Gregus-Type Common Fixed Point Theorems for Tangential Multivalued Mappings of Integral Type in Metric Spaces

The concept of tangential for single-valued mappings is extended to multivalued mappings and used to prove the existence of a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. Consequently, several known fixed point results generalized and improved the corresponding recent result of Pathak and Shahzad 2009 and many authors.


Introduction
The first important result on fixed points for contractive-type mappings was the well-known Banach contraction principle, published for the first time in 1922 in 1 see also 2 .Banach contraction principle has been extended in many different directions, see 3-5 , and so forth.Many authors in 3, 5-12 established fixed point theorems involving more general contractive conditions.In 1969, Nadler 13 combines the ideas of set-valued mapping and Lipschitz mapping and prove some fixed point theorems about multivalued contraction mappings.Afterward, the study of fixed points for multivalued contractions using the Hausdorff metric was initiated by Markin 14 .Later, an interesting and rich fixed point theory for such maps was developed see 15-18 .The theory of multivalued maps has applications in optimization problems, control theory, differential equations, and economics.Sessa 19 introduced the concept of weakly commuting maps.Jungck 20 defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true.The aim of this paper is to define a tangential property for multivalued mappings which generalize the concept of tangential property for single-valued mappings of Pathak and Shahzad 24 and prove a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type.

Preliminary
Throughout this paper, X, d denotes a metric space.We denote by CB X , the class of all nonempty bounded closed subsets of X.
The set of all fixed points of f resp.T is denoted by F f resp.F T .A point x ∈ X is a coincidence point of f and T if fx ∈ Tx.The set of all coincidence points of f and T is denoted by C f, T .A point x ∈ X is a common fixed point of f and T if x fx ∈ Tx.The set of all common fixed points of f and T is denoted by F f, T .
Definition 2.1.The maps f : X → X and g : X → X are said to be commuting if fgx gfx, for all x ∈ X. Definition 2.2 see 19 .The maps f : X → X and g : X → X are said to be weakly commuting if d fgx, gfx ≤ d fx, gx , for all x ∈ X.
Definition 2.4 see 26 .The maps f : X → X and g : X → X are said to be weakly compatible fgx gfx, for all x ∈ C f, g .Definition 2.5 see 22 .Let f : X → X and g : X → X.The pair f, g satisfies property E.A if there exist the sequence By n z ∈ X. 2.5

Main Results
In this section, we first introduce the notion of tangential property for two single-valued and two multivalued mappings.Throughout this section, Ê denotes the set of nonnegative real numbers.

3.9
So, the mapping f is tangential with respect to the mapping A. Now, we state and prove our main result.Then, f and A have a common fixed point in X.

International
Journal of Mathematics and Mathematical Sciences This concept was further improved by Jungck and Rhoades 21 with the notion of weakly compatible mappings.In 2002, Aamri and Moutawakil 22 defined property E.A .This concept was frequently used to prove existence theorems in common fixed point theory.Three years later, Liu et al. 23 introduced common property E.A .The class of E.A maps contains the class of noncompatible maps.Branciari 3 studied contractive conditions of integral type, giving an integral version of the Banach contraction principle, that could be extended to more general contractive conditions.Recently, Pathak and Shahzad 24 introduced the new concept of weak tangent point and tangential property for single-valued mappings and established common fixed point theorems.Very recently, Vetro 25 obtained an interesting theorem for mappings satisfying a contractive condition of integral type which is a generalization of Branciari 3, Theorem 2 .
for all x, y ∈ X for which the right-hand side of 3.27 is positive, where 0 < a < 1.If the following conditions (a)-(d) holds: a there exists sequence {x n } in X such that lim n → ∞ fx n ∈ X, b f is tangential with respect to A, c ffa fa for a ∈ C f, A , d the pairs f, A is weakly compatible.
Definition 2.6 see 23 .Let f, g, A, B : X → X.The pair f, g and A, B satisfy a common property E.A if there exist sequences {x n } and {y n } in X such that lim Remark 2.7.If A f, B g, and {x n } {y n } in 2.3 , then we get the definition of property E.A .Definition 2.8 see 24 .Let f, g : X → X.A point z ∈ X is said to be a weak tangent point to f, g if there exist sequences {x n } and {y n } in X such that lim n → ∞ fx n lim n → ∞ gy n z ∈ X. 2.4 Remark 2.9.If {x n } {y n } in 2.4 , we get the definition of property E.A .Definition 2.10 see 24 .Let f, g, A, B : X → X.The pair f, g is called tangential with respect to the pair A, B if there exist sequences {x n } and {y n } in X such that lim n → ∞ International Journal of Mathematics and Mathematical Sciences Example 3.2.Let Ê , d be a metric space with usual metric d.Let f, g : Ê → Ê and A, B : Ê → CB Ê be mappings defined by fx x 1, gx x 2, Ax {x 2 /2 1}, and Bx {x 2 2}, for all x ∈ Ê .Clearly, there exists two sequences {x n 2 1/n} and So, the pair f, g is tangential with respect to the pair A, B .Definition 3.3.Let f : X → X and A : X → CB X .The mapping f is called tangential with respect to the mapping A if lim n → ∞ Ax n lim n → ∞ Ay n D ∈ CB X ,3.5 whenever sequences {x n } and {y n } in X such that lim Example 3.4.Let Ê , d be a metric space with usual metric d.Let f : Ê → Ê and A : Ê → CB Ê be mappings defined by fx x 1, Ax x 2 1 .
It follows from v ∈ C g, B , u ∈ C f, A that ggv gv, ffu fu, and Afu Bgv.Hence, gz z, fz z and Az Bz.Since the pair g, B is weakly compatible, gBv Bgv.Thus gz ∈ gBv Bgv Bz.Similarly, we can prove that fz ∈ Az.Consequently, z fz gz ∈ Az Bz.Therefore the maps f, g, A, and B have a common fixed point.If α 0 in Theorem 3.5, we get the following corollary.International Journal of Mathematics and Mathematical SciencesCorollary 3.6.Let f, g : X → X and A, B : X → CB X satisfy ∈ X for which the right-hand side of 3.21 is positive, where 0 < a < 1 and ψ : Ê → Ê ∈ X for which the right-hand side of 3.23 is positive, where 0 < a < 1 and ψ : Ê → Ê is a Lebesgue integrable mapping which is a summable nonnegative and such that International Journal of Mathematics and Mathematical Sciences for all x, y ∈ X for which the right-hand side of 3.26 is positive, where 0 < a < 1.If the following conditions (a)-(d) hold:a there exists a point z ∈ f X ∩ g Xwhich is a weak tangent point to f, g , b f, g is tangential with respect to A, B , c ffa fa, ggb gb and Afa Bgb for a ∈ C f, A and b ∈ C g, B , d the pairs f, A and g, B are weakly compatible.Then, f, g, A, and B have a common fixed point in X.If ψ t1, α 0, g f and B A in Theorem 3.5, we get the following corollary.
Theorem 3.5.Let f, g : X → X and A, B : X → CB X satisfy for all x, y ∈ X for which the right-hand side of 3.10 is positive, where 0 < a < 1, α ≥ 0 andψ : Ê → Ê is aLebesgue integrable mapping which is a summable nonnegative and such that 0 ψ t dt > 0, 3.11 for each > 0. If the following conditions (a)-(d) hold: a there exists a point z ∈ f X ∩ g X which is a weak tangent point to f, g , b f, g is tangential with respect to A, B , c ffa fa, ggb gb, and Afa Bgb for a ∈ C f, A and b ∈ C g, B , d the pairs f, A and g, B are weakly compatible.Then, f, g, A, and B have a common fixed point in X. Proof.Since z ∈ f X ∩ g X , z fu gv for some u, v ∈ X.It follows from a point z which is a weak tangent point to f, g that there exist sequences {x n } and {y n } in X such that lim n → ∞ fx n lim Corollary 3.7.Let f : X → X and A : X → CB X satisfy Corollary 3.8.Let f, g : X → X and A, B : X → CB X satisfy 1 αd fx, gy H Ax, By < α d Ax, fx d By, gy d Ax, gy d fx, By ad fx, gy 1 − a max d Ax, fx , d By, gy , d Ax, fx for all x, y ∈ X for which the right-hand side of 3.25 is positive, where 0 < a < 1 and α ≥ 0. If the following conditions (a)-(d) holds: a there exists a point z ∈ f X ∩ g X which is a weak tangent point to f, g , b f, g is tangential with respect to A, B , c ffa fa, ggb gb and Afa Bgb for a ∈ C f, A and b ∈ C g, B , d the pairs f, A and g, B are weakly compatible.Then, f, g, A, and B have a common fixed point in X. Corollary 3.10.Let f : X → X and A : X → CB X satisfy H Ax, Ay < ad fx, fy 1 − a max d Ax, fx , d Ay, fy , d Ax, fx 1/2 d Ax, fy 1/2 , d fx, Ay 1/2 d Ax, fy