Introduction

The Banach Contraction Mapping Principle, appeared in explicit form in Banach's thesis in 1922 [1] (see also [2]) where it was used to establish the existence of a solution for an integral equation. Since then, because of its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [3–5], etc. In 1969, the Banach's Contraction Mapping Principle extended nicely to set-valued or multivalued mappings, a fact first noticed by Nadler [6]. Afterward, the study of fixed points for multi-valued contractions using the Hausdorff metric was initiated by Markin [7]. Later, an interesting and rich fixed point theory for such mappings was developed (see [[8–13]]). The theory of multi-valued mappings has applications in optimization problems, control theory, differential equations, and economics.

In 1982, Sessa [14] introduced the notion of weakly commuting mappings. Jungck [15] defined the notion of compatible mappings to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true [15]. In recent years, a number of fixed point theorems have been obtained by various authors utilizing this notion. Jungck further weakens the notion of compatibility by introducing the notion of weak compatibility and in [16] Jungck and Rhoades further extended weak compatibility to the setting of single-valued and multivalued maps. In 2002, Aamri and Moutawakil [17] defined property (E.A). This concept was frequently used to prove existence theorems in common fixed point theory. Three years later, Liu et al.[18] introduced common property (E.A). The class of (E.A) maps contains the class of noncompatible maps. Recently, Pathak and Shahzad [19] introduced the new concept of weak tangent point and tangential property for single-valued mappings and established common fixed point theorems.

The aim of this article is to develop a tangential property, which can be used only single-valued mappings, based on the work of Pathak and Shahzad [19]. We define a tangential property, which can be used for both single-valued mappings and multi-valued mappings, and prove common fixed point theorems of Gregus type for four mappings satisfying a strict general contractive condition of integral type.

Preliminaries

Throughout this study (X, d) denotes a metric space. We denote by CB(X), the class of all nonempty bounded closed subsets of X. The Hausdorff metric induced by d on CB(X) is given by

for every A, B ∈ CB(X), where d(a, B) = d(B, a) = inf{d(a, b): b ∈ B} is the distance from a to B ⊆ X.

Definition 2.1. Let f : X → X and T : X → CB(X).

  1. 1.

    A point x ∈ X is a fixed point of f (respecively T ) iff fx = x (respecively x ∈ Tx).

The set of all fixed points of f (respecively T) is denoted by F (f) (respecively F (T)).

  1. 2.

    A point x ∈ X is a coincidence point of f and T iff fx ∈ Tx.

The set of all coincidence points of f and T is denoted by C(f, T).

  1. 3.

    A point x ∈ X is a common fixed point of f and T iff x = fx ∈ Tx.

The set of all common fixed points of f and T is denoted by F (f, T).

Definition 2.2. Let f : X → X and g : X → X. The pair (f, g) is said to be

  1. (i)

    commuting if fgx = gfx for all x ∈ X;

  2. (ii)

    weakly commuting [14] if d(fgx, gfx) ≤ d(fx, gx) for all x ∈ X;

  3. (iii)

    compatible [15] if limn→∞d(fgx n , gfx n ) = 0 whenever {x n } is a sequence in X such that

for some z ∈ X;

  1. (iv)

    weakly compatible [20] fgx = gfx for all x ∈ C(f, g).

Definition 2.3. [16] The mappings f : X → X and A : X → CB(X) are said to be weakly compatible fAx = Afx for all x ∈ C(f, A).

Definition 2.4. [17] Let f : X → X and g : X → X. The pair (f, g) satisfies property (E.A) if there exist the sequence {x n } in X such that

(1)

See example of property (E.A) in Kamran [21, 22] and Sintunavarat and Kumam [23].

Definition 2.5. [18] 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

(2)

Remark 2.6. If A = f, B = g and {x n } = {y n } in (2), then we get the definition of property (E.A).

Definition 2.7. [19] Let f, g : X → X. A point z ∈ X is said to be a weak tangent point to (f, g) if there exists sequences {x n } and {y n } in X such that

(3)

Remark 2.8. If {x n } = {y n } in (3), we get the definition of property (E.A).

Definition 2.9. [19] Let f, g, A, B : X → X. The pair (f, g) is called tangential w.r.t. the pair (A, B) if there exists sequences {x n } and {y n } in X such that

(4)

Main results

We first introduce the definition of tangential property for two single-valued and two multi-valued mappings.

Definition 3.1. Let f, g : X → X and A, B : X → CB(X). The pair (f, g) is called tangential w.r.t. the pair (A, B) if there exists two sequences {x n } and {y n } in X such that

(5)

for some z ∈ X, then

(6)

Throughout this section, ℝ+ denotes the set of nonnegative real numbers.

Example 3.2. Let (ℝ+, d) be a metric space with usual metric d, f, g : ℝ+ → ℝ+ and A, B : ℝ+ → CB(ℝ+) mappings defined by

Since there exists two sequences and such that

and

Thus the pair (f, g) is tangential w.r.t the pair (A, B).

Definition 3.3. Let f : X → X and A : X → CB(X). The mapping f is called tangential w.r.t. the mapping A if there exist two sequences {x n } and {y n } in X such that

(7)

for some z ∈ X, then

(8)

Example 3.4. Let (ℝ+, d) be a metric space with usual metric d, f : ℝ+ → ℝ+ and A : ℝ+ → CB(ℝ+) mappings defined by

Since there exists two sequences and such that

and

Therefore the mapping f is tangential w.r.t the mapping A.

Define Ω = {w : (ℝ+)4 → ℝ+| w is continuous and w(0, x, 0, x) = w(x, 0, x, 0) = x}. There are examples of w ∈ Ω:

  1. (1)

    w1(x1, x2, x3, x4) = max{x1, x2, x3, x4};

  2. (2)

    ;

  3. (3)

    .

Next, we prove our main results.

Theorem 3.5. Let f, g : X → X and A, B : X → CB(X) satisfy

(9)

for all x, y ∈ X for which the righthand side of (9) is positive, where 0 < a < 1, α ≥ 0, p ≥ 1, w ∈ Ω and ψ : ℝ+ → ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that

(10)

for each ε > 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    ffa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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. It follows from z ∈ f(X) ∩ g(X) that z = fu = gv for some u, v ∈ X. Using that a point z is a weak tangent point to (f, g), there exist two sequences {x n } and {y n } in X such that

(11)

Since the pair (f, g) is tangential w.r.t the pair (A, B) and (11), we get

(12)

for some D ∈ CB(X). Using the fact z = fu = gv, (11) and (12), we get

(13)

We show that z ∈ Bv. If not, then condition (9) implies

(14)

Letting n → ∞, we get

(15)

Since

(16)

which is a contradiction. Therefore z ∈ Bv. Again, we claim that z ∈ Au. If not, then condition (9) implies

(17)

Letting n → ∞, we get

(18)

Since

(19)

which is a contradiction. Thus z ∈ Au.

Now we conclude z = gv ∈ Bv and z = fu ∈ Au. 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 we setting w in Theorem 3.5 by then we get the following corollary:

Corollary 3.6. Let f, g : X → X and A, B : X → CB(X) satisfy

(20)

for all x, y ∈ X for which the righthand side of (20) is positive, where 0 < a < 1, α ≥ 0, p ≥ 1 and ψ : ℝ+ → ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that

(21)

for each ε > 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    f fa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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 we setting w in Theorem 3.5 by and p = 1, then we get the following corollary:

Corollary 3.7. Let f, g : X → X and A, B : X → CB(X) satisfy

(22)

for all x, y ∈ X for which the righthand side of (22) is positive, where 0 < a < 1, α ≥ 0 and ψ : ℝ+ → ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that

(23)

for each ε > 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    f fa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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 α = 0 in Corollary 3.7, we get the following corollary:

Corollary 3.8. Let f, g : X → X and A, B : X → CB(X) satisfy

(24)

for all x, y ∈ X for which the righthand side of (24) is positive, where 0 < a < 1 and ψ : ℝ+ → ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that

(25)

for each ε > 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    f fa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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 α = 0, g = f and B = A in Corollary 3.7, we get the following corollary:

Corollary 3.9. Let f : X → X and A : X → CB(X) satisfy

(26)

for all x, y ∈ X for which the righthand side of (26) is positive, where 0 < a < 1 and ψ : ℝ+ → ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that

(27)

for each ε > 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a sequence {x n } in X such that limn→∞fx n ∈ X,

  2. (b)

    f is tangential w.r.t A,

  3. (c)

    f fa = fa for a ∈ C(f, A),

  4. (d)

    the pair (f, A) is weakly compatible.

Then f and A have a common fixed point in X.

If ψ (t) = 1 in Corollary 3.7, we get the following corollary:

Corollary 3.10. Let f, g : X → X and A, B : X → CB(X) satisfy

(28)

for all x, y ∈ X for which the righthand side of (28) is positive, where 0 < a < 1 and α ≥ 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    f fa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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(t) = 1 and α = 0 in Corollary 3.7, we get the following corollary:

Corollary 3.11. Let f, g : X → X and A, B : X → CB(X) satisfy

(29)

for all x, y ∈ X for which the righthand side of (29) is positive, where 0 < a < 1. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    f fa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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(t) = 1, α = 0, g = f, and B = A in Corollary 3.7, we get the following corollary:

Corollary 3.12. Let f : X → X and A : X → CB(X) satisfy

(30)

for all x, y ∈ X for which the righthand side of (30) is positive, where 0 < a < 1. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a sequence {x n } in X such that limn→∞fx n ∈ X,

  2. (b)

    f is tangential w.r.t A,

  3. (c)

    f fa = fa for a ∈ C(f, A),

  4. (d)

    the pair (f, A) is weakly compatible.

Then f and A have a common fixed point in X.

Define Λ = {λ : (ℝ+)5 → ℝ+| λ is continuous and λ(0, x, 0, x, 0) = λ(x, 0, x, 0, 0) = kx where 0 < k < 1}.

Theorem 3.13. Let f, g : X → X and A, B : X → CB(X) satisfy

(31)

for all x, y ∈ X for which the righthand side of (31) is positive, where α ≥ 0, p ≥ 1, λ ∈ Λ and ψ : ℝ+ → ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that

(32)

for each ε > 0. If the following conditions (a)-(d) holds:

  1. (a)

    there exists a point z ∈ f(X) ∩ g(X) which is a weak tangent point to (f, g),

  2. (b)

    (f, g) is tangential w.r.t (A, B),

  3. (c)

    f fa = fa, ggb = gb and Afa = Bgb for a ∈ C(f, A) and b ∈ C(g, B),

  4. (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 is a weak tangent point to (f, g) and the pair (f, g) is tangential w.r.t the pair (A, B). It follows similarly Theorem 3.5 that there exist sequences {x n } and {y n } in X such that

(33)

for some D ∈ CB(X). We claim that z ∈ Bv. If not, then condition (31) implies

(34)

Letting n → ∞, we get

(35)

Since

(36)

which is a contradiction. Therefore z ∈ Bv. Again, we claim that z ∈ Au. If not, then condition (31) implies

(37)

Letting n →∞, we get

(38)

Since

(39)

which is a contradiction. Thus z ∈ Au.

Now we conclude z = gv ∈ Bv and z = fu ∈ Au. It follows from Theorem 3.5 that z = fz = gz ∈ Az = Bz. Therefore the maps f, g, A and B have a common fixed point.    □