The (CLR g )-property for coincidence point theorems and Fredholm integral equations in modular metric spaces

Transcription

1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No., 17, ISSN Published by New York Business Global The (CLR g )-property for coincidence point theorems and Fredholm integral equations in modular metric spaces Phumin Sumalai 1, Poom Kumam 1,,5,, Y. J. Cho 3,4, Anantachai Padcharoen 1, 1 KMUTT-Fixed Point Research Laboratory, Department of Mathematics, Fixed Point Laboratory, Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), Thrung Khru, Bangkok, Thailand KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), Bangkok, Thailand 3 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, Korea 4 Center for General Education, China Medical University, Taichung, Taiwan 5 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Abstract. In paper, we prove some common fixed point theorems for the pair of self-mappings with the g-quasi-condition in modular metric spaces. Also, we modify and prove some common fixed point theorems by using the (CLR g )-property along with the weakly compatible mapping. Finally, we give some applications on integral equations to illustrate our main results. 1 Mathematics Subject Classifications: 47H9, 47H1, 54H5, 37C5. Key Words and Phrases: Coincidence point, fixed point, g-quasi-condition, (CLR g )-property, modular metric space. 1. Introduction In 1998, Jungck and Rhoades [1] introduced the notion of weakly compatible mappings as follows: Let X be a nonempty set. Two mappings f, g : X X are said to be weakly compatible if fx = gx implies fgx = gfx for any x X. In 11, Sintunavarat and Kumam [4] introduced a new relax condition is called the (CLR g )-property as follows: Corresponding author. addresses: phumin.su8@mail.kmutt.ac.th (P. Sumalai), poom.kum@kmutt.ac.th (P. Kumam), yjcho@gnu.ac.kr (Y. J. Cho), apadcharoen@yahoo.com (A. Padcharoen) 38 c 17 EJPAM All rights reserved.

2 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Suppose that (X, d) is a metric space and f, g : X X be two mappings. The mappings f and g are said to satisfy the common limit in the range of g (shortly, (CLR g )- property) if there exists a sequence {x n } in X such that lim fx n = lim gx n = gx n n for some x X. The importance of (CLR g )-property ensures that one does not require the closeness of range subspaces On the other hand, in 1, Chistyakov [] introduced the notion of a modular metric space which is a new generalization of a metric space. In the same way, Mongkolkeha et al. [3] proved the existence of fixed point theorems for contraction mappings as following: Let ω be a metric modular on X and X ω be a modular metric space induced by ω. If X ω is a complete modular metric space and T : X ω X ω be a mapping such there exists k [, 1) with ω (T x, T y) kω (x, y) for all x, y X ω and >, then T has a unique fixed point in X ω. Currently Aydi et al. [5] established some coincidence and common fixed point results for three self-mappings on a partially ordered cone metric space satisfying a contractive condition and proved an existence theorem of a common solution of integral equations. In the same way, Shatanawi et al.[6] studied some new real generalizations on coincidence points for weakly decreasing mappings satisfying a weakly contractive condition in an ordered metric space. Many author studies in modular metric spaces [11, 1, 13, 14, 15, 16, 17]. In this paper, we study and prove the existence of some coincidence point theorems for generalized contraction mappings in modular metric spaces and give some applications on integral equations for our main results.. Preliminaries In this section, we give some definitions and their properties for our main results. Definition 1. [7] Let (X, d) be a metric space. Two mappings f : X X and g : X X are said to satisfy the (E.A)-property if there exist a sequences {x n } in X such that for some t X. lim fx n = lim gx n = t n n Next, we introduce the notion of a modular metric space as follows: Definition. Let X be a linear space over R with θ X as its zero element. A functional ρ : X [, + ] is called a modular on X if, for all x, y, z X, the following conditions hold: (M1) ρ(x) = if and only if x = θ;

3 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), (M) ρ(x) = ρ( x); (M3) ρ(αx + βy) ρ(x) + ρ(y) whenever α, β and α + β = 1. { } The linear subspace X ρ := x X : lim ρ(x) = is called a modular space. Definition 3. [] Let X be a nonempty set. (1) A function ω : (, ) X X [, ] is called a metric modular on X if, for all x, y, z X, the following conditions hold: (MM1) ω (x, y) = for all > if and only if x = y; (MM) ω (x, y) = ω (y, x) for all > ; (MM3) ω +µ (x, y) ω (x, z) + ω µ (z, y) for all, µ >. () If, instead of the condition (MM1), we have the following condition: (MM1 ) ω (x, x) = for all >, then ω is called a (metric) pseudo-modular on X. Remark 1. A modular ω on a set X, the function < ω (x, y) [, ] for all x, y X, is a non-increasing on (, ). In fact, if < µ <, then the conditions (MM3), (MM1 ) and (MM) imply ω (x, y) ω µ (x, x) + ω µ (x, y) = ω µ (x, y). (1) It follows that, at each point >, the right limit ω + (x, y) := lim ω +ɛ(x, y) and ɛ + the left limit ω (x, y) := lim ω ε(x, y) exist in [, ] and the following two inequalities ε + hold: ω + (x, y) ω (x, y) ω (x, y). () for all x, y X. We know that, if x X, the set X ω = {x X : lim ω (x, x ) = } is a metric space, which is called a modular space, whose metric is given by d ω(x, y) = inf{ > : ω (x, y) } for all x, y X ω. Also, it follows that, if X is a real linear space, ρ : X [, ] and ( x y ) ω (x, y) = ρ for all > and x, y X, then ρ is a modular on X if and only if ω is a metric modular on X (see []). Example 1. [8] The following indexed objects ω are simple examples of a modular on a set X. Let > and x, y X. Then we have (1) ω (x, y) = if d(x, y), and ω (x, y) = if > d(x, y); () ω (x, y) = if < d(x, y), and ω (x, y) = if d(x, y).

4 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Definition 4. [3] Let X ω be a modular metric space. (1) The sequence {x n } in X ω is said to be ω-convergent to a point x X ω if ω (x n, x) as n for all > ; () The sequence {x n } in X ω is called an ω-cauchy sequence if ω (x m, x n ) as m, n for all > ; (3) A subset C of X ω is said to be ω-closed if the limit of a convergent sequence {x n } of C always belongs to C; (4) A subset C of X ω is said to be ω-complete if any ω-cauchy sequence {x n } in C is ω-convergent to a point is in C; (5) A subset C of X ω is said to be ω-bounded if, for all >, δ ω (C) = sup{ω (x, y) : x, y C} <. Definition 5. Let X ω be a modular metric space and f, g : X X be two mappings. The mappings f and g are said to satisfy the common limit in the range of g (shortly, (CLR g )-property) if lim fx n = lim gx n = gx n n for some x X ω. Definition 6. [9] Let X ω be a modular metric space. We say that ω satisfies the - condition if, for any sequence {x n } X ω and x X w, there exists a number >, possibly depending on {x n } and x, such that lim ω (x n, x) = for some > implies n lim ω (x n, x) = for all >. n Note that, in this paper, we suppose that ω is a modular on X and satisfies the -condition on X. 3. Fixed point results for the contractive condition Lemma 1. Let f and g be weakly compatible self-mappings of a set X ω. If f and g have a unique coincidence point, that is, t = fx = gx, then t is the common fixed point of f and g. Theorem 1. Let X ω be a modular metric space and f, g : X ω X ω be weakly compatible mappings such that f(x ω ) g(x ω ) and g(x ω ) is a ω-complete subspace of X ω. Suppose there exists number a [, 1 4 ) for all x, y X ω and > such that (a) there exists x, x 1 X ω such that ω (fx, gx 1 ) < ; (b) ω (fx, fy) a[ω (fx, gy) + ω (fy, gx) + ω (fx, gx) + ω (fy, gy)]. Then f and g have a coincidence point.

5 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Proof. Let x be an arbitrary point in X ω. Since f(x ω ) g(x ω ), there exists a sequence {x n } in X ω such that gx n = fx n 1 for all n 1. Now, setting x = x n and y = x n+1 in (b), we have ω (fx n, fx n+1 ) a[ω (fx n, fx n ) + ω (fx n+1, fx n 1 ) + ω (gx n+1, gx n ) + ω (fx n+1, fx n )] = a[ω (fx n+1, fx n 1 ) + ω (gx n+1, gx n ) + ω (fx n+1, fx n )] for all >. On the other hand, we have and so ω (fx n+1, fx n 1 ) ω (fx n+1, fx n ) + ω (fx n, fx n 1 ) = ω (fx n+1, fx n ) + ω (gx n+1, gx n ) ω (fx n, fx n+1 ) a[ω (fx n+1, fx n ) + ω (gx n+1, gx n ) + ω (gx n+1, gx n ) + ω (fx n+1, fx n )]. This implies that ω (fx n, fx n+1 ) a 1 a ω (gx n, gx n+1 ) for all n N, where α = a < 1. By induction, we have 1 a ω (fx n, fx n+1 ) α n ω (gx, gx 1 ) (3) for all n N. By (a), it follows that {fx n } is a ω-cauchy sequence. Since g(x ω ) is ω-complete, there exists u, v X ω such that u = g(v) and fx n u as n. Since ω satisfy the -condition on X, we have lim n ω (fx n, u) = for all > and hence lim ω (fx n, u) = lim ω (gx n, u) = (4) n n for all >. Letting x = x n and y = v in (b), we have ω (fx n, fv) a[ω (fx n, gv) + ω (fv, gx n ) + ω (fx n, gx n ) + ω (fv, gv)] a[ω (fx n, gv) + ω (fv, fx n ) + ω (fx n, gx n ) + ω (fv, gv)] and, by Remark 1, since the function ω (x, y) is non-increasing, we have ω (fx n, fv) a[ω (fx n, gv) + ω (fv, fx n ) + ω (fx n, gx n ) + ω (fv, gv)]. By (b), letting n in the above inequality, we have ω (fv, gv) [ω (fv, gv) + ω (fv, fv) + ω (fv, gv) + ω (fv, gv)]. Thus (1 4k)ω (fv, gv) for all > and so gv = fv = u, which proves that g and f have a coincidence point. Now, we generalize Theorem 1 by using (CLRg)-property for weakly compatible mappings as follows:

6 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Theorem. Let X ω be a modular metric space and f, g : X ω X ω be weakly compatible mappings such that f(x ω ) g(x ω ). Suppose there exists a number a [, 1 4 ) for all x, y X ω and > such that (a) there exists x, x 1 X ω such that ω (fx, gx 1 ) < ; (b) ω (fx, fy) a[ω (fx, gy) + ω (fy, gx) + ω (fx, gx) + ω (fy, gy)]. If f and g satisfy the (CLR g )-property, then f and g have a unique common fixed point. Proof. Since f and g satisfy the (CLR g )-property, there exists a sequence {x n } in X ω such that lim fx n = lim gx n = gx for some x X ω. From (b), we have n n ω (fx n, fx) a[ω (fx n, gx) + ω (fx, gx n ) + ω (fx n, gx n ) + ω (fx, gx)] for all n 1. Letting n, we have gx = fx. Let t = fx = gx. Since f and g are weakly compatible mappings, fgx = gfx implies that ft = fgx = gfx = gt. Now, we claim that ft = t. From (b), we have ω (ft, t) = ω (ft, fx) a[ω (ft, gx) + ω (fx, gt) + ω (ft, gt) + ω (fx, gx)] = a[ω (ft, gx) + ω (fx, gt)] = a[ω (ft, t) + ω (t, ft)] and, by Remark 1, since the function ω (x, y) is non-increasing, we have ω (ft, t) a[ω (ft, t) + ω (t, ft)]. This implies that (1 a)ω (ft, t) for all >, that is, ω (ft, t) = and so ft = t = gt. Thus t is a common fixed point of f and g. For the uniqueness of the common fixed point, we suppose that u is another common fixed point in X ω such that fu = gu. From (b), we have ω (gu, gt) = ω (fu, ft) a[ω (fu, gt) + ω (ft, gu) + ω (fu, gu) + ω (ft, gt)] = a[ω (fu, gt) + ω (ft, gu)] = a[ω (gu, gt) + ω (gt, gu)] and, by Remark 1, since the function ω (x, y) is non-increasing, we have ω (gu, gt) a[ω (gu, gt) + ω (gt, gu)]. This implies gu = gt. Thus, by Lemma 1, we have f and g have a unique common fixed point.

7 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Theorem 3. Let X ω be a modular metric space and f, g : X ω X ω be weakly compatible mappings such that f(x ω ) g(x ω ). Suppose that there exist a 1, a, a 3, a 4, a 5 [, 1 4 ) and 5 a i < 1 such that, for all x, y X ω and >, i=1 (a) there exists x, x 1 X ω such that ω (fx, gx 1 ) < ; (b) ω (fx, fy) a 1 ω (fx, gx)+a ω (fy, gy)+a 3 ω (fy, gx)+a 4 ω (fx, gy)+a 5 ω (gy, gx). If f and g satisfy (CLR g )-property, then f and g have a unique common fixed point. Proof. Since f and g satisfy the (CLR g )-property, there exists a sequence {x n } in X ω such that lim fx n = lim gx n = gx for some x X ω. From (b), we have n n ω (fx n, fx) a 1 ω (fx n, gx n ) + a ω (fx, gx) + a 3 ω (fx, gx n ) + a 4 ω (fx n, gx) + a 5 ω (gx, gx n ) for all n 1. By taking the limit n, we have ω (gx, fx) a 1 ω (gx, gx) + a ω (fx, gx) + a 3 ω (fx, gx) + a 4 ω (gx, gx) + a 5 ω (gx, gx) = (a + a 3 )ω (fx, gx). This implies that (1 a a 3 )ω (fx, gx) for all >, which is a contradiction. Thus fx = gx. Now, let t = fx = gx. Since f and g are weakly compatible mappings, we have fgx = gfx, which implies that ft = fgx = gfx = gt. Now, we show that gt = t. Suppose ω (gt, t) >. Then, from (b), we have ω (gt, t) = ω (ft, fx) a 1 ω (ft, gt) + a ω (fx, gx) + a 3 ω (fx, gt) + a 4 ω (ft, gx) + a 5 ω (gx, gt) a 3 ω (t, gt) + a 4 ω (gt, t) + a 5 ω (t, gt) = (a 3 + a 4 + a 5 )ω (gt, t). This implies that (1 a 3 a 4 a 5 )ω (gt, t) for all >, which is a contradiction. Thus t is a common fixed point of f and g. For the uniqueness of the common fixed point, we suppose that u is another common fixed point in X ω such that fu = gu. From (b), we have ω (u, t) = ω (gu, gt) = ω (fu, ft) a 1 ω (fu, gu) + a ω (ft, gt) + a 3 ω (ft, gu) + a 4 ω (fu, gt) + a 5 ω (gt, gu) a 3 ω (ft, gu) + a 4 ω (gu, ft) + a 5 ω (ft, gu) = (a 3 + a 4 + a 5 )ω (u, t).

8 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), This implies that (1 a 3 a 4 a 5 )ω (u, t) for all >, which is a contradiction. Thus ω (u, t) = and so u = t. Hence f and g have a unique common fixed point. By setting g = I Xω, we deduce the following result of fixed point for one self-mapping from Theorem 3. Corollary 1. Let X ω be an ω-complete modular metric space and f : X ω X ω such that, for all > and x, y X ω, ω (x, fx ) < and ω (fx, fy) a 1 ω (fx, x) + a ω (fy, y) + a 3 ω (fy, x) + a 4 ω (fx, y) + a 5 ω (x, y) where a 1, a, a 3, a 4, a 5 [, 1 4 ) with 5 i=1 a i < 1. Then f has a unique fixed point z. Further, for any x X ω, the Picard sequence {fx n } with an initial point x is ω-convergent to the fixed point z. Corollary. Let X ω be an ω-complete modular metric space and f : X ω X ω such that, for all > and x, y X ω, ω (x, fx ) < and ω (fx, fy) a 1 ω (fx, x) + a ω (fy, y) + a 3 ω (x, y) where a 1, a, a 3 [, 1 4 ) with a 1 + a + a 3 < 1. Then f has a unique fixed point. Corollary 3. Let X ω be an ω-complete modular metric space and f : X ω X ω such that, for all > and x, y X ω, ω (x, fx ) < and ω (fx, fy) aω (x, y) where a < 1. Then f has a unique fixed point. Now, we give some examples of the (CLR g )-property as follows: Example. Let X ω = [, ) be a modular metric space. Define two mappings f, g : X ω X ω by fx = x + 4 and gx = 5x for all x X ω, respectively. Now, we consider the sequence {x n } defined by x n = {1 + 1 n } for each n 1. Since f and g satisfy the (CLR g )-property. lim fx n = lim gx n = 5 = g(1) X ω, n n Example 3. The conclusion of Example remains true if the self-mappings f and g is defined on X ω by f(x) = x 4 and g(x) = x for all x X ω, respectively. Let a sequence {x n } be defined by x n = { 1 n } in X ω. Since f and g satisfy the (CLR g )-property. lim fx n = lim gx n = = g() X ω, n n

9 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Fixed point results for the strict contractive condition Definition 7. Let (X, ω) be a modular metric space and (f, g) be a pair of self-mappings on X ω. For any x, y X ω, consider the following sets: (x, y) = {ω (gx, gy), ω (gx, fx), ω (gy, fy), ω (gx, fy), ω (gy, fx)}, ω (gx, gy), ω (gx, fx), ω (gy, fy), ω (gx, fy) + ω (gy, fx) { (x, y) = ω (gx, gy), ω (gx, fx) + ω (gy, fy), ω (gx, fy) + ω (gy, fx) and define the following conditions: (C1) for any x, y X ω, there exists α (x, y) (x, y) such that 1 (x, y) = { ω (fx, fy) < α (x, y), (C) for any x, y X ω, there exists α 1 (x, y) 1 (x, y) such that ω (fx, fy) < α 1 (x, y), (C3) for any x, y X ω, there exists α (x, y) (x, y) such that ω (fx, fy) < α (x, y). These conditions are called the strict contractive conditions. Definition 8. Let (X, ω) be a modular metric space. Let f, g be self-mappings on X ω. Then f is called a g-quasi-contraction if, for some constant a (, 1), there exists α(x, y) (x, y) such that ω (fx, fy) aα(x, y) }, }. for all x, y X ω. Theorem 4. Let X ω be a modular metric space and f, g : X ω X ω are weakly compatible mappings such that f(x ω ) g(x ω ) satisfies the condition (C3) for all x, y X ω and >. If f and g satisfy the (CLR g )-property, then f and g have a unique common fixed point. Proof. Since f and g satisfy the (CLR g )-property, there exists a sequence {x n } in X ω such that lim fx n = lim gx n = gx for some x X ω. From (C3), we have n n ω (fx n, fx) < α (x n, x), where α (x n, x) (x n, x). Therefore, we have { (x n, x) = ω (gx n, gx), ω (gx n, fx n ) + ω (gx, fx), ω (gx n, fx) + ω (gx, fx n ) }.

10 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Now, we show that fx = gx. Suppose that fx gx. From (C3), we have the following three cases: Case 1. ω (fx n, fx) < α (x n, x). Taking limit as n, we have ω (gx, fx) < ω (gx, gx) =, which is a contradiction. Case. ω (fx n, fx) < ω (gx n, fx n ) + ω (gx, fx). Taking limit as n, we have ω (gx, fx) < ω (gx, gx) + ω (gx, fx) = 1 ω (gx, fx), which is a contradiction. Case 3. ω (fx n, fx) < ω (gx n, fx) + ω (gx, fx n ). Taking limit as n, we have ω (gx, fx) < ω (gx, fx) + ω (gx, gx) = 1 ω (gx, fx), which is a contradiction. Hence gx = fx in all the cases. Let t = fx = gx. Since f and g are weakly compatible mappings, fgx = gfx, which implies that ft = fgx = gfx = gt. Now, we show that ft = t. Suppose that ft t. From (C3), we have ω (ft, t) = ω (ft, fx) < α (t, x), where α (t, x) (t, x). Therefore, we have { (t, x) = ω (gt, gx), ω (gt, ft) + ω (gx, fx) = {ω (ft, t),, ω (ft, t)}., ω (gt, fx) + ω (gx, ft) } So, we have only two possible cases: Case 4. ω (ft, t) < ω (ft, t), which is a contradiction. Case 5. ω (ft, t) <, which is a contradiction. Hence ft = t = gt. Therefor, t is a common fixed point of f and g. For the uniqueness of the common fixed point, we suppose that u is another common fixed point in X ω such that fu = gu. From (C3), we have ω (t, u) = ω (gt, gu) = ω (ft, fu) < α (t, u), where α (t, u) (t, u). Therefore, we have { (t, u) = ω (gt, gu), ω (gt, ft) + ω (gu, fu) = {ω (gt, gu),, ω (gt, gu)}., ω (gt, fu) + ω (gu, ft) } So, we have only two possible cases. Case 6. ω (gu, gt) < ω (gu, gt), which is a contradiction. Case 7. ω (gu, gt) <, which is a contradiction. Hence gu = gt. implies u = t and so f and g have a unique common fixed point.

11 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Theorem 5. Let X ω be a modular metric space and f, g : X ω X ω be weakly compatible mappings such that f is the g-quasi-contraction for all x, y X ω and >. If f and g satisfy (CLR g )-property, then f and g have a unique common fixed point. Proof. Since f and g satisfy the (CLR g )-property, there exists a sequence {x n } in X ω such that lim fx n = lim gx n = gx for some x X ω. Since f is the g-quasi-contraction, n n we have ω (fx n, fx) aα (x n, x), where α (x n, x) (x n, x). Therefore, we have (x n, x) = {ω (gx n, gx), ω (gx, fx), ω (gx n, fx n ), ω (gx n, fx), ω (gx, fx n )}. Now, we have the following five cases: Case 1. ω (fx n, fx) aω (gx n, gx). Taking the limit as n, we have gx = fx. Case. ω (fx n, fx) aω (gx n, fx n ). Taking the limit as n, we have gx = fx. Case 3. ω (fx n, fx) aω (gx, fx). Taking the limit as n, we have gx = fx. Case 4. ω (fx n, fx) aω (gx, fx n ). Taking the limit as n, we have gx = fx. Case 5. ω (fx n, fx) aω (gx, fx n ). Taking the limit as n, we have gx = fx. Hence, in all the possible cases, gx = fx. Now, let t = fx = gx. Since f and g are weakly compatible mappings, it follows that fgx = gfx, which implies that ft = fgx = gfx = gt. Now, we claim that ft = t. Since f is the g-quasi-contraction, we have ω (ft, t) = ω (ft, fx) aα (t, x), where α (t, x) (t, x). Therefore, we have (t, x) = {ω (gt, gx), ω (gt, ft), ω (gx, fx), ω (gt, fx), ω (gx, ft)} = {ω (ft, t),,, ω (ft, t), ω (t, ft)}. Now, we have two cases. Case 6. ω (ft, t) aω (ft, t). This implies ft = t. Case 7. ω (ft, t). This implies ft = t. Hence ft = t = gt and so t is a common fixed point of f and g. For the uniqueness of the common fixed point t, we suppose that u is another common fixed point in X ω such that fu = gu. Since f is the g-quasi-contraction, we have ω (gt, gu) = ω (ft, fu) aα (t, u), where α (t, u) (t, u). Therefore, we have (t, u) = {ω (gt, gu), ω (gt, ft), ω (gu, fu), ω (gt, fu), ω (gu, ft)} = {ω (ft, fu),,, ω (ft, fu), ω (fu, ft)}

12 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), So, we have only two possible cases. Case 8. ω (ft, fu) aω (ft, fu). This implies ft = fu. Case 9. ω (ft, fu). This implies ft = fu. Therefore, f and g have a unique common fixed point. Example 4. Let X ω = (, 1] with ω (x, y) = 1 x y for all >. Consider the functions f and g defined by 4 5, if x (, 4 5 ], fx = 1 5, if x (4 5, 1]. 1 x 4, if x (, 4 5 ], gx = 9 1, if x (4 5, 1]. Choosing a sequences {x n } = { }, we can see that f and g enjoy the (CLRg)-property n lim fx n = lim gx n = 4 n n 5 = g(4 5 ). Also, f( 4 5 ) = g(4 5 ) implies fg(4 5 ) = gf(4 5 ), which shows that f and g are weakly compatible. Case 1. For each x, y (, 4 ], we have 5 and ω (fx, fy) = 1 fx fy = (x, y) = {ω (gx, gy), ω (gx, fx), ω (gy, fy), ω (gx, fy), ω (gy, fx)} = { 1 gx gy, 1 gx fx, 1 gy fy, 1 gx fy, 1 gy fx } = { 1 1 x 4 (1 y 4 ), 1 1 x 4 4 5, 1 1 y 4 4 5, 1 1 x 4 4 5, 1 1 y }. Thus, we obtain ω (fx, fy) aα (x, y), where a (, 1).

13 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), Case. For x (, 4 5 ] and y (4, 1] we have 5 and ω (fx, fy) = 1 fx fy = (x, y) = {ω (gx, gy), ω (gx, fx), ω (gy, fy), ω (gx, fy), ω (gy, fx)} = { 1 gx gy, 1 gx fx, 1 gy fy, 1 gx fy, 1 gy fx } = { 1 1 x 4 9 1, 1 1 x 4 4 5, , 1 1 x 4 1 5, }. Thus, we obtain ω (fx, fy) aα (x, y), where a (, 1). Case 3. For x ( 4 5, 1] and y (, 4 ], we have 5 and ω (fx, fy) = 1 fx fy = (x, y) = {ω (gx, gy), ω (gx, fx), ω (gy, fy), ω (gx, fy), ω (gy, fx)} = { 1 gx gy, 1 gx fx, 1 gy fy, 1 gx fy, 1 gy fx } = { (1 y 4 ), , 1 1 y 4 4 5, , 1 1 y }. Thus, we obtain ω (fx, fy) aα (x, y), where a (, 1). Case 4. For each x, y ( 4, 1], we have 5 ω (fx, fy) = 1 fx fy =

14 P. Sumalai, P. Kumam, Y. J. Cho, A. Padcharoen / Eur. J. Pure Appl. Math, 1 () (17), and (x, y) = {ω (gx, gy), ω (gx, fx), ω (gy, fy), ω (gx, fy), ω (gy, fx)} = { 1 gx gy, 1 gx fx, 1 gy fy, 1 gx fy, 1 gy fx } = { , , , , }. Thus, we obtain ω (fx, fy) aα (x, y), where a (, 1). Therefore, f and g satisfy all conditions of Theorem 5 are satisfied and x = 4 5 unique common fixed point of f and g. is the 5. Some applications to Fredholm integral equations The purpose of this section is to show the existence and uniqueness of a solution of Fredholm integral equations in modular metric spaces with a function space (C(I, R), ω ) and a contraction by using our main results. Consider the integral equation: fx(t) µ K(t, s)hx(s)ds = g(t), (5) where x : I R is an unknown function, g : I R and h, f : R R are two functions, µ is a parameter. The kernel K of the integral equation is defined by I R R, where I = [, r]. Theorem 6. Let K, f, g, h be continuous. Suppose that C R is such that, for all t, s I, K(t, s) C and, for each x (C(I, R), ω ), there exists y (C(I, R), ω ) such that (fy)(t) = g(t) + µ K(t, s)hx(s)ds for all r C(I, R). If f is injective, there exists L R such that, for all x, y R, hx hy L fx fy and {fx : x (C(I, R), ω )} is complete, then, for any µ ( 1 w (C(I, R), ω ) such that, for any x (C(I, R), ω ), [ fw(t) = lim fx n(t) = lim g(t) + µ x x and w is the unique solution of the equation (5). CrL, 1 ] K(t, s)hx n 1 (s)ds CrL), there exists (6)

15 REFERENCES 5 Proof. Let X ω = Y ω = (C(I, R), ω ) and define d(x, y) = max t I x(t) y(t) for all x, y X ω. Let T, S X ω X ω be the mappings defined as follows: (T x)(t) = g(t) + µ K(t, s)(hx)(s)ds, Sx = fx. Then, by the assumptions, S(X ω ) = {Sx : x X ω } is complete. Let x T (X ω ) for any x X ω and x (t) = T x(t). By the assumptions, there exists y X ω such that T x(t) = fy(t) and hence T (X ω ) S(X ω ). Since ω (T x, T y) = µ Therefore, for any µ µ L µ C L µ C [K(t, s)(hx)(s)ds] c (hx)(s) (hy)(s) ds (fx)(s) (fy)(s) ds (Sx)(s) (Sy)(s) ds ( ) sup (Sx)(t) (Sy)(t) t I L µ C L µ Crd(Sx, Sy). [K(t, s)(hy)(s)ds] ( ) 1 CrL, 1 CrL, there exists a unique w X ω such that ds fw(t) = lim x Sx n(t) = lim x T x n 1(t) = T (w)(t), x X ω for all t I, which is the unique solution of the equation (5). So, S and T have a coincidence point in X ω. Moreover, if either T or S is injective, then S and T have a unique coincidence point in X ω. Acknowledgements This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT. References [1] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (1) [] V.V. Chistyakov, Modular metric spaces I: Basic concepts, Nonlinear Anal. 7(1), 1-14.

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