_x₋

u → 0 asn → ∞. Next, we prove thatuis a unique fixed point ofT. Indeed,

ρc

2u−Tu

ρc

2

u−Tn1xTn1x−Tu

≤ρcu−Tn1xρcTn1x−Tu,

3.13

_ρcTn1_{x−T u}

0

ϕtdt≤

_ρlTn_x−u

0

ϕtdt−φ

ρlTn_x−u

0

ϕtdt

≤

_ρlTn_x−u

0

ϕtdt.

3.14

SinceρTnx−u → 0 asn → ∞, we obtain

lim

n→ ∞

_ρcTn1_{x−T u}

0

ϕtdt≤0, 3.15

which implies that

ρcTn1x−Tu−→0 asn−→ ∞. 3.16

So, we have

(10)

Thusρc/2u−Tu 0 andTuu. Suppose that there existsw∈Xρsuch thatTwwand

w /u, we have₀ρcw−uϕtdt >0 and

_ρcw−u

0

ϕtdt

_{ρcT w−T u}

0

ϕtdt

≤

_ρlw−u

0

ϕtdt−φ

_ρlw−u

0

ϕtdt

<

_ρlw−u

0

ϕtdt

<

_ρcw−u

0

ϕtdt,

3.18

which is a contradiction. Hence,uwand the proof is complete.

Corollary 3.2. LetXρbe aρ-completemodular space, whereρsatisfies theΔ2-condition. Letf :

Xρ → Xρbe a mapping such that there exists anλ ∈0,1andc, l∈ wherel < cand for each

x, y∈Xρ,

_{ρcf x−f y}

0

ϕtdt≤λ

_ρlx−y

0

ϕtdt 3.19

whereϕ:0,∞ → 0,∞is a Lebesgue integrable which is summable, nonnegative, and for allε >0,

ε

oϕtdt > o. Then,T has a unique fixed point inXρ.

Corollary 3.3see24. LetXρbe aρ-completemodular space whereρsatisfies theΔ2-condition.

Assume thatψ: → 0,∞is an increasing and upper semicontinuous function satisfyingψt< t

for allt >0. Letϕ:0,∞ → 0,∞be a Lebesgue integrable which is summable, nonnegative, and for allε > 0,₀εϕtdt > 0 and letf :Xρ → Xρbe a mapping such that there arec, l ∈ where

l < c,

_{ρcT x−T y}

0

ϕtdt≤ψ

_ρlx−y

0

ϕtdt , 3.20

for eachx, y∈Xρ. Then,T has a unique fixed point inXρ.

Acknowledgments

(11)

References

1 S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

2 G. Jungck, “Compatible mappings and common fixed points,” International Journal of Mathematics and

Mathematical Sciences, vol. 9, no. 4, pp. 771–779, 1986.

3 G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,” Far East

Journal of Mathematical Sciences, vol. 4, no. 2, pp. 199–215, 1996.

4 G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,” Indian Journal

of Pure and Applied Mathematics, vol. 29, no. 3, pp. 227–238, 1998.

5 W. Sintunavarat and P. Kumam, “Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1877–1881, 2009.

6 W. Sintunavarat and P. Kumam, “Weak condition for generalized multi-valuedf, α, β-weak contrac-tion mappings,” Applied Mathematics Letters, vol. 24, no. 4, pp. 460–465, 2011.

7 Y. I. Alber and S. Guerre-Delabriere, “Principle of weakly contractive maps in Hilbert spaces,” in New

Results in Operator Theory and Its Applications, vol. 98 of Operator Theory: Advances and Applications, pp.

7–22, Birkh¨auser, Basel, Switzerland, 1997.

8 B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 47, no. 4, pp. 2683–2693, 2001.

9 A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531– 536, 2002.

10 B. E. Rhoades, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 63, pp. 4007– 4013, 2003.

11 W. Sintunavart and P. Kumam, “Gregus-type common fixed point theorems for tangential multival-ued mappings of integral type in metric spaces,” International Journal of Mathematics and Mathematical

Sciences, vol. 2011, Article ID 923458, 12 pages, 2011.

12 W. Sintunavarat and P. Kumam, “Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type,” Journal of Inequalities and Applications. In Press.

13 H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo Mathematical Book Series, Maruzen Co. Ltd, Tokyo, Japan, 1950.

14 J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959.

15 T. Dominguez Benavides, M. A. Khamsi, and S. Samadi, “Uniformly Lipschitzian mappings in modular function spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 2, pp. 267– 278, 2001.

16 M. A. Khamsi, “Quasicontraction mappings in modular spaces withoutΔ2-condition,” Fixed Point

Theory and Applications, vol. 2008, Article ID 916187, 6 pages, 2008.

17 M. A. Khamsi, W. M. Kozłowski, and S. Reich, “Fixed point theory in modular function spaces,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 11, pp. 935–953, 1990.

18 P. Kumam, “On nonsquare and Jordan-von Neumann constants of modular spaces,” Southeast Asian

Bulletin of Mathematics, vol. 30, no. 1, pp. 69–77, 2006.

19 P. Kumam, “On uniform Opial condition, uniform Kadec-Klee property in modular spaces and application to fixed point theory,” Journal of Interdisciplinary Mathematics, vol. 8, no. 3, pp. 377–385, 2005.

20 P. Kumam, “Fixed point theorems for nonexpansive mappings in modular spaces,” Archivum

Mathe-maticum, vol. 40, no. 4, pp. 345–353, 2004.

21 P. Kumam, “Some geometrical properties and fixed point theorems in modular spaces,” in

Interna-tional Conference on Fixed Point Theory and Applications, pp. 173–188, Yokohama Publishers, Yokohama,

Japan, 2004.

22 A. Razani, E. Nabizadeh, M. B. Mohamadi, and S. H. Pour, “Fixed points of nonlinear and asymptotic contractions in the modular space,” Abstract and Applied Analysis, vol. 2007, Article ID 40575, 10 pages, 2007.

23 A. Razani and R. Moradi, “Common fixed point theorems of integral type in modular spaces,” Bulletin

(12)

24 M. Beygmohammadi and A. Razani, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space,” International Journal of Mathematics and

Mathematical Sciences, vol. 2010, Article ID 317107, 10 pages, 2010.

(13)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

Discrete Mathematics

Journal of

Fixed Point and Common Fixed Point Theorems for Generalized Weak Contraction Mappings of Integral Type in Modular Spaces

Research Article