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Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)

Volume 4 (2009), 215 – 238

FIXED POINT THEOREMS FOR GENERALIZEDWEAKLY CONTRACTIVE MAPPINGS

Ramendra Krishna Bose and Mrinal Kanti Roychowdhury

Abstract. In this paper several fixed point theorems for generalized weakly contractive

mappings in a metric space setting are proved. The set of generalized weakly contractive mappings

considered in this paper contains the family of weakly contractive mappings as a proper subset.

Fixed point theorems for single and multi-valued mappings, approximating scheme for common

fixed point for some mappings, and fixed point theorems for fuzzy mappings are presented. It

extends the work of several authors including Bose and Roychowdhury.

1 Introduction

Weakly contractive mappings in a Hilbert space setting was first introduced by Alberand Guerre-Delabriere (cf. [1]). Rhoades proved that most of the results in [1] holdin a Banach space setting, and Bae considered these type of multi-valued mappings(cf. [6]). Kamran, Zhang and Song, Beg and Abbas, Bose and Roychowdhuryconsidered some generalized versions of these mappings and proved some fixed pointtheorems (cf. [17, 27, 7, 9]). Recently, Dutta and Choudhury have given anothergeneralization of the weakly contractive mappings (for single-valued mappings) (cf.[12]). We have considered a family of generalized weakly contractive mappings whichcontains the class of mappings considered by Dutta and Choudhury, and also theclass of weakly contractive mappings. We have proved several fixed point theoremsfor both single-valued as well as multi-valued mappings of this type which extendsthe work of several authors (cf. [12, 5, 7, 24, 9]).

Approximating fixed points of some mappings by an iterative scheme is an areaof active research work. Mann (cf. [19]) introduced a one-step iterative scheme,Isikawa (cf. [15]) introduced a two-step iterative scheme, and Noor (cf. [21])introduced a three-step iterative scheme. Bose and Mukherjee (cf. [8]) and othersused Mann iterative scheme to approximate a fixed point of some mappings, andGhose and Debnath (cf. [13]) and others considered Ishikawa iterative scheme toapproximate a fixed point of some mappings. In this paper, we considered the

2000 Mathematics Subject Classification: 03E72; 47H04; 47H10; 54H25; 54C60.Keywords: Fixed point; Weakly contractive mapping; Weakly compatible mapping; Mann

iteration; Fuzzy mapping.

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216 R. K. Bose and M. K. Roychowdhury

(modified) Mann iterative scheme and (modified) Ishikawa iterative scheme whichwere first introduced by Rhoades (cf. [24]), and later on by Beg and Abbas (cf.[7]). Azam and Shakeel (cf. [5]) considered these schemes in slightly modifiedform for weakly contractive mappings with respect to f . We have extended theirresults to generalized weakly contractive mappings with respect to f and provedtheir results under less conditions, i.e., without imposing extra conditions on theconstant sequences in the iterative scheme.

Azam and Beg (cf. [4]) considered weakly contractive fuzzy mappings andproved a common fixed point theorem for a pair of such fuzzy mappings. Next,Bose and Roychowdhury considered such fuzzy mappings and its two generalizedversions, and proved some fixed point theorems (cf. [9]). The work in this paperextends/generalizes the work of Azam and Beg (cf. [4]), and Bose and Roychowdhury(cf. [9]).

In this paper in Section 2, we give all the basic definitions and lemmas thatare used. In Section 3, we have presented all fixed point theorems for single-valuedand multi-valued mappings and approximating scheme for common fixed point ofsome mappings. In Section 4, we have considered some fixed point theorems forgeneralized fuzzy weakly contractive mappings.

2 Basic Definitions and Lemmas

In this section first we give the following basic definitions for single and multi-valuedmappings, and then that for the fuzzy mappings. (X, d) always represents a metricspace, H represents the Haudorff distance induced by the metric d, and K(X) thefamily of nonempty compact subsets of X. A point x in a metric space (X, d) iscalled a fixed point of a multi-valued mapping T : X → 2X if x ∈ T (x). Note that, xis a fixed point of a multi-valued mapping T if and only if d(x, T (x)) = 0, wheneverT (x) is a closed subset of X.

Definition 1. (generalized weakly contractive single-valued mappings). A mappingT : X → X is said to be generalized weakly contractive with respect to f : X → X iffor all x, y ∈ X,

ψ(d(T (x), T (y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y))),

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). If f is the identity mapping on X, then themapping is said to be generalized weakly contractive.

Remark 2. The class of generalized weakly contractive mappings considered byDutta and Choudhury (cf. [12]) used an additional condition which is monotonicityof φ, which is not required anywhere. If ψ(t) = t for all t ∈ [0,∞), then the mapping

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Fixed point theorems for generalized weakly contractive mappings 217

T : X → X satisfying the above inequality is said to be weakly contractive withrespect to f and if f is the identity mapping, T is said to be weakly contractive.

Definition 3. (generalized weakly contractive multi-valued mappings). A multi-valued mapping T : X → K(X) is said to be generalized weakly contractive withrespect to f : X → X if for all x, y ∈ X,

ψ(H(T (x), T (y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y))),

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψis monotonically increasing (strictly). If f is the identity mapping on X, thenthe multi-valued mapping T : X → K(X) is simply said to be generalized weaklycontractive. If ψ(t) = t for all t ∈ [0,∞), then the mapping is said to be weaklycontractive (with respect to f).

Definition 4. Let (X, d) be a metric space and let f and g be self-mappings of X.The mappings f and g are called R-weakly commuting, provided there exists somepositive real number R such that

d(fgx, gfx) ≤ Rd(fx, gx)

for each x ∈ X. For details see Pant [22].

Note that R-weakly commuting mappings commute at their coincidence points.Jungck and Rhoades (cf. [16]) then defined a pair of self-mappings to be weaklycompatible if they commute at their coincidence points.

Definition 5. (cf. [26]) Let (X, d) be a metric space and I = [0, 1]. A mappingW : X × X × I → X is said to be a convex structure on X if for each (x, y, λ) ∈X ×X × I and u ∈ X,

d(u,W (x, y, λ)) ≤ λd(u, x) + (1− λ)d(u, y).

A metric space X together with the convex structure W is called a convex metricspace.

Definition 6. Let X be a convex metric space. A nonempty subset C ⊂ X is said tobe convex if W (x, y, λ) ∈ C whenever (x, y, λ) ∈ C ×C × [0, 1]. Takahashi (cf. [26])has shown that open spheres B(x, r) = {y ∈ X : d(x, y) < r} and closed spheresB[x, r] = {y ∈ X : d(x, y) ≤ r} are convex. Also if {Cα : α ∈ A} is a family ofconvex subsets of X, then

⋂{Cα : α ∈ A} is convex. All normed spaces and their

convex subsets are convex metric spaces.





Definition 7. (generalized modified Mann iterative scheme). Let (X, d) be a convexcomplete metric space (or Banach space) and let T, f be self-mappings on X suchthat for all x, y ∈ X


where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψis monotonically increasing (strictly). Assume that T (X) ⊂ f(X) and f(X) is aconvex subset of X. Define a sequence {yn} in f(X) as

yn = f(xn+1) = W (T (xn), f(xn), αn) (or (1−αn)f(xn) +αnT (xn)) x0 ∈ X,n ≥ 0,

where 0 ≤ αn ≤ 1 for each n ≥ 0. The sequence {yn} thus obtained is calledgeneralized modified Mann iterative scheme.

Remark 8. In the above definition if we put ψ(t) = t for all t ∈ [0,∞), then itreduces to the definition of modified Mann iterative scheme (cf. [7, 5]). If ψ(t) = tfor all t ∈ [0,∞) and f is the identity mapping on X, then the reduced definition iscalled Mann iterative scheme (cf. [24]).

Definition 9. (generalized modified Ishikawa iterative scheme). Let (X, d) be aconvex complete metric space (or Banach space) and let T, f be self-mappings on Xsuch that for all x, y ∈ X


where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψis monotonically increasing (strictly). Assume that T (X) ⊂ f(X) and f(X) is aconvex subset of X. Suppose two sequences {yn} and {zn} in f(X) are defined as

zn = f(xn+1) = W (T (vn), f(xn), αn) (or (1− αn)f(xn) + αnT (vn))yn = f(vn) = W (T (xn), f(xn), βn) (or (1− βn)f(xn) + βnT (xn)),

where x0 ∈ X and 0 ≤ αn, βn ≤ 1 for each n ≥ 0. Then the sequence {zn} thusobtained is called generalized modified Ishikawa iterative scheme.

Remark 10. In the above definition if we put ψ(t) = t for all t ∈ [0,∞), then thereduced definition we call as modified Ishikawa iterative scheme. If ψ(t) = t for allt ∈ [0,∞) and f is the identity mapping on X, then the reduced definition is calledIshikawa iterative scheme (cf. [24]).

Lemma 11. (cf. [20]) Let A and B be nonempty compact subsets of a metric space(X, d). If a ∈ A, then there exists b ∈ B such that d(a, b) ≤ H(A,B).





A real linear space X with a metric d is called a metric linear space if d(x+z, y+z) = d(x, y) and αn → α, xn → x =⇒ αnxn → αx. Let (X, d) be a metric linearspace. A fuzzy set A in a metric linear space X is a function from X into [0, 1]. Ifx ∈ X, the function value A(x) is called the grade of membership of x in A. Theα-level set of A, denoted by Aα, is defined by

Aα = {x : A(x) ≥ α} if α ∈ (0, 1],A0 = {x : A(x) > 0}.

Here B̄ denotes the closure of the (non-fuzzy) set B.

Definition 12. A fuzzy set A is said to be an approximate quantity if and only ifAα is compact and convex in X for each α ∈ [0, 1] and supx∈X A(x) = 1.

When A is an approximate quantity and A(x0) = 1 for some x0 ∈ X, Ais identified with an approximation of x0. For x ∈ X, let {x} ∈ W (X) withmembership function equal to the characteristic function χx of the set {x}.

Let F(X) be the collection of all fuzzy sets in X and W (X) be a sub-collectionof all approximate quantities.

Definition 13. Let A,B ∈W (X), α ∈ [0, 1]. Then we define

pα(A,B) = infx∈Aα, y∈Bα

d(x, y),

p(A,B) = supαpα(A,B),

Dα(A,B) = H(Aα, Bα),D(A,B) = sup

αDα(A,B).

where H is the Hausdorff distance induced by the metric d.

The function Dα(A,B) is called an α-distance between A,B ∈ W (X), and Da metric on W (X). We note that pα is a non-decreasing function of α and thusp(A,B) = p1(A,B). In particular if A = {x}, then p({x}, B) = p1(x,B) = d(x,B1).Next we define an order on the family W (X), which characterizes the accuracy of agiven quantity.

Definition 14. Let A,B ∈ W (X). Then A is said to be more accurate than B,denoted by A ⊂ B (or B includes A), if and only if A(x) ≤ B(x) for each x ∈ X.

The relation ⊂ induces a partial order on the family W (X).

Definition 15. Let X be an arbitrary set and Y be any metric linear space. F iscalled a fuzzy mapping if and only if F is a mapping from the set X into W (Y ).





Definition 16. For F : X → W (X), we say that u ∈ X is a fixed point of F if{u} ⊂ F (u), i.e. if u ∈ F (u)1.Lemma 17. (cf. [14]) Let x ∈ X and A ∈ W (X). Then {x} ⊂ A if and only ifpα(x,A) = 0 for each α ∈ [0, 1].Remark 18. Note that from the above lemma it follows that for A ∈W (X), {x} ⊂A if and only if p({x}, A) = 0. If no confusion arises instead of p({x}, A) we willwrite p(x,A).

Lemma 19. (cf. [14]) pα(x,A) ≤ d(x, y) + pα(y,A) for each x, y ∈ X.Lemma 20. (cf. [14]) If {x0} ⊂ A, then pα(x0, B) ≤ Dα(A,B) for each B ∈W (X).Lemma 21. (cf. [18]) Let (X, d) be a complete metric linear space, F : X →W (X)be a fuzzy mapping and x0 ∈ X. Then there exists x1 ∈ X such that {x1} ⊂ F (x0).Remark 22. Let f : X → X be a self map and T : X →W (X) be a fuzzy mappingsuch that ∪{T (X)}α ⊆ f(X) for α ∈ [0, 1]. Then from Lemma 21, it follows thatfor any chosen point x0 ∈ X there exist points x1, y1 ∈ X such that y1 = f(x1) and{y1} ⊂ T (x0). Here T (x)α = {y ∈ X : T (x)(y) ≥ α}.Definition 23. (generalized weakly contractive fuzzy mappings). A fuzzy mappingsT : X →W (X) is said to be generalized weakly contractive with respect to f : X → Xif for all x, y ∈ X,

ψ(D(T (x), T (y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y))),

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). If f is the identity mapping on X, then thefuzzy mapping T : X → W (X) is simply said to be generalized weakly contractive.If ψ(t) = t for all t ∈ [0,∞), then the mapping is said to be weakly contractive (withrespect to f).

3 Fixed point theorems for single and multi-valued mappings

In this section we prove the following main theorems of this paper concerning singleand multi-valued mappings.

Theorem 24. Let (X, d) be a complete metric space and let T, S : X → X beself-mappings such that for all x, y ∈ X

ψ(d(T (x), S(y))) ≤ ψ(d(x, y))− φ(d(x, y)), (3.1)

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Then there exists a unique point u ∈ X suchthat u = T (u) = S(u).





Proof. Let x0 ∈ X, we construct a sequence {xk} in X by taking x2k+1 = T (x2k)and x2k+2 = S(x2k+1) for all k ≥ 0. Hence by (3.1) we have,

ψ(d(x2k+1, x2k+2)) = ψ(d(T (x2k), S(x2k+1)))≤ ψ(d(x2k, x2k+1))− φ(d(x2k, x2k+1)) (3.2)≤ ψ(d(x2k, x2k+1)).

Again by (3.1) we have,

ψ(d(x2k+2, x2k+3)) = ψ(d(S(x2k+1), T (x2k+2)))= ψ(d(T (x2k+2), S(x2k+1)))≤ ψ(d(x2k+2, x2k+1))− φ(d(x2k+2, x2k+1)) (3.3)≤ ψ(d(x2k+2, x2k+1))= ψ(d(x2k+1, x2k+2)).

Thus for n ≥ 0 we have, ψ(d(xn+1, xn+2)) ≤ ψ(d(xn, xn+1)). As ψ is monotonicallyincreasing, from this inequality we have d(xn+1, xn+2) ≤ d(xn, xn+1), which impliesthat {d(xn, xn+1)} is a non-increasing sequence of positive real numbers and thereforetends to a limit ` ≥ 0. If possible, let ` > 0. By (3.2) and (3.3) for any n ≥ 0 wehave,

ψ(d(xn+1, xn+2)) ≤ ψ(d(xn, xn+1))− φ(d(xn, xn+1)).

Taking n→∞ and using the continuity of ψ and φ, we obtain

ψ(`) ≤ ψ(`)− φ(`) =⇒ φ(`) ≤ 0,

which is a contradiction as ` > 0, and φ(t) > 0 for t > 0. Therefore, d(xn, xn+1)→ 0as n → ∞. Now proceeding in the same way as in Theorem 3.4 in [9], it can beshown that the sequence {xn} is a Cauchy sequence. From the completeness ofX, it follows that there exists u ∈ X such that xn → u as n → ∞. Moreover,x2n → u and x2n+1 → u as n→∞. Now we show that u = T (u) = S(u). We knowx2n+1 = T (x2n) and x2n+2 = S(x2n+1). Note that,

ψ(d(x2n+1, S(u))) = ψ(d(T (x2n), S(u))) ≤ ψ(d(x2n, u))− φ(d(x2n, u)).

Letting n→∞ and using the continuity of both ψ and φ we have, ψ(d(u, S(u))) ≤ψ(0)− φ(0) = 0 =⇒ ψ(d(u, S(u))) ≤ 0. Now using the fact ψ(t) > 0 for t > 0 andψ(0) = 0 we have, ψ(d(u, S(u))) = 0 =⇒ d(u, S(u))) = 0 =⇒ u = S(u). Similarlywe can show u = T (u). Thus, we have u = T (u) = S(u).

If there exists another point v ∈ X such that v = T (v) = S(v), then we haveψ(d(u, v)) = ψ(d(T (u), S(v))) ≤ ψ(d(u, v)) − φ(d(u, v)) =⇒ φ(d(u, v)) ≤ 0, andhence u = v. Thus, the proof is complete.





Corollary 25. If we take T = S, then we have Theorem 2.1 of Dutta and Choudhury(cf. [12]).

Corollary 26. If we take ψ(t) = t for all t ∈ [0,∞), then Theorem 24 reduces tousual weak contraction theorem (cf. [24, Theorem 1]).

Theorem 27. Let (X, d) be a complete metric space and T, S : X → K(X) be twomappings such that for all x, y ∈ X

ψ(H(Tx, Sy)) ≤ ψ(d(x, y))− φ(d(x, y)), (3.4)

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Then T and S have a common fixed point.

Proof. Let {xk} be a sequence in X such that x2k+1 ∈ T (x2k) and x2k+2 ∈ S(x2k+1)for all k ≥ 0, and d(x2k+1, x2k+2) ≤ H(T (x2k), S(x2k+1)) (by Lemma 11). Hence,

ψ(d(x2k+1, x2k+2)) ≤ ψ(H(T (x2k), S(x2k+1)))≤ ψ(d(x2k, x2k+1))− φ(d(x2k, x2k+1))≤ ψ(d(x2k, x2k+1)).

Similarly we can show that

ψ(d(x2k+2, x2k+3)) ≤ ψ(d(x2k+1, x2k+2))− φ(d(x2k+1, x2k+2)) ≤ ψ(d(x2k+1, x2k+2)).

Thus for n ≥ 0 we have ψ(d(xn+1, xn+2)) ≤ ψ(d(xn, xn+1)). As ψ is monotonicallyincreasing, from this inequality it follows that for n ≥ 0, d(xn+1, xn+2) ≤ d(xn, xn+1),which shows that {d(xn, xn+1)} is a non-increasing sequence of positive real numbers.Now proceeding in the same way as in Theorem 3.4 in [9], it can be shown that thesequence {xn} is a Cauchy sequence in X. Hence from the completeness of X itfollows that xn → u for some u ∈ X. Moreover, x2n → u and x2n+1 → u asn → ∞. Now we show that u ∈ T (u) and u ∈ S(u). We know x2n+1 ∈ T (x2n) andx2n+2 ∈ S(x2n+1). Note that,

ψ(d(x2n+1, S(u))) ≤ ψ(d(T (x2n), S(u))) ≤ ψ(d(x2n, u))− φ(d(x2n, u)).

Letting n→∞ and using the continuity of both ψ and φ we have, ψ(d(u, S(u))) ≤ψ(0)− φ(0) = 0 =⇒ ψ(d(u, S(u))) ≤ 0. Now using the fact ψ(t) > 0 for t > 0 andψ(0) = 0 we have, ψ(d(u, S(u))) = 0 =⇒ d(u, S(u))) = 0 =⇒ u ∈ S(u). Similarly,we can show u ∈ T (u), i.e., T and S have a common fixed point.

Corollary 28. If T = S, then T has a fixed point.





Theorem 29. Let (X, d) be a metric space and T, S, f be self-mappings on X suchthat for all x, y ∈ X,

ψ(d(T (x), S(y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y)),

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). If T (X)∪S(X) ⊂ f(X) and f(X) is a completesubspace of X, then there exists a point p ∈ X such that f(p) = T (p) = S(p) (thepoint p is unique if f is one-to-one).

Proof. Let x0 be a point in X. Choose a point x1 ∈ X such that f(x1) = T (x0). Thiscan be done since T (X) ⊂ f(X). Similarly, choose x2 ∈ X such that f(x2) = S(x1).In general, having chosen xk ∈ X, we obtain xk+1 ∈ X such that f(x2k+1) = T (x2k)and f(x2k+2) = S(x2k+1) for any integer k ≥ 0. Hence, by the given hypothesis

ψ(d(f(x2k+1), f(x2k+2))) = ψ(d(T (x2k), S(x2k+1)))≤ ψ(d(f(x2k), f(x2k+1)))− φ(d(f(x2k), f(x2k+1))) (3.5)≤ ψ(d(f(x2k), f(x2k+1))),

and similarly,

ψ(d(f(x2k+2), f(x2k+3))) = ψ(d(S(x2k+1), T (x2k+2))) ≤ ψ(d(f(x2k+1), f(x2k+2))).(3.6)

As ψ is monotonically increasing, from (3.5) and (3.6) it follows {d(f(xn), f(xn+1))}is a non-increasing sequence of positive real numbers and therefore, tends to a limit` ≥ 0. By (3.5) and (3.6),

ψ(d(f(xn+1), f(xn+2))) ≤ ψ(d(f(xn), f(xn+1)))−φ(d(f(xn), f(xn+1))) for all n ≥ 0.


ψ(`) ≤ ψ(`)− φ(`) =⇒ φ(`) ≤ 0,

which is a contradiction as ` > 0, and φ(t) > 0 for t > 0. Therefore ` = 0, i.e.,limn→∞ d(f(xn), f(xn+1)) = 0. Now proceeding in the same way as in Theorem 3.4in [9], it can be shown that the sequence {f(xn)} is a Cauchy sequence in f(X). Asf(X) is a complete subspace of X, {f(xn)} has a limit q in f(X). Consequently, weobtain a point p in X such that f(p) = q. Thus, f(x2k+1) → q and f(x2k) → q ask →∞. Now by the given hypothesis,

ψ(d(f(x2k+1), S(p))) ≤ ψ(d(f(x2k), f(p)))− φ(d(f(x2k), f(p))).





Letting n→∞ and using the continuity of both ψ and φ we have, ψ(d(q, S(p))) ≤ψ(d(q, f(p))) − φ(d(q, f(p))) = ψ(0) − φ(0) = 0 =⇒ ψ(d(q, S(p))) ≤ 0. Nowusing the fact ψ(t) > 0 for t > 0 and ψ(0) = 0 we have, ψ(d(q, S(p))) = 0 =⇒d(q, S(p)) = 0 =⇒ q = S(p). Similarly, we can show q = T (p). Thus, we haveq = f(p) = T (p) = S(p) for a point p ∈ X. Clearly the point p is unique if f isone-to-one.

Remark 30. The above theorem extends Theorem 2.1 of Beg and Abbas (cf. [7]),and Theorem 2.3 of Azam and Shakeel (cf. [5])

Theorem 31. In the above theorem Theorem 29, if further we have the pairs (f, T )and (f, S) are weakly compatible (or R-weakly commuting) (see Definition 4), thenf, T and S have a common fixed point (and the point is unique if f is one-to-one).

Proof. Note thatR-weakly commuting mappings commute at their coincidence point.We have q = f(p) = T (p) = S(p) and this implies that f(T (p)) = T (f(p)) andf(S(p)) = S(f(p)). Also we have T (q) = f(q) = S(q). We claim that f(q) = q. Wehave,

ψ(f(q), q) = ψ(T (q), S(p)) ≤ ψ(d(f(q), f(p)))− φ(d(f(q), f(p)))=⇒ φ(d(f(q), f(p))) ≤ 0 =⇒ φ(d(f(q), f(p))) = 0=⇒ d(f(q), f(p)) = 0 =⇒ f(q) = f(p) = q,

which implies q is a fixed point of f , and then from f(T (p)) = T (f(p)) and f(S(p)) =S(f(p)) we have q is also a fixed point of both T and S. Hence, f, T and S have acommon fixed point. Clearly the point is unique if f is one-to-one.

Corollary 32. In the above theorem if T = S, then we have that f and T have acommon (unique if f is one-to-one) fixed point in X.

Remark 33. The above theorem extends Theorem 2.5 of Beg and Abbas (cf. [7]),and Theorem 2.5 of Azam and Shakeel (cf. [5]).

Proceeding in the same way as Theorem 29 (taking the sequence {xn} such thatxn+1 = Tn+1(xn) for n = 0, 1, 2, · · · ), the following theorem can also be proved.

Theorem 34. Let (X, d) be a metric space and {Ti}∞i=1 be a sequence of self-mappings on X such that for all x, y ∈ X,

ψ(d(Ti(x), Tj(y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y)) for all i, j ≥ 1,

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψis monotonically increasing (strictly). If Ti(X) ⊂ f(X) for all i and f(X) is acomplete subspace of X, then there exists a point p ∈ X such that f(p) = Ti(p) forall i ≥ 1 (the point p is unique if f is one-to-one).





Theorem 35. Let (X, d) be a convex metric space and let T, f be self-mappings onX such that for all x, y ∈ X


where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing, and ψis both monotonically increasing (strictly) and convex. If the pair (f, T ) is weaklycompatible (or R-weakly commuting) and T (X) ⊂ f(X) and f(X) is a convex andcomplete subspace of X, then the generalized modified Mann iterative scheme (seeDefinition 7) converges to a common fixed point of f and T .

Proof. By Corollary 32, we obtain a common fixed point q of f and T . Now consider

ψ(d(yn, q) = ψ(d(f(xn+1), f(p))) = ψ(d(W (T (xn), f(xn), αn), f(p)))≤ ψ((1− αn)d(f(xn), f(p)) + αnd(T (xn), f(p)))≤ (1− αn)ψ(d(f(xn), f(p))) + αnψ(d(T (xn), f(p)))≤ (1− αn)ψ(d(f(xn), f(p))) + αn(ψ(d(f(xn), f(p)))− φ(d(f(xn), f(p)))≤ ψ(d(f(xn), f(p)))− φ(d(f(xn), f(p))) (3.7)= ψ(d(yn−1, q)).

As ψ is monotonically increasing from the above inequality it follows that d(yn, q) ≤d(yn−1, q), i.e., {d(yn, q)} is a non-increasing sequence of positive real numbers, whichgives limn→∞ d(yn, q) = ` ≥ 0. Now if ` > 0, then from (3.7) as both ψ and φ arecontinuous, taking n → ∞ we have, limn→∞ ψ(d(yn, q)) ≤ limn→∞ ψ(d(yn−1, q)) −φ(d(yn−1, q)) =⇒ ψ(`) ≤ ψ(`) − φ(`) =⇒ φ(`) ≤ 0, which is a contradiction as` > 0 and φ(t) > 0 for t > 0. Therefore, ` = 0. Hence, the generalized modifiedMann iterative scheme converges to a common fixed point of f and T .

Corollary 36. If we take ψ(t) = t for all t ∈ [0,∞), then the modified Manniterative scheme converges to a common fixed point of f and T , which is the workof Azam and Shakeel (cf. [5, Theorem 2.7]) without the condition

∑αn =∞.

Remark 37. With reference to Theorem 31 (when f is one-to-one), one can showthat the generalized modified Mann iterative scheme (either for the mapping T orfor the mapping S) converges to a unique common fixed point of f , T and S.

Theorem 38. Let (X, d) be a normed linear space and let T, f be self-mappings onX such that for all x, y ∈ X


where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing, and ψ is both





monotonically increasing (strictly) and convex. If the pair (f, T ) is weakly compatible(or R-weakly commuting) and T (X) ⊂ f(X) and f(X) is a complete subspace of X,then the generalized modified Mann iterative scheme (see Definition 7) converges toa common fixed point of f and T .

Proof. By Corollary 32, we obtain a common fixed point q of f and T . Writingd(yn, q) = ||yn − q|| and yn = (1− αn)f(xn) + αnT (xn), and then proceeding in thesame way as in Theorem 35 we can show that limn→∞ ||yn − q|| = 0, that is, thegeneralized modified Mann iterative scheme converges to a common fixed point off and T .

Corollary 39. If we take ψ(t) = t for all t ∈ [0,∞), then the modified Manniterative scheme converges to a common fixed point of f and T , which is the workof Beg and Abbas (cf. [7, Theorem 2.6]) without the condition

∑αn =∞.

Corollary 40. If we take ψ(t) = t for all t ∈ [0,∞) and f is the identity mapping onX, then it reduces to the work of Rhoades (cf. [24]) without the condition

∑αn =∞.

Remark 41. Rhoades, Beg and Abbas, Azam and Shakeel in their proofs used anextra condition

∑αn =∞ on the sequence {αn}, which we do not need in our proof.

We can also prove the following two theorems.

Theorem 42. Let (X, d) be a convex metric space and let T, f be self-mappings onX such that for all x, y ∈ X


where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing, and ψis both monotonically increasing (strictly) and convex. If the pair (f, T ) is weaklycompatible (or R-weakly commuting) and T (X) ⊂ f(X) and f(X) is a convex andcomplete subspace of X, then the generalized modified Ishikawa iterative scheme (seeDefinition 9) converges to a common fixed point of f and T .

Corollary 43. If we take ψ(t) = t for all t ∈ [0,∞), then the modified Ishikawaiterative scheme converges to a common fixed point of f and T , which is the workof Azam and Shakeel (cf. [5, Theorem 2.8]) without the condition

∑αnβn =∞.

Theorem 44. Let (X, d) be a normed linear space and let T, f be self-mappings onX such that for all x, y ∈ X


where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing, and ψ is both





monotonically increasing (strictly) and convex. If the pair (f, T ) is weakly compatible(or R-weakly commuting) and T (X) ⊂ f(X) and f(X) is a complete subspace of X,then the generalized modified Ishikawa iterative scheme (see Definition 9) convergesto a common fixed point of f and T .

Corollary 45. If we take ψ(t) = t for all t ∈ [0,∞), then the modified Ishikawaiterative scheme converges to a common fixed point of f and T , which is the workof Beg and Abbas (cf. [7, Theorem 2.7]) without the condition

∑αnβn =∞ on the

sequence {αn}.

Corollary 46. If we take ψ(t) = t for all t ∈ [0,∞) and f is the identity mapping onX, then it reduces to the work of Rhoades (cf. [24]) without the condition

∑αnβn =

∞.

Remark 47. Rhoades, Beg and Abbas, Azam and Shakeel in their proofs used anextra condition

∑αnβn =∞, which we do not need in our proof.

Theorem 48. Let (X, d) be a metric space, f : X → X be a self-mapping andT, S : X → K(X) be multi-valued mappings such that for all x, y ∈ X,

ψ(H(T (x), S(y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y)),

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψis monotonically increasing (strictly). If ∪x∈XT (x) ⊂ f(X), ∪x∈XS(x) ⊂ f(X)and f(X) is a complete subspace of X, then there exists a point p ∈ X such thatf(p) ∈ T (p) and f(p) ∈ S(p).

Proof. Let x0 be a point in X. Choose a point x1 ∈ X such that f(x1) ∈ T (x0). Thiscan be done since ∪x∈XT (x) ⊂ f(X). Similarly, choose x2 ∈ X such that f(x2) ∈S(x1) and such that d(f(x1), f(x2)) ≤ H(T (x0), S(x1)). In general, having chosenxk ∈ X, we obtain xk+1 ∈ X such that f(x2k+1) ∈ T (x2k), f(x2k+2) ∈ S(x2k+1),and such that d(f(x2k+1), f(x2k+2)) ≤ H(T (x2k), S(x2k+1)) for any integer k ≥ 0(by Lemma 11). Hence, by the given hypothesis

ψ(d(f(x2k+1), f(x2k+2))) ≤ ψ(H(T (x2k), S(x2k+1)))≤ ψ(d(f(x2k), f(x2k+1)))− φ(d(f(x2k), f(x2k+1))) (3.8)≤ ψ(d(f(x2k), f(x2k+1))),

and similarly,

ψ(d(f(x2k+2), f(x2k+3))) ≤ ψ(H(S(x2k+1), T (x2k+2))) ≤ ψ(d(f(x2k+1), f(x2k+2))),(3.9)





which show that {d(f(xn), f(xn+1))} is a non-increasing sequence of positive realnumbers and therefore, tends to a limit ` ≥ 0. By (3.8) and (3.9),

ψ(d(f(xn+1), f(xn+2))) ≤ ψ(d(f(xn), f(xn+1)))−φ(d(f(xn), f(xn+1))) for all n ≥ 0.


ψ(`) ≤ ψ(`)− φ(`) =⇒ φ(`) ≤ 0,

which is a contradiction as ` > 0, and φ(t) > 0 for t > 0. Therefore ` = 0, i.e.,limn→∞ d(f(xn), f(xn+1)) = 0. Now proceeding in the same way as in Theorem 3.4in [9], it can be shown that the sequence {f(xn)} is a Cauchy sequence in f(X). Asf(X) is a complete subspace of X, {f(xn)} has a limit q in f(X). Consequently, weobtain p in X such that f(p) = q. Thus, f(x2k+1) → q and f(x2k) → q as k → ∞.Now by the given hypothesis,

ψ(d(f(x2k+1), S(p))) ≤ ψ(d(f(x2k), f(p)))− φ(d(f(x2k), f(p))).

Letting n→∞ and using the continuity of both ψ and φ we have, ψ(d(q, S(p))) ≤ψ(d(q, f(p))) − φ(d(q, f(p))) = ψ(0) − φ(0) = 0 =⇒ ψ(d(q, S(p))) ≤ 0. Nowusing the fact ψ(t) > 0 for t > 0 and ψ(0) = 0 we have, ψ(d(q, S(p))) = 0 =⇒d(q, S(p)) = 0 =⇒ q ∈ S(p). Similarly, we can show q ∈ T (p) and hence is thetheorem.

Remark 49. If we take ψ(t) = t for all t ∈ [0,∞), then the above theorem reducesto Theorem 3.7 of Bose and Roychowdhury (cf. [9]).

Proceeding in the same way as Theorem 48 (taking the sequence {xn} such thatxn+1 ∈ Tn+1(xn) for n ∈ Z+), the following theorem can also be proved.

Theorem 50. Let (X, d) be a metric space, f : X → X be a self-mapping and{Ti : X → K(X)}i∈Z+ be a sequence of multi-valued mappings such that for allx, y ∈ X,

ψ(H(Ti(x), Tj(y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y)) for all i, j ≥ 1,

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). If ∪x∈XTi(x) ⊂ f(X) for all i and f(X) is acomplete subspace of X, then there exists a point p ∈ X such that f(p) ∈ Ti(p) forall i.

Theorem 51. Let (X, d) be a complete metric space and T, S : X → K(X) be twomappings such that for all x, y ∈ X

ψ(H(Tx, Sy)) ≤ ψ(M(x, y))− φ(M(x, y)), (3.10)





where

M(x, y) = max{d(x, y), d(T (x), x), d(S(y), y), 12

[d(y, T (x)) + d(x, S(y))]},

and ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Then there there exists a point u ∈ X such thatu ∈ T (u) and u ∈ S(u).

Proof. Clearly M(x, y) = 0 if and only if x = y is a common fixed point of T andS. Let {xk} be a sequence in X such that x2k+1 ∈ T (x2k) and x2k ∈ S(x2k+1) forall k ≥ 0, and d(x2k+1, x2k) ≤ H(T (x2k), S(x2k+1)). Hence as ψ is monotonicallyincreasing we have,

ψ(d(x2k+1, x2k+2)) ≤ ψ(H(T (x2k), S(x2k+1)))≤ ψ(M(x2k, x2k+1))− φ(M(x2k, x2k+1))≤ ψ(M(x2k, x2k+1))≤ ψ(d(x2k, x2k+1)) [as M(x2k, x2k+1) ≤ d(x2k, x2k+1)]

=⇒ d(x2k+1, x2k+2) ≤M(x2k, x2k+1) ≤ d(x2k, x2k+1).

Similarly we have, d(x2k+2, x2k+3) ≤ M(x2k+1, x2k+2) ≤ d(x2k+1, x2k+2). Thus forn ≥ 0 we have d(xn+1, xn+2) ≤ d(xn, xn+1), which shows that {d(xn, xn+1)} is anon-increasing sequence of positive real numbers. Now proceeding in the same wayas in the previous theorem, and in Theorem 3.4 in [9], it can be shown that {xn} isCauchy sequence in X. Hence from the completeness of X it follows that xn → ufor some u ∈ X. Now we show that u ∈ T (u) and u ∈ S(u). We have,

d(S(u), u) ≤M(x2k, u) = max{d(x2k, u), d(T (x2k), x2k), d(S(u), u),12

[d(u, T (x2k)) + d(x2k, S(u))]}

≤ max{d(x2k, u), d(x2k+1, x2k) + d(T (x2k), x2k+1), d(S(u), u),12

[d(u, x2k+1) + d(x2k+1, T (x2k)) + d(x2k, u) + d(u, S(u))]}

= max{d(x2k, u), d(x2k+1, x2k), d(S(u), u),12

[d(u, x2k+1) + d(x2k, u) + d(u, S(u))]}.

and hence, taking k →∞ we have

d(u, S(u)) ≤M(x2k, u) ≤ max{0, 0, d(S(u), u),12

[0 + 0 + d(u, S(u))]} = d(S(u), u),

and so limk→∞M(x2k, u) = d(S(u), u). We have,

d(u, S(u)) ≤ d(u, x2k+1) + d(x2k+1, S(u)) ≤ d(u, x2k+1) +H(T (x2k), S(u)).





As limk→∞ d(u, x2k+1) = 0, ψ is monotonically increasing and both ψ and φ arecontinuous, we have by the given hypothesis (3.10),

ψ(d(u, S(u))) ≤ 0 + limk→∞

ψ(H(T (x2k), S(u)))

≤ limk→∞

ψ(M(x2k, u))− limk→∞

φ(M(x2k, u))

= ψ(d(u, S(u)))− φ(d(u, S(u))),

which implies φ(d(u, S(u))) ≤ 0, i.e. φ(d(u, S(u))) = 0 =⇒ d(u, S(u)) = 0 =⇒u ∈ S(u). Similarly, we can show u ∈ T (u).


Using the techniques used in Theorem 48 and Theorem 51 of this paper, andTheorem 3.6 in [9] we can also prove the following theorem.

Theorem 53. Let K be a nonempty closed subset of a complete and convex metricspace (X, d) and T : K → K(X) be a mapping such that for all x, y ∈ K

ψ(H(Tx, Ty)) ≤ ψ(M(x, y))− φ(M(x, y)),

where

M(x, y) = max{d(x, y), d(T (x), x), d(T (y), y), 12

[d(y, T (x)) + d(x, T (y))]},

and ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Suppose that T (x) ⊂ K for each x ∈ ∂K (theboundary of K). Then there there exists a point u ∈ K such that u ∈ T (u).


4 Fixed point theorems concerning fuzzy mappings

In this section we give the main theorems of this paper concerning fuzzy mappings.

Theorem 55. Let (X, d) be a complete metric linear space and T, S : X → W (X)be a pair of fuzzy mappings such that for all x, y ∈ X

ψ(D(T (x), S(y))) ≤ ψ(d(x, y))− φ(d(x, y)), (4.1)

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Then there there exists a point u ∈ X such that{u} ⊂ T (u) and {u} ⊂ S(u).





Proof. Let x0 be an arbitrary but fixed element of X. We shall construct a sequence{xn} of points of X as follows. By Lemma 21, there exists x1 ∈ X such that{x1} ⊂ T (x0). By Lemma 21 and Lemma 11, we can choose x2 ∈ X such that{x2} ⊂ S(x1) and

d(x1, x2) ≤ H(T (x0)1, S(x1)1).

This in view of the inequality (4.1) we have,

ψ(d(x1, x2)) ≤ ψ(D1(T (x0), S(x1))) ≤ ψ(D(T (x0), S(x1))) ≤ ψ(d(x0, x1))−φ(d(x0, x1)).

Continuing this process, having chosen xk ∈ X, we obtain xk+1 ∈ X such that forall k ≥ 0, {x2k+1} ⊂ T (x2k), {x2k+2} ⊂ S(x2k+1), and

ψ(d(x2k+1, x2k+2)) ≤ ψ(D(T (x2k), S(x2k+1))) ≤ ψ(d(x2k, x2k+1)))−φ(d(x2k, x2k+1))),

ψ(d(x2k+2, x2k+3)) ≤ ψ(D(S(x2k+1), T (x2k+2))) ≤ ψ(d(x2k+1, x2k+2)))−φ(d(x2k+1, x2k+2))).

As ψ is monotonically increasing, from the above two inequalities it follows thatfor n ≥ 0, {d(xn, xn+1)} is a non-increasing sequence of positive real numbers andtherefore tends to a limit ` ≥ 0. If possible, let ` > 0. We have

ψ(d(xn+1, xn+2)) ≤ ψ(d(xn, xn+1))− φ(d(xn), xn+1)) for all n ≥ 0.


ψ(`) ≤ ψ(`)− φ(`) =⇒ φ(`) ≤ 0,

which is a contradiction as ` > 0, and φ(t) > 0 for t > 0. Therefore ` = 0, i.e.,limn→∞ d(xn, xn+1) = 0. Now proceeding in the same way as in Theorem 3.4 in[9], it can be shown that the sequence {xn} is a Cauchy sequence in X. From thecompleteness of X, it follows that there exists u ∈ X such that xn → u as n→∞.Moreover, x2n → u and x2n+1 → u as n→∞. Now we prove that {u} ⊂ T (u) and{u} ⊂ S(u). We know {x2k+1} ⊂ T (x2k) and {x2k} ⊂ S(x2k+1) for k ≥ 0. Notethat,

d(x2k+1, S(u)1) ≤ H(T (x2k)1, S(u)1) = D1(T (x2k), S(u)) ≤ D(T (x2k), S(u)),

and hence as ψ is monotonically increasing we have,

ψ(d(x2k+1, S(u)1)) ≤ ψ(D(T (x2k), S(u))) ≤ ψ(d(x2k, u))− φ(d(x2k, u)).

Letting n→∞ and using the continuity of both ψ and φ we have, ψ(d(u, S(u)1)) ≤ψ(d(u, u)) − φ(d(u, u)) = ψ(0) − φ(0) = 0 =⇒ ψ(d(u, S(u)1)) ≤ 0. Now using thefact ψ(t) > 0 for t > 0 and ψ(0) = 0 we have, ψ(d(u, S(u)1)) = 0 =⇒ d(u, S(u)1) =0 =⇒ u ∈ S(u)1, i.e., {u} ⊂ S(u). Similarly, we can show {u} ⊂ T (u) and hence isthe theorem.





Theorem 56. Let (X, d) be a complete metric linear space and T, S : X → W (X)be a pair of mappings such that for all x, y ∈ X

ψ(D(T (x), S(y))) ≤ ψ(M(x, y))− φ(M(x, y)), (4.2)

where

M(x, y) = max{d(x, y), p(T (x), x), p(S(y), y), 12

[p(y, T (x)) + p(x, S(y))]}.

and ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Then there there exists a point u ∈ X such that{u} ⊂ T (u) and {u} ⊂ S(u).

Proof. Note that M(x, y) = 0 if and only if x = y is a common fixed point of T andS (cf. Theorem 3.4 in [9]).

Let x0 be an arbitrary but fixed element of X. We shall construct a sequence{xn} of points of X as follows. By Lemma 21, there exists x1 ∈ X such that{x1} ⊂ T (x0). By Lemma 21 and Lemma 11, we can choose x2 ∈ X such that{x2} ⊂ S(x1) and

d(x1, x2) ≤ H(T (x0)1, S(x1)1) = D1(T (x0), S(x1)) ≤ D(T (x0), S(x1)).

Continuing this process, having chosen xn ∈ X, we obtain xn+1 ∈ X such that{x2k+1} ⊂ T (x2k), {x2k+2} ⊂ S(x2k+1) and

d(x2k+1, x2k+2) ≤ D(T (x2k), S(x2k+1)) and d(x2k+2, x2k+3) ≤ D(S(x2k+1), T (x2k+2)).

Hence by the given hypothesis and as ψ is monotonically increasing we have,

ψ(d(x2k+1, x2k+2)) ≤ ψ(M(x2k, x2k+1))− φ(M(x2k, x2k+1)) ≤ ψ(M(x2k, x2k+1)),(4.3)

ψ(d(x2k+2, x2k+3)) ≤ ψ(M(x2k+1, x2k+2))− φ(M(x2k+1, x2k+2)) ≤ ψ(M(x2k+1, x2k+2)).(4.4)

As done in Theorem 3.4 in [9], we have

M(x2k, x2k+1) ≤ d(x2k, x2k+1) and M(x2k+1, x2k+2) ≤ d(x2k+1, x2k+2). (4.5)

As ψ is monotonically increasing by (4.3), (4.4) and (4.5) for n ≥ 0 we have,

d(xn+1, xn+2) ≤M(d(xn, xn+1)) ≤ d(xn, xn+1), (4.6)





which shows that {d(xn, xn+1)} is a non-increasing sequence of positive real numbersand therefore tends to a limit ` ≥ 0. If possible, let ` > 0. Now taking n → ∞from (4.6) we have, limn→∞M(xn, xn+1) = `. Again by (4.3) and (4.4) for n ≥ 0we have,

ψ(d(xn+1, xn+2) ≤ ψ(M(xn, xn+1))− φ(M(xn, xn+1)).Now taking n → ∞ and using the continuity of both ψ and φ we have, ψ(`) ≤ψ(`) − φ(`) =⇒ φ(`) ≤ 0, which is a contradiction as ` > 0, and φ(t) > 0 fort > 0. Therefore, d(xn, xn+1) → 0 as n → ∞. Proceeding in the same way as inTheorem 3.4 in [9], we can show that {xn} is a Cauchy sequence. It follows fromthe completeness of X, there exists u ∈ X such that xn → u as n→∞. Moreover,x2n → u and x2n+1 → u as n→∞.

Now we prove that {u} ⊂ T (u) and {u} ⊂ S(u). We have {x2k+1} ⊂ T (x2k) and{x2k} ⊂ S(x2k+1), and

p(u, S(u)) ≤M(x2k, u)= max{d(x2k, u), p(T (x2k), x2k), p(S(u), u),

12

[p(u, T (x2k)) + p(x2k, S(u))]}

≤ max{d(x2k, u), d(x2k, x2k+1) + p(T (x2k), x2k+1), p(S(u), u),12

[d(u, x2k+1) + p(x2k+1, T (x2k)) + d(x2k, u) + p(u, S(u))]}

≤ max{d(x2k, u), d(x2k, x2k+1), p(S(u), u),12

[d(u, x2k+1) + d(x2k, u) + p(u, S(u))]}.

Now taking k →∞ we have,

p(u, S(u)) ≤M(x2k, u) ≤ max{0, 0, p(S(u), u),12

[0 + 0 + p(u, S(u))]} = p(u, S(u)),

and so limk→∞M(x2k, u) = p(u, S(u)). Note that

p(u, S(u)) ≤ d(u, x2k+1) + p(x2k+1, S(u)) ≤ d(u, x2k+1) +D(T (x2k), S(u)).

Hence taking k →∞ we have,

p(u, S(u)) ≤ 0 + limk→∞

D(T (x2k), S(u)) = limk→∞

D(T (x2k), S(u)).

As ψ is non-decreasing, using the continuity of ψ and φ and the given hypothesis(4.2) we have,

ψ(p(u, S(u)) ≤ limk→∞

ψ(D(T (x2k), S(u)))

≤ limk→∞

ψ(M(x2k, u))− limk→∞

φ(M(x2k, u))

= ψ(p(u, S(u))− φ(p(u, S(u)))=⇒ φ(p(u, S(u))) ≤ 0.





Now using the fact φ(t) > 0 for t > 0 and φ(0) = 0 we have, φ(p(u, S(u))) = 0 =⇒p(u, S(u)) = 0 =⇒ {u} ⊂ S(u). Similarly we can show, {u} ⊂ T (u).


Theorem 58. Let (X, d) be a complete metric linear space. Let f : X → X be aself-mapping, and T : X →W (X) be a fuzzy mapping such that for all x, y ∈ X

ψ(D(T (x), T (y))) ≤ ψ(d(f(x), f(y)))− φ(d(f(x), f(y))),

where ψ, φ : [0,∞) → [0,∞) are both continuous functions such that ψ(t), φ(t) > 0for t ∈ (0,∞) and ψ(0) = 0 = φ(0). In addition, φ is non-decreasing and ψ ismonotonically increasing (strictly). Suppose ∪{T (X)}α ⊆ f(X) for α ∈ [0, 1], andf(X) is complete. Then there exists u ∈ X such that u is a coincidence point of fand T , that is {f(u)} ⊂ T (u). Here T (x)α = {y ∈ X : T (x)(y) ≥ α}.

Proof. Let x0 ∈ X and y0 = f(x0). Since ∪{T (X)}α ⊂ f(X) for each α ∈ [0, 1],by Remark 22 for x0 ∈ X there exist points x1, y1 ∈ X such that y1 = f(x1) and{y1} ⊂ T (x0). Again by Remark 22 and Lemma 11, for x1 ∈ X there exist pointsx2, y2 ∈ X such that y2 = f(x2) and {y2} ⊂ T (x1), and

d(y1, y2) ≤ H(T (x0)1, T (x1)1) ≤ D(T (x0), T (x1)).

By repeating this process for any k ≥ 1, we can select points xk, yk ∈ X such thatyk = f(xk) and {yk} ⊂ T (xk−1), and

d(yk, yk+1) ≤ H(T (xk−1)1, T (xk)1) ≤ D(T (xk−1), T (xk)).

As ψ is monotonically increasing, from the above inequalities for k ≥ 0 we have,

ψ(d(yk+1, yk+2)) ≤ ψ(D(T (xk), T (xk+1)))≤ ψ(d(f(xk), f(xk+1)))− φ(d(f(xk), f(xk+1))) (4.7)≤ ψ(d(yk, yk+1)).

As ψ is monotonically increasing from the above inequality we have, d(yn+1, yn+2) ≤d(yn, yn+1) for n ≥ 0, which shows that {d(yn, yn+1)} is a non-increasing sequenceof positive real numbers and therefore tends to a limit ` ≥ 0. If possible, let ` > 0.For any n ≥ 0 by (4.7) we have, ψ(d(yn+1, yn+2)) ≤ ψ(d(yn, yn+1))−φ(d(yn, yn+1)).Taking n→∞ and using the continuity of ψ and φ, we obtain

ψ(`) ≤ ψ(`)− φ(`) =⇒ φ(`) ≤ 0,

which is a contradiction as ` > 0, and φ(t) > 0 for t > 0. Therefore ` = 0, i.e.,limn→∞ d(yn, yn+1) = 0. Now proceeding in the same way as in Theorem 3.4 in





[9], it can be shown that the sequence {yn} is a Cauchy sequence. Since f(X) iscomplete, {yn} converges to some point in f(X). Let y = limn→∞ yn and u ∈ X besuch that y = f(u). Note that,

d(yk+1, T (u)1) ≤ H(T (xk)1, T (u)1) = D1(T (xk), T (u)) ≤ D(T (xk), T (u)),

and hence as ψ is monotonically increasing we have,

ψ(d(yk+1, T (u)1)) ≤ ψ(D((T (xk), T (u))) ≤ ψ(d(f(xk), f(u)))− φ(d(f(xk), f(u))).

Letting n→∞ and using the continuity of both ψ and φ we have, ψ(d(y, T (u)1)) ≤ψ(d(y, y)) − φ(d(y, y)) = ψ(0) − φ(0) = 0 =⇒ ψ(d(y, T (u)1)) ≤ 0. Now using thefact ψ(t) > 0 for t > 0 and ψ(0) = 0 we have, ψ(d(y, T (u)1)) = 0 =⇒ d(y, S(u)1) =0 =⇒ y = f(u) ∈ T (u)1, i.e., {f(u)} ⊂ T (u), and hence is the theorem.


Remark 60. Dutta and Choudhury in their paper ([12, Theorem 2.1]) assumedthat ψ, φ : [0,∞) → [0,∞) are both continuous and monotone non-decreasing withψ(t) = 0 = φ(t) if and only if t = 0. In the proof they wrote that ψ(d(xn, xn+1)) ≤ψ(d(xn−1, xn)) =⇒ d(xn, xn+1) ≤ d(xn−1, xn), that means by ‘monotone non-decreasing’ they meant that reverse implication holds. But this is not the way thatmonotone functions are defined. In the example they gave, the function φ is notone-to-one, which creates confusion (cf. [12, Example 2.2]). Their example clearlyfits in our paper, as in our case ψ is monotonically increasing (strictly), i.e., ψ isone-to-one and φ is non-decreasing.

Example: Let X = [0, 1] ∪ {2, 3, 4, · · · } and

d(x, y) =

|x− y|, if x, y ∈ [0, 1], x 6= y,x+ y, if at least one of x or y /∈ [0, 1] and x 6= y,0, if x = y.

Then (X, d) is a complete metric space (cf. [11]). Let ψ : [0,∞)→ [0,∞) be definedas

ψ(t) ={t, if 0 ≤ t ≤ 1,t2, if t > 1,

and let φ : [0,∞)→ [0,∞) be defined as

φ(t) ={

12 t

2, if 0 ≤ t ≤ 1,12 , if t > 1.

Let T : X → X be defined as

Tx ={x− 12x

2, if 0 ≤ x ≤ 1,x− 1, if x ∈ {2, 3, 4, · · · }.

Then it is seen that T has a unique fixed point which is 0 (cf. [12]).





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Ramendra Krishna Bose Mrinal Kanti Roychowdhury

Department of Mathematics, Department of Mathematics,

The University of Texas-Pan American, The University of Texas-Pan American,

1201 West University Drive, 1201 West University Drive,

Edinburg, TX 78539-2999, USA. Edinburg, TX 78539-2999, USA.

e-mail: [email protected], [email protected] e-mail: [email protected]



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IntroductionBasic Definitions and LemmasFixed point theorems for single and multi-valued mappingsFixed point theorems concerning fuzzy mappings

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