Weakly compatible and quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations

<jats:p>In this paper, we present some weakly compatible and quasi-contraction results for self-mappings in fuzzy cone metric spaces and prove some coincidence point and common fixed point theorems in the said space. Moreover, we use two Urysohn type integral equations to get the existence theorem for common solution to support our results. The two Urysohn type integral equations are as follows:
<jats:disp-formula><jats:alternatives><jats:tex-math> $$\begin{aligned} &x(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &y(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned}$$ </jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtable><mml:mtr><mml:mtd /><mml:mtd><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>l</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msub><mml:mi>K</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mspace /><mml:mi>d</mml:mi><mml:mi>v</mml:mi><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>l</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mi>y</mml:mi><mml:mo>(</mml:mo><mml:mi>l</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msub><mml:mi>K</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mspace /><mml:mi>d</mml:mi><mml:mi>v</mml:mi><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>l</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></jats:alternatives></jats:disp-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$l\in [0,1]$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>l</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$x,y,g\in \mathbf{E}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi></mml:math></jats:alternatives></jats:inline-formula>, where <jats:bold>E</jats:bold> is a real Banach space and <jats:inline-formula><jats:alternatives><jats:tex-math>$K_{1},K_{2}:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>:</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo><mml:mo>×</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo><mml:mo>×</mml:mo><mml:mi>R</mml:mi><mml:mo>→</mml:mo><mml:mi>R</mml:mi></mml:math></jats:alternatives></jats:inline-formula>.</jats:p>

The initial version of fuzzy set theory was given by Zadeh [12], while Kramosil et al. in [13] introduced the fuzzy metric space or (shortly FM-space). Later on, a stronger form of the metric fuzziness was given by George et al. [14]. Some more related results in the context of fuzzy metric space can be found (e.g., see [15][16][17][18][19]).
Recently, Oner et al. in [20] introduced the concept of fuzzy cone metric space or shortly FCM-space. They presented some basic properties and a fuzzy cone Banach contraction theorem in a fuzzy cone metric space with the assumption that fuzzy cone contractive sequences are Cauchy. Some more properties and fixed point results in FCM-spaces can be found (e.g., see [21][22][23][24][25][26] and the references therein).
The aim of this paper is to obtain some coincidence point and common fixed point results for weakly compatible self-mappings in FCM-spaces. We also give the concept of quasi-contraction for weakly compatible self-mappings and establish some common fixed point theorems. Moreover, we present an integral type application from which we obtained the existence of fixed point results. The application of integral equations in fuzzy cone metric spaces is the new direction in the theory of fixed point. This new concept of application will be very fruitful for finding the existence solution of integral value problems on FCM-spaces. For this purpose, we use the two Urysohn integral type equations for common solution to support our results. We also present some illustrative examples to support our work.

Preliminaries
In this section, we present some basic definitions and a helpful concept for our main results. (1) * is commutative, associative, and continuous.

Definition 2.2 ([14])
A three-tuple (X, M, * ) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous s-norm, and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions: For more details, we shall refer the readers to study [14].

Definition 2.3 ([1]) A subset P of a real Banach space E is called a cone if
(1) P is closed, nonempty and P = {ϑ}, where ϑ is the zero element of E.
A partial ordering " " for a given cone P on E is defined as y x iff xy ∈ P. y ≺ x stands for y x and y = x, while y x stands for xy ∈ int(P). Throughout this paper, all the cones have nonempty interior.
is said to be complete if every Cauchy sequence is convergent in X. (iv) (x i ) is said to be a fuzzy cone contractive if ∃β ∈ (0, 1) such that ∀x, y, z ∈ X and each s ϑ.  for each x, y ∈ X and s ϑ. β is called the contraction constant of F 1 .
then v is called a coincidence point of F 1 and , and u is called a point of coincidence of F 1 and . The self-mappings F 1 and are said to be weakly compatible if they commute at their coincidence point, i.e.,

Lemma 2.13 ([28])
Let F 1 and be occasionally weakly compatible self-maps of a set X. If F 1 and have a unique point of coincidence, F 1 v = v = u, then u is a unique common fixed point of and F 1 .
"A self-mapping F 1 in a complete FCM-space in which the contractive sequences are Cauchy and hold (2.1), then F 1 has a unique fixed point in X" is a Banach contraction principle, which has been obtained in [20].
We note that fuzzy cone contractive sequences can be proved to be Cauchy sequences for weakly compatible self-mappings in FCM-spaces (see the proof of Theorem 3.1). In this paper we use the concept of complete FCM-spaces given by Rehman and Li [25] and prove some coincidence point and common fixed point theorems for weakly compatible three self-mappings and some quasi-contraction results in FCM-spaces. Moreover, we present some illustrative examples and the application of two Urysohn's integral type equations for the existence of common solution to support our work.

Weakly compatible mapping results in FCM-space
Theorem 3.1 Let F 1 , F 2 , : X → X be three self-maps and M be triangular in a complete FCM-space (X, M, * ) satisfying ∀x, y ∈ X, Proof Fix x 0 ∈ X and use the condition F 1 (X) ∪ F 2 (X) ⊂ (X). We define some iterative sequences in X such that which shows that a sequence ( x i ) i≥0 is fuzzy cone contractive. Hence, which shows that a sequence ( x i ) is Cauchy sequence and (X) is a complete subspace of X. Hence ∃u, v ∈ X such that Then, Thus, from (3.3) and (3.4), we have Similarly, we can prove that u = v = F 2 v. It follows that u is a common coincidence point of the mappings , F 1 , and F 2 in X such that u Now we prove the uniqueness of the point of coincidence in X for the mappings F 1 , F 2 , and . Let u * be the other point in X such that for some v * ∈ X. Then, by using (3.1) for s ϑ, β + 2δ < 1, since β + 2γ + 2δ < 1. Thus we get that M(u, u * , t) = 1, that is, u = u * . By using the weak compatibility of (F 1 , ), (F 2 , ) and Proposition 2.11, we can get a unique common fixed point of F 1 , F 2 , and , that is By using the map = I x and by taking into account that every self-mapping is weakly compatible with identity map, i.e., I x , we can get the following corollary.
for all x, y ∈ X, s ϑ, and β, γ , δ ∈ [0, ∞) with β + 2γ + 2δ < 1. Then F 1 and F 2 have a unique common fixed point in X. Moreover, the fixed point of F 1 is to be a fixed point of F 2 and conversely.

Theorem 4.2 Let F 1 , : X → X be two self-maps and M be triangular in a complete FCMspace (X, M, * ) such that F 1 (X) ⊂ (X) and (X) is closed. If F 1 is an -quasi-contraction with constant q c ∈ [0, 1), then and F 1 have a unique point of coincidence. Moreover, if a pair ( , F 1 ) is occasionally weakly compatible, then F 1 and have a unique common fixed point in X.
Proof Fix x 0 ∈ X and use the condition F 1 (X) ⊂ (X). We construct a sequence (y i ) in X such that Now, we have to show that (y i ) is a Cauchy sequence. First, we prove that for all i ≥ 1 and s ϑ. Indeed, Then we may have the following four cases: Thus (4.4) holds as q c < q c /(1q c ) since q c ∈ [0, 1). (ii) Second, by using the M triangular property, we have It follows that (4.4) holds. (iii) Third, Hence (4.4) holds. (iv) Fourth, which implies M(y i , y i+1 , s) = 1 for s ϑ.
In this case, immediately (4.4) follows since q c ∈ [0, 1). Now, we may assume that δ = q c 1-q c < 1, then we have that , for s ϑ.
In view of (4.4), (y 0 , y 1 , s) for all i ≥ 1 and s ϑ, which shows that (y i ) is a fuzzy cone contractive sequence in X such that lim i→∞ M(y i , y i+1 , s) = 1 for s ϑ. (4.7) Since M is triangular, then for all j > i ≥ i 0 , which shows that (y i ) is a Cauchy sequence in X. Since (X, M, * ) is complete and (X) is closed, ∃v ∈ X such that  Now we have to show that v = F 1 v. By using the triangularity of M, we have By the definition of -quasi-contraction, we have that where This implies for s ϑ. Then we have the following two cases: Then from (4.8), (4.9), and (4.10), we get that Then from (4.10) we have that lim sup Now, this together with (4.8) and (4.9) gives, F 1 v, s) -1 , for s ϑ.
Since q c < 1, therefore M( v, F 1 v, s) = 1, i.e., v = F 1 v = u. Thus from both cases we get that v = F 1 v = u. Hence, the same as in Theorem 3.1, v is the coincidence point of ( , F 1 ) and u is its coincidence point in X. The uniqueness of the coincidence point can be shown by the standard way. By using Lemma 2.13, one can readily obtain that, when ( , F 1 ) is occasionally weakly compatible, then u is a unique common fixed point of and F 1 in X.

Theorem 4.3
Let be a self-map on X and M be triangular in a complete FCM-space (X, M, * ) such that 2 is continuous. Let the self-map F 1 : X → X that commutes with . Further, we assume that F 1 and satisfy F 1 (X) ⊂ 2 (X), (4.11) and let F 1 be an -quasi-contraction. Then F 1 and have a unique common fixed point in X.
Proof By condition (4.11), starting with fix x 0 ∈ (X), define a sequence (x i ) in X such that as in Theorem 4.2. Now The same as in Theorem 4.2, we can get that (v i ) is a Cauchy sequence and convergent to some point v ∈ X such that lim i→∞ M( y i+1 , v, s) = 1 for s ϑ.
Further, we have to show that 2 v = F 1 v. Since, (4.12) by the continuity of 2 , it follows that Now, by the triangular property of M, we have -1 , for s ϑ (4.14) and This together with (4.13) and (4.14) leads to Since 0 ≤ q c < 1, this implies that M( 2 v, F 1 v, s) = 1, that is, F 1 v = 2 v. Thus from both cases we get that F 1 v = 2 v. This implies that v is the common fixed point of and F 1 . Now we prove the uniqueness. Assume that v = w such that F 1 w = w, and let w * be the other common fixed point of the mappings and F 1 such that F 1 w * = w * . Then, by the standard way of -quasi-contraction, easily we can get that w = w * . This completes the proof.

Application
In this section, we present an integral type application, which is the new direction in FCMspaces. For this purpose, we present the two Urysohn integral type equations, or shortly UITEs, to prove the existence result for common solution. Assume that X = [0, 1], and let E be the real-valued functions on X. Then E is a vector space over R under the following operations: , where l ∈ [0, 1] and x, y, g ∈ E.
Now, we present a special type of example for UITEs. , where i = 1, 2.

Conclusion
We defined weakly compatible self-mappings in fuzzy cone metric spaces and proved some coincidence point and common fixed point theorems under the fuzzy cone contraction condition without the assumption that fuzzy cone contractive sequences are Cauchy by using the "M triangular condition". This change, to use "M triangular condition" to weaken the "fuzzy cone contractive sequences are Cauchy", is expected to bring a wider range of applications of fixed point theorems in fuzzy cone metric spaces. We also gave the concept of quasi-contraction and proved some common fixed point theorems in fuzzy cone metric spaces. Moreover, we presented an application of the two Urysohn integral type equations for common solution to support our result. We also presented some illustrative examples to support our theoretical work.