Common fixed point theorems for pairs of single and multivalued D-maps satisfying an integral type

Annales Mathematicae et Informaticae 35 (8p. 43 59 http://www.ektf.hu/ami Common fixed point theorems for pairs of single and multivalued D-maps satisfying an integral type H. Bouhadjera A. Djoudi Laboratoire de Mathématiques Appliquées Université Badji Mokhtar Annaba Algérie Submitted 4 March 8; Accepted 8 June 8 Abstract This contribution is a continuation of [1 3 14]. The concept of subcompatibility between single maps and between single and multivalued maps is used as a tool for proving existence and uniqueness of common fixed points on complete metric and symmetric spaces. Extensions of known results in particularly results given by Djoudi and Aliouche Elamrani and Mehdaoui Pathak et al. are thereby obtained. Keywords: Commuting and weakly commuting maps compatible and compatible maps of type (A) (B) (C) and (P) weakly compatible maps δ- compatible maps subcompatible maps D-maps integral type common fixed point theorems metric space. MSC: 47H1 54H5 1. Introduction and preliminaries Let (X d) be a metric space and let B(X) be the class of all nonempty bounded subsets of X. For all A B in B(X) define δ(a B) = sup {d(a b) : a A b B}. If A = {a} we write δ(a B) = δ(a B). Also if B = {b} it yields that δ(a B) = d(a b). From the definition of δ(a B) for all A B C in B(X) it follows that δ(a B) = δ(b A) δ(a B) δ(a C) + δ(c B) 43

44 H. Bouhadjera A. Djoudi δ(a A) = diama δ(a B) = iff A = B = {a}. In his paper [15] Sessa introduced the notion of weak commutativity which generalized the notion of commutativity. Later on Jungck [6] gave a generalization of weak commutativity by introducing the concept of compatibility. Again to generalize weakly commuting maps the same author with Murthy and Cho [8] introduced the concept of compatible maps of type (A). Extending type (A) Pathak and Khan [13] made the notion of compatible maps of type (B). In [11] the concept of compatible maps of type (P) was introduced and compared with compatible and compatible maps of type (A). In 1998 Pathak Cho Kang and Madharia [1] defined the notion of compatible maps of type (C) as another extension of compatible maps of type (A). In his paper [7] Jungck generalized all the concepts of compatibility by giving the notion of weak compatibility (subcompatibility). The authors of [9] extended the concept of compatible maps to the setting of single and multivalued maps by giving the notion of δ-compatible maps. Also the same authors [1] extended the definition of weak compatibility to the setting of single and multivalued maps by introducing the concept of subcompatible maps. In their paper [] Djoudi and Khemis introduced the notion of D-maps which is a generalization of δ-compatible maps. Definition 1.1 ([4]). A sequence {A n } of nonempty subsets of X is said to be convergent to a subset A of X if: (i) each point a A is the limit of a convergent sequence {a n } where a n A n for n N (ii) for arbitrary ǫ > there exists an integer m such that A n A ǫ for n > m where A ǫ denotes the set of all points x in X for which there exists a point a in A depending on x such that d(x a) < ǫ. Lemma 1. ([4 5]). If {A n } and {B n } are sequences in B(X) converging to A and B in B(X) respectively then the sequence {δ(a n B n )} converges to δ(a B). Lemma 1.3 ([5]). Let {A n } be a sequence in B(X) and y be a point in X such that δ(a n y). Then the sequence {A n } converges to the set {y} in B(X). Definition 1.4 ([15]). The self-maps f and g of a metric space X are said to be weakly commuting if d(fgx gfx) d(gx fx) for all x X. Definition 1.5 ([6 8 13 1 11]). The self-maps f and g of a metric space X are said to be (1) compatible if lim n d(fgx n gfx n ) =

Common fixed point theorems for pairs of single and multivalued D-maps... 45 () compatible of type (A) if lim d(fgx n g x n ) = and lim d(gfx n f x n ) = n n (3) compatible of type (B) if lim d(fgx n g x n ) 1 [ n [ lim n d(gfx n f x n ) 1 (4) compatible of type (C) if ] lim d(fgx n ft) + lim d(ft n n f x n ) ] lim d(gfx n gt) + lim d(gt n n g x n ) lim d(fgx n g x n ) 1 [ lim n 3 d(fgx n ft) n ] + lim d(ft n f x n ) + lim d(ft n g x n ) lim d(gfx n f x n ) 1 [ lim n 3 d(gfx n gt) n (5) compatible of type (P) if + lim n d(gt g x n ) + lim n d(gt f x n ) lim n d(f x n g x n ) = whenever {x n } is a sequence in X such that lim fx n = lim gx n = t for some n n t X. Definition 1.6 ([7]). The self-maps f and g of a metric space X are called weakly compatible if fx = gx x X implies fgx = gfx. Definition 1.7 ([9]). The maps f : X X and F : X B(X) are δ-compatible if lim n δ(ffx n ffx n ) = whenever {x n } is a sequence in X such that ffx n B(X) fx n t and Fx n {t} for some t X. Definition 1.8 ([1]). Maps f : X X and F : X B(X) are subcompatible if they commute at coincidence points; i.e. for each point u X such that Fu = {fu} we have Ffu = ffu. Definition 1.9 ([]). The maps f : X X and F : X B(X) are said to be D-maps iff there exists a sequence {x n } in X such that for some t X lim fx n = t and n lim Fx n = {t}. n ]

46 H. Bouhadjera A. Djoudi Recently in 7 Pathak et al. [14] established a general common fixed point theorem for two pairs of weakly compatible maps satisfying integral type implicit relations. The first main object of this paper is to prove a common fixed point theorem for a quadruple of maps satisfying certain integral type implicit relations. Our result extended the result of [14] to the setting of single and multivalued maps. For this consideration we need the following: Let Φ = {ϕ : R + R is a Lebesgue-integrable map which is summable} and let F be the set of all continuous functions F : R 6 + R + satisfying the following conditions: (F a ) F(uuu) ϕ(t)dt implies u = ; (F b ) F(uuu) ϕ(t)dt implies u =. The function F satisfies the condition (F 1 ) if F(uuuu) ϕ(t)dt > for all u >.. Main results Theorem.1. Let f g be self-maps of a metric space (X d) and let F G: X B(X) be two multivalued maps such that (1) F X gx and GX fx () F(δ(FxGy)d(fxgy)δ(fxFx)δ(gyGy)δ(fxGy)δ(gyFx)) ϕ(t)dt for all x y in X where F F and ϕ Φ. If either (3) f and F are subcompatible D-maps; g and G are subcompatible and F X is closed or (3 ) g and G are subcompatible D-maps; f and F are subcompatible and GX is closed. Then f g F and G have a unique common fixed point t X such that Ft = Gt = {ft} = {gt} = {t}. Proof. Suppose that f and F are D-maps then there exists a sequence {x n } in X such that fx n t and Fx n {t} for some t X. Since F X is closed and F X gx then there is a point u X such that gu = t. We show that Gu = {gu} = {t}. Using inequality () we have F(δ(FxnGu)d(fx ngu)δ(fx nfx n)δ(gugu)δ(fx ngu)δ(gufx n)) Since F is continuous we get at infinity F(δ(guGu)δ(guGu)δ(guGu)) ϕ(t) dt ϕ(t)dt.

Common fixed point theorems for pairs of single and multivalued D-maps... 47 which implies by using condition (F a ) δ (gu Gu) = ; i.e. Gu = {gu} = {t}. Since the pair (g G) is subcompatible it follows that Ggu = ggu; i.e. Gt = {gt}. If t gt using () we have F(δ(FxnGt)d(fx ngt)δ(fx nfx n)δ(gtgt)δ(fx ngt)δ(gtfx n)) Taking limit as n we get F(d(tgt)d(tgt)d(tgt)d(gtt)) ϕ(t)dt ϕ(t)dt. which contradicts (F 1 ). Hence Gt = {gt} = {t}. Since GX fx there is v X such that {t} = Gt = {fv}. If Fv {t} using () again we have = F(δ(FvGt)d(fvgt)δ(fvFv)δ(gtGt)δ(fvGt)δ(gtFv)) F(δ(Fvt)δ(tFv)δ(tFv)) ϕ(t)dt ϕ(t) dt which implies by using condition (F b ) that δ (Fv t) = hence Fv = {t} = {fv}. Since F and f are subcompatible it follows that Ffv = ffv; i.e. Ft = {ft}. If t ft using () we have = F(δ(FtGt)d(ftgt)δ(ftFt)δ(gtGt)δ(ftGt)δ(gtFt)) F(d(ftt)d(ftt)d(ftt)d(tft)) ϕ(t)dt ϕ(t)dt which contradicts (F 1 ). Thus {ft} = {t} = Ft. We get the same conclusion if we use (3 ) instead of (3). The uniqueness of the common fixed point follows easily from conditions () and (F 1 ). Corollary.. Let f be a map from a metric space (X d) into itself and let F be a map from X into B(X). If (i) F X fx (ii) f and F are subcompatible D-maps (iii) F(δ(FxFy)d(fxfy)δ(fxFx)δ(fyFy)δ(fxFy)δ(fyFx)) ϕ(t)dt for all x y in X where ϕ Φ and F is continuous and satisfies conditions (F a ) and (F 1 ) or (F b ) and (F 1 ). If F X is closed then f and F have a unique common fixed point in X.

48 H. Bouhadjera A. Djoudi The next Theorem is a generalization of Theorem.1. Theorem.3. Let f g be self-maps of a metric space (X d) and let F n : X B(X) where n = 1... be multivalued maps such that (i) F n X gx and F n+1 X fx (ii) F(δ(FnxF n+1y)d(fxgy)δ(fxf nx)δ(gyf n+1y)δ(fxf n+1y)δ(gyf nx)) ϕ(t)dt for all x y in X where F F and ϕ Φ. If either (iii) f and F n are subcompatible D-maps; g and F n+1 are subcompatible and F n X is closed or (iii) g and F n+1 are subcompatible D-maps; f and F n are subcompatible and F n+1 X is closed. Then f g and F n have a unique common fixed point t X such that F n t = {ft} = {gt} = {t}. Now let Ψ be the set of all maps ψ: R + R + such that ψ is a Lebesgueintegrable which is summable nonnegative and satisfies ǫ > for each ǫ >. In [3] a common fixed point theorem for a pair of generalized contraction selfmaps and a pair of multivalued maps in a complete metric space was obtained. Our second main subject is to complement and improve the result of [3] by relaxing the notion of δ-compatibility to subcompatibility removing the assumption of continuity imposed on at least one of the four maps and deleting some conditions required on the functions Φ a b and c by using an integral type in a metric space instead of a complete metric space. Theorem.4. Let f g be self-maps of a metric space (X d) and let F G be maps from X into B(X) satisfying the following conditions (1 ) f and g are surjective ( ) (δ(fxgy)) a (d (fx gy)) + b (d (fx gy)) + c (d (fx gy)) (d(fxgy)) (δ(fxfx))+ (δ(gygy)) min{ (δ(fxgy)) (δ(gyfx))} for all x y in X where : [ ) [ ) is an upper semi-continuous map such that (t) = iff t = ; a b c: [ ) [ 1) are upper semi-continuous such that a(t) + c(t) < 1 for every t > and ψ Ψ. If either (3 ) f and F are subcompatible D-maps; g and G are subcompatible or

Common fixed point theorems for pairs of single and multivalued D-maps... 49 (3 ) g and G are subcompatible D-maps; f and F are subcompatible. Then f g F and G have a unique common fixed point t X such that Ft = Gt = {ft} = {gt} = {t}. Proof. Suppose that f and F are D-maps then there is a sequence {x n } in X such that lim fx n = t and lim Fx n = {t} for some t X. By condition (1 ) there n n exist points u v in X such that t = fu = gv. First we show that Gv = {gv} = {t}. Using inequality ( ) we get (δ(fxngv)) a (d (fx n gv)) + b (d (fx n gv)) + c (d (fx n gv)) (d(fxngv)) Taking the limit as n one obtains (δ(gvgv)) (δ(fxnfx n))+ (δ(gvgv)) min{ (δ(fxngv)) (δ(gvfx n))} b () (δ(gvgv)) <. (δ(gvgv)) this contradiction implies that Gv = {gv} = {t}. Since the pair (g G) is subcompatible then Ggv = ggv; i.e. Gt = {gt}. We claim that Gt = {gt} = {t}. Suppose not then by condition ( ) we have (δ(fxngt)) When n we obtain (δ(tgt)) a (d (fx n gt)) + b (d (fx n gt)) + c (d (fx n gt)) = (d(tgt)) (d(fxngt)) (δ(fxnfx n))+ (δ(gtgt)) min{ (δ(fxngt)) (δ(gtfx n))} [a (d (t gt)) + c (d (t gt))] < (d(tgt)) (d(tgt)).

5 H. Bouhadjera A. Djoudi which is a contradiction. Hence {gt} = {t} = Gt. Next we claim that Fu = {fu} = {t}. If not then by ( ) we get (δ(fufu)) = (δ(fugt)) a (d (fu gt)) + b (d (fu gt)) + c (d (fu gt)) = b () (δ(fufu)) (d(fugt)) (δ(fufu))+ (δ(gtgt)) min{ (δ(fugt)) (δ(gtfu))} < (δ(fufu)) which is a contradiction. Thus F u = {f u} = {t}. Since F and f are subcompatible then Ffu = ffu; i.e. Ft = {ft}. Suppose that ft t. Then the use of ( ) gives (d(ftt)) = (δ(ftgt)) a (d (ft gt)) + b (d (ft gt)) + c (d (ft gt)) (d(ftgt)) (δ(ftft))+ (δ(gtgt)) min{ (δ(ftgt)) (δ(gtft))} = [a (d (ft t)) + c (d (ft t))] < (d(ftt)) (d(ftt)) this contradiction implies that f t = t and hence F t = {f t} = {t}. Therefore t is a common fixed point of both f g F and G. The uniqueness of the common fixed point follows easily from condition ( ). We get the same conclusion if we consider (3 ) in lieu of (3 ). Remark.5. Theorem 3.1 of [3] becomes a special case of Theorem.4 with ψ(x) = 1. If we put f = g in Theorem.4 we get the next corollary. Corollary.6. Let (X d) be a metric space and let f : X X; F G: X B(X) be maps. Suppose that

Common fixed point theorems for pairs of single and multivalued D-maps... 51 (i) f is surjective (ii) (δ(fxgy)) a (d (fx fy)) + b (d (fx fy)) + c (d (fx fy)) (d(fxfy)) (δ(fxfx))+ (δ(fygy)) min{ (δ(fxgy)) (δ(fyfx))} for all x y in X where ψ a b c are as in Theorem.4. If either (iii) f and F are subcompatible D-maps; f and G are subcompatible or (iii) f and G are subcompatible D-maps; f and F are subcompatible. Then f F and G have a unique common fixed point t X such that Ft = Gt = {ft} = {t}. For a single map f : X X (resp. a multivalued map F : X B(X)) F f (resp. F F ) will denote the set of fixed point of f (resp. F). Theorem.7. Let F G: X B(X) be multivalued maps and let f g: X X be single maps on the metric space X. If inequality ( ) holds for all x y in X then (F f F g ) F F = (F f F g ) F G. Proof. We can check the above equality by using inequality ( ). Theorems.4 and.7 imply the next one. Theorem.8. Let f g be self-maps of a metric space (X d) and let F n where n = 1... be maps from X into B(X) such that (i) f and g are surjective (ii) (δ(fnxf n+1y)) a (d (fx gy)) + b (d (fx gy)) + c (d (fx gy)) (d(fxgy)) (δ(fxfnx))+ (δ(gyf n+1y)) min{ (δ(fxfn+1y)) (δ(gyf nx))} for all x y in X where ψ a b c are as in Theorem.4. If either (iii) f and F 1 are subcompatible D-maps; g and F are subcompatible or

5 H. Bouhadjera A. Djoudi (iii) g and F are subcompatible D-maps; f and F 1 are subcompatible. Then f g and F n have a unique common fixed point t X such that F n t = {ft} = {gt} = {t} for n = 1.... Let Ω be the family of all maps ω: R + R + such that ω is upper semicontinuous and ω(t) < t for each t >. In [1] Djoudi and Aliouche proved a common fixed point theorem of Greguš type for four maps satisfying a contractive condition of integral type in a metric space using the concept of weak compatibility. Our aim henceforth is to extend this result to multivalued maps by using the concept of D-maps. Theorem.9. Let (X d) be a metric space and let f g: X X; F k : X B(X) be single and multivalued maps respectively. Suppose that (i) F k X gx and F k+1 X fx (ii) ( δ(fk xf k+1 y) ( ( p { ( d(fxgy) p δ(fxfk x) ω a ) + (1 a)max α ) ( p ( δ(gyfk+1 y) δ(fxfk x) β ) ( δ(gyfk x) ( ( δ(gyfk x) δ(fxfk+1 y) 1 (( p ( δ(fxfk x) )}) δ(gyfk+1 y) ) + for all x y in X where k N = {1... } ω Ω ψ Ψ < a < 1 < α β 1 and p is an integer such that p 1. If either (iii) f and F k are subcompatible D-maps; g and F k+1 are subcompatible and F k X is closed or (iii) g and F k+1 are subcompatible D-maps; f and F k are subcompatible and F k+1 X is closed. Then f g and F k have a unique common fixed point t X such that F k t = {ft} = {gt} = {t}. Proof. Suppose that f and F k are D-maps then there exists a sequence {x n } in X such that lim fx n = t and lim F kx n = {t} for some t X. Since F k X is n n closed and F k X gx then there is u X such that gu = t. If F k+1 u {gu}

Common fixed point theorems for pairs of single and multivalued D-maps... 53 using inequality (ii) we get ( δ(fk x nf k+1 u) ( ( d(fxngu) ω a { ( p ( δ(fxnf k x n) p δ(gufk+1 u) + (1 a)max α ) β ) ( ( δ(fxnf k x n) δ(gufk x n) ( ( δ(gufk x n) δ(fxnf k+1 u) 1 (( p ( δ(fxnf k x n) )}) δ(gufk+1 u) ) +. Letting n we obtain ( δ(gufk+1 u) ( { ω (1 a)max β 1 } ( ) δ(gufk+1 u) { < (1 a)max β 1 } ( p ( δ(gufk+1 u) δ(gufk+1 u) ) < which is a contradiction. Then F k+1 u = {gu} = {t}. Since the pair (g F k+1 ) is subcompatible we have F k+1 gu = gf k+1 u; i.e. F k+1 t = {gt}. If t gt using inequality (ii) we obtain ( δ(fk x nf k+1 t) ( ( d(fxngt) ω a { ( p ( δ(fxnf k x n) p δ(gtfk+1 t) + (1 a)max α ) β ) ( ( δ(fxnf k x n) δ(gtfk x n)

54 H. Bouhadjera A. Djoudi ( ( δ(gtfk x n) δ(fxnf k+1 t) 1 (( p ( δ(fxnf k x n) )}) δ(gtfk+1 t) ) +. At infinity we get ( p d(tgt) ) ω (( ) ( d(tgt) d(tgt) < which is a contradiction. Therefore F k+1 t = {gt} = {t}. Since F k+1 X fx there exists v X such that F k+1 t = {t} = {fv}. We claim that F k v = {fv} suppose not then by condition (ii) we have ( δ(fk vf k+1 t) ( ( p { ( d(fvgt) p δ(fvfk v) ω a ) + (1 a)max α ) ( p ( δ(gtfk+1 t) δ(fvfk v) β ) ( δ(gtfk v) ( ( δ(gtfk v) δ(fvfk+1 t) 1 (( p ( δ(fvfk v) )}) δ(gtfk+1 t) ) + that is ( p ( ( δ(fk vfv) ) δ(fk vfv) ) ω (1 a) ( δ(fk vfv) < (1 a) ( δ(fk vfv) < which is a contradiction. Hence F k v = {fv} = {t}. Since the pair (f F k ) is subcompatible then F k fv = ff k v; i.e. F k t = {ft}. The use of (ii) gives ( δ(fk tf k+1 t)

Common fixed point theorems for pairs of single and multivalued D-maps... 55 ( ( p { ( d(ftgt) p δ(ftfk t) ω a ) + (1 a)max α ) ( p ( δ(gtfk+1 t) δ(ftfk t) β ) ( δ(gtfk t) ( ( δ(gtfk t) δ(ftfk+1 t) 1 (( p ( δ(ftfk t) )}) δ(gtfk+1 t) ) + i.e. ( p (( d(ftt) ) ( d(ftt) d(ftt) ) ω < this contradiction implies that {ft} = {t} = F k t. Thus t is a common fixed point of f g and F k. The uniqueness of the common fixed point follows from inequality (ii). If one uses condition (iii) instead of (iii) one gets the same conclusion. Theorem.1. Let (X d) be a metric space and let f g: X X; F n : X B(X) be single and multivalued maps such that (i) F n X gx and F n+1 X fx (ii) ( δ(fnxf n+1y) ( ( p { d(fxgy) δ(fxfnx) ω a ) + (1 a)max δ(gyfn+1y) ( ( δ(fxfnx) δ(gyfnx) ( ( δ(gyfnx) δ(fxfn+1y) p for all x y in X where ω Ω ψ Ψ < a < 1 and p is an integer such that p 1. If either (iii) f and F n are subcompatible D-maps; g and F n+1 are subcompatible and F n X is closed or (iii) g and F n+1 are subcompatible D-maps; f and F n are subcompatible and F n+1 X is closed.

56 H. Bouhadjera A. Djoudi Then f g and F n have a unique common fixed point t X such that F n t = {ft} = {gt} = {t} for n = 1.... Proof. It is similar to the proof of Theorem.9. Now we prove a unique common fixed point theorem of Greguš type by using a strict contractive condition of integral type for two pairs of single and multivalued maps in a metric space. Theorem.11. Let f and g be self-maps of a metric space (X d) and let {F n } n = 1... be multivalued maps from X into B(X) such that (1 ) f and g are surjective ( ) δ(f1xf k y) b < α d(fxgy) δ(gyfk y) + (1 α) max { a δ(fxf1x) ( ( δ(fxf1x) δ(gyf1x) c ( ( δ(gyf1x) δ(fxfk y) d for all x y in X and some k > 1 for which the right hand side is positive where ψ Ψ < α a b c d < 1 and α + d(1 α) < 1. If either (3 ) f and F 1 are subcompatible D-maps; g and F k are subcompatible or (3 ) g and F k are subcompatible D-maps; f and F 1 are subcompatible. Then f g and {F n } have a unique common fixed point t X such that F n t = {ft} = {gt} = {t} for n = 1.... Proof. Suppose that condition (3 ) holds then there is a sequence {x n } in X such that fx n t and F 1 x n {t} as n for some t X. By condition (1 ) there are two elements u and v in X such that t = fu = gv. We show that {t} = F k v. Indeed using inequality ( ) we get δ(f1x nf k v) b < α d(fxngv) δ(gvfk v) + (1 α) max { a δ(fxnf 1x n) ( ( δ(fxnf 1x n) δ(gvf1x n) c

Common fixed point theorems for pairs of single and multivalued D-maps... 57 ( ( δ(gvf1x n) δ(fxnf k v) d Taking limit as n we obtain δ(tfk v) b (1 α) δ(tfk v) <. δ(tfk v) thus we have F k v = {t} = {gv} and since g and F k are subcompatible we have F k gv = gf k v; that is F k t = {gt}. Again by ( ) we obtain δ(f1x nf k t) b < α d(fxngt) δ(gtfk t) + (1 α) max { a δ(fxnf 1x n) ( ( δ(fxnf 1x n) δ(gtf1x n) c ( ( δ(gtf1x n) δ(fxnf k t) d When n we get d(tgt) [α + d (1 α)] d(tgt). < d(tgt) this contradiction implies that {t} = {gt} = F k t = {fu}. We claim that F 1 u = {t}. By condition ( ) we have δ(f1ut) b < α d(fugt) δ(gtfk t) = δ(f1uf k t) { + (1 α)max a δ(fuf1u) ( ( δ(fuf1u) δ(gtf1u) c ( ( δ(gtf1u) δ(fufk t) d = (1 α) max {a c} δ(f1ut) < δ(f1ut)

58 H. Bouhadjera A. Djoudi this contradiction demands that F 1 u = {t} = {fu}. Since f and F 1 are subcompatible then F 1 fu = ff 1 u; that is F 1 t = {ft}. Moreover by ( ) one may get d(ftt) b < α = d(ftgt) δ(gtfk t) δ(f1tf k t) { + (1 α)max a δ(ftf1t) ( ( δ(ftf1t) δ(gtf1t) c ( ( δ(gtf1t) δ(ftfk t) d = [α + d (1 α)] d(ftt) < d(ftt) which is a contradiction. Thus {ft} = {t} = F 1 t. Therefore F 1 t = F k t = {ft} = {gt} = {t}. Uniqueness follows easily from condition ( ). The proof is thus completed. Important remark. Every contractive or strict contractive condition of integral type automatically includes a corresponding contractive or strict contractive condition not involving integrals by setting ϕ(t) = 1 (resp. ψ(t) = 1) over R +. So our results extend generalize and complement several various results existing in the literature. References [1] Djoudi A. Aliouche A. Common fixed point theorems of Greguš type for weakly compatible mappings satisfying contractive conditions of integral type J. Math. Anal. Appl. 39 (7) no. 1 31 45. [] Djoudi A. Khemis R. Fixed points for set and single valued maps without continuity Demonstratio Math. 38 (5) no. 3 739 751. [3] Elamrani M. Mehdaoui B. Common fixed point theorems for compatible and weakly compatible mappings Rev. Colombiana Mat. 34 () no. 1 5 33. [4] Fisher B. Common fixed points of mappings and set-valued mappings Rostock. Math. Kolloq. 18 (1981) 69 77. [5] Fisher B. Sessa S. Two common fixed point theorems for weakly commuting mappings Period. Math. Hungar. (1989) no. 3 7 18. [6] Jungck G. Compatible mappings and common fixed points Internat. J. Math. Math. Sci. 9 (1986) no. 4 771 779.

Common fixed point theorems for pairs of single and multivalued D-maps... 59 [7] Jungck G. Common fixed points for noncontinuous nonself maps on nonmetric spaces Far East J. Math. Sci. 4 (1996) no. 199 15. [8] Jungck G. Murthy P.P. Cho Y.J. Compatible mappings of type (A) and common fixed points Math. Japon. 38 (1993) no. 381 39. [9] Jungck G. Rhoades B.E. Some fixed point theorems for compatible maps Internat. J. Math. Math. Sci. 16 (1993) no. 3 417 48. [1] Jungck G. Rhoades B.E. Fixed points for set valued functions without continuity Indian J. Pure Appl. Math. 9 (1998) no. 3 7 38. [11] Pathak H.K. Cho Y.J. Kang S.M. Lee B.S. Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming Matematiche (Catania) 5 (1995) no. 1 15 33. [1] Pathak H.K. Cho Y.J. Kang S.M. Madharia B. Compatible mappings of type (C) and common fixed point theorems of Greguš type Demonstratio Math. 31 (1998) no. 3 499 518. [13] Pathak H.K. Khan M.S. Compatible mappings of type (B) and common fixed point theorems of Greguš type Czechoslovak Math. J. 45(1) (1995) no. 4 685 698. [14] Pathak H.K. Tiwari R. Khan M.S. A common fixed point theorem satisfying integral type implicit relations Appl. Math. E-Notes 7 (7) 8. [15] Sessa S. On a weak commutativity condition of mappings in fixed point considerations Publ. Inst. Math. (Beograd) (N.S.) 3(46) (19849 153. H. Bouhadjera A. Djoudi Laboratoire de Mathématiques Appliquées Université Badji Mokhtar B. P. 1 3 Annaba Algérie e-mail: b_hakima@yahoo.fr