Existence of a common solution of an integral equations system by ( ψ , α , β )-weakly contractions

In this paper, we consider a system of integral equations and apply the coincidence and common fixed point theorems for four mappings satisfying a (ψ ,α,β)-weakly contractive condition in ordered metric spaces to prove the existence of a common solution to integral equations. Also we furnish suitable examples to demonstrate the validity of the hypotheses of our results. MSC: 54H25; 47H10


Introduction and preliminary
Fixed point theory has wide and endless applications in many fields of engineering and science. Its core, the Banach contraction principle (see []), has attracted many researchers who tried to generalize it in different aspects. In particular, Alber and Guerre-Delabriere In [], Rhoades proved that if X is complete, then every weak contraction has a unique fixed point.
The weak contraction principle, its generalizations and extensions and other fixed point results for mappings satisfying weak contractive type inequalities have been considered in a number of recent works.
In , Dutta and Choudhury [] proved the following theorem. In [], Eslamian and Abkar introduced the concept of (ψ, α, β)-weak contraction. They stated the following theorem as a generalization of Theorem ..
Aydi et al. [] proved that Theorem . is a consequence of Theorem .. (Define ϕ : It is also known that common fixed point theorems are generalizations of fixed point theorems. Recently, many researchers have been interested in generalizing fixed point theorems to coincidence point theorems and common fixed point theorems.
On the other hand, compatibility of two mappings introduced by Jungck [, ] is an important concept in the context of common fixed point problems in metric spaces.
Definition . ([]) Let (X, d) be a metric space and f , g : X → X be given self-mappings on X. The pair (f , g) is said to be compatible if lim n→∞ d(fgx n , gfx n ) = , whenever {x n } is a sequence in X such that lim n→∞ fx n = lim n→∞ gx n = t, for some t ∈ X.
and ψ(t) =  if and only if t =  and α() = β() = . Also, if any non-decreasing sequence {x n } in X converges to z, then we assume x n z for all n ∈ N ∪ {}. If there exists x  ∈ X such that gx  fx  , then f and g have a coincidence point.

Altun and
Simsek [] introduced the concept of weakly increasing mappings as follows.
Definition . Let f , g be two self-maps on a partially ordered set (X, ). A pair (f , g) is said to be (i) weakly increasing if fx g(fx) and gx f (gx) for all x ∈ X [], (ii) partially weakly increasing if fx g(fx) for all x ∈ X [].
Note that a pair (f , g) is weakly increasing if and only if the ordered pairs (f , g) and (g, f ) are partially weakly increasing.
Nashine and Samet [] introduced weakly increasing mappings with respect to another map as follows.
Definition . ([]) Let (X, ) be a partially ordered set and f , g, h : X → X be given mappings such that fX ⊆ hX and gX ⊆ hX. We say that f and g are weakly increasing with respect to h if and only if for all x ∈ X, we have fx gy, ∀y ∈ h - (fx) and gx fy, ∀y ∈ h - (gx), where h - (x) := {u ∈ X | hu = x} for x ∈ X. http://www.journalofinequalitiesandapplications.com/content/2014/1/517 If f = g, we say that f is weakly increasing with respect to h.
If h : X → X is the identity mapping (hx = x for all x ∈ X), then f and g being weakly increasing with respect to h implies that f and g are weakly increasing mappings.
[] proved some new coincidence point and common fixed point theorems for a pair of weakly increasing mappings with respect to another map.
In Suppose that for every x, y ∈ X such that Sx and Ty are comparable, () holds. Then the pairs (f , S) and (g, T) have a coincidence point u ∈ X.
In this paper, an attempt is made to derive some coincidence and common fixed point theorems for four mappings on complete ordered metric spaces, satisfying a (ψ, α, β)weak contractive condition, which generalizes the existing results. Our results are supported by some examples.

Coincidence and common fixed point results
We begin our study with the following result.
Theorem . Let (X, d, ) be a partially ordered complete metric space. Let f , g, S, T : X → X be given mappings satisfying: is partially weakly increasing with respect to T and (g, f ) is partially weakly increasing with respect to S. Suppose that for every x, y ∈ X such that Sx and Ty are comparable, we have Then the pairs (f , S) and (g, T) have a coincidence point u ∈ X; that is, fu = Su and gu = Tu. Moreover, if Su and Tu are comparable, then u ∈ X is a coincidence point of f , g, S and T.
Proof Let x  be an arbitrary point in X. Since fX ⊆ TX, there exists x  ∈ X such that Tx  = fx  . Since gX ⊆ SX, there exists x  ∈ X such that Sx  = gx  . Continuing this process, we can construct sequences {x n } and {y n } in X defined by By construction we have x n+ ∈ T - (fx n ). Then, using the fact that (f , g) is partially weakly increasing with respect to T, we obtain Tx n+ = fx n gx n+ = Sx n+ , ∀n ∈ N ∪ {}. http://www.journalofinequalitiesandapplications.com/content/2014/1/517 On the other hand, we have x n+ ∈ S - (gx n+ ). Then, using the fact that (g, f ) is partially weakly increasing with respect to S, we obtain Therefore, we can then write We will prove our result in four steps.
Since Sx n and Tx n+ are comparable, by applying inequality (), we have where If d(y n- , y n ) < d(y n , y n+ ), then it follows from () and () that Therefore, () implies that By (), we have d(y n , y n+ ) = ; that is, y n = y n+ , and consequently we obtain Now, by applying inequality (), we have and () implies that d(y n+ , y n+ ) = ; that is, y n+ = y n+ . Repeating the above process inductively, one obtains y k = y n for all k ≥ n, which implies that () holds. On the other hand, if Taking the upper limit on both sides of () and using (), (), the upper semi-continuity of α, the lower semi-continuity of β and the continuity of ψ , we obtain ψ(r) ≤ α(r)β(r), which by () implies that r = . So equation () holds and the proof of Step  is completed.
Step . We claim that {y n } is a Cauchy sequence in X. By (), it suffices to show that the subsequence {y n } of {y n } is a Cauchy sequence in X. If not, then there exists >  for which we can find two subsequences {y m(k) } and {y n(k) } of {y n } such that n(k) is the smallest integer and, for all k > , Therefore we use (), () and the triangular inequality to get ≤ d(y m(k) , y n(k) ) ≤ d(y m(k) , y n(k)- ) + d(y n(k)- , y n(k)- ) + d(y n(k)- , y n(k) ) < + d(y n(k)- , y n(k)- ) + d(y n(k)- , y n(k) ).
Letting again k → ∞ in the above inequality and using () and (), we obtain By (), we have = , which is a contradiction. Thus {y n } is a Cauchy sequence in X, and hence {y n } is a Cauchy sequence.
Step . Existence of a coincidence point for (f , S) and (g, T). From the completeness of (X, d), there is u ∈ X such that lim n→∞ y n = u.
From () and (), we obtain Since the pairs (f , S) and (g, T) are compatible, Now, we relax the conditions of Theorem ., the continuity of f , g, S and T and the compatibility of the pairs (f , S) and (g, T), and we replace them by other conditions in order to find the same result. This will be the purpose of the next theorems. (X, d, ) be a partially ordered complete metric space such that X is regular. Let f , g, S, T : X → X be given mappings satisfying:

Theorem . Let
(i) fX ⊆ TX, gX ⊆ SX, (ii) SX and TX are closed subsets of (X, d), (iii) pairs (f , S) and (g, T) are weakly compatible, (iv) (f , g) is partially weakly increasing with respect to T and (g, f ) is partially weakly increasing with respect to S. Suppose that for every x, y ∈ X such that Sx and Ty are comparable, () holds. http://www.journalofinequalitiesandapplications.com/content/2014/1/517 Then the pairs (f , S) and (g, T) have a coincidence point u ∈ X; that is, fu = Su and gu = Tu. Moreover, if Su and Tu are comparable, then u ∈ X is a coincidence point of f , g, S and T .
Proof We take the same sequences {x n } and {y n } as in the proof of Theorem .. In particular, {y n } is a Cauchy sequence in (X, d). Hence, there exists v ∈ X such that lim n→∞ y n = v.
(   ) Since SX and TX are closed subsets of (X, d), there exist u  , u  ∈ X such that y n = Tx n+ → Tu  , y n+ = Sx n+ → Su  .
Therefore v = Tu  = Su  . Since {y n } is a non-decreasing sequence and X is regular, it follows from () that y n v for all n ∈ N ∪ {}. Hence, Applying inequality (), we have and Letting n → ∞ in () and using (), we obtain Therefore we can apply inequality () to obtain and Letting n → ∞ in () and using (), we obtain By (), we have d(v, gu  ) =  and hence v = gu  . Therefore we have obtained Now, if (f , S) and (g, T) are weakly compatible, then fv = fSu  = Sfu  = Sv and gv = gTu  = Tgu  = Tv, and v is a coincidence point of (f , S) and (g, T). The rest of the conclusion follows as in the proof of Theorem ..

Definition . ([]
) Let (X, d) be a metric space and f , g : X → X be given self-mappings on X. The pair (f , g) is said to be semi-compatible if the two conditions hold: (i) ft = gt implies fgt = gft, (ii) lim n→∞ fx n = lim n→∞ gx n = t for some t ∈ X, implies lim n→∞ fgx n = gt.
Singh and Jain [] observe that (ii) implies (i). Hence, they defined the semi-compatibility by condition (ii) only. It is clear that if the pair (f , g) is semi-compatible, then (f , g) is weakly compatible. http://www.journalofinequalitiesandapplications.com/content/2014/1/517 Definition . ([]) Let (X, d) be a metric space and f , g : X → X be given selfmappings on X. The pair (f , g) is said to be reciprocally continuous if lim n→∞ fgx n = ft and lim n→∞ gfx n = gt whenever {x n } is a sequence such that lim n→∞ fx n = lim n→∞ gx n = t for some t ∈ X.
Definition . ([]) Let (X, d) be a metric space and f , g : X → X be given self-mappings on X. The pair (f , g) is said to be f -weak reciprocally continuous if lim n→∞ fgx n = ft whenever {x n } is a sequence such that lim n→∞ fx n = lim n→∞ gx n = t for some t ∈ X.
In the next theorem, the concepts of semi-compatibility and f -weakly reciprocal continuity are used. (X, d, ) be a partially ordered complete metric space. Let f , g, S, T : X → X be given mappings satisfying:

Theorem . Let
(i) fX ⊆ TX, gX ⊆ SX, (ii) the pair (f , S) is f -weak reciprocally continuous and semi-compatible, (iii) the pair (g, T) is g-weak reciprocally continuous and semi-compatible, (iv) (f , g) is partially weakly increasing with respect to T and (g, f ) is partially weakly increasing with respect to S. Suppose that for every x, y ∈ X such that Sx and Ty are comparable, () holds.
Then the pairs (f , S) and (g, T) have a coincidence point u ∈ X; that is, fu = Su and gu = Tu. Moreover, if Su and Tu are comparable, then u ∈ X is a coincidence point f , g, S and T.
Proof We take the same sequences {x n } and {y n } as in the proof of Theorem .. In particular, {y n } is a Cauchy sequence in (X, d). Hence, there exists u ∈ X such that Hence, by () and (), we deduce that w = fw = gw = Sw = Tw. Therefore w is a common fixed point of f , g, S and T. Now, suppose that the set of fixed points of f is totally ordered. Assume on the contrary that fp = gp = Sp = Tp = p and fq = gq = Sq = Tq but p = q. Since p and q contain a set of fixed points of f , we obtain p = Sp and q = Tq are comparable, by inequality (), we have

Some examples
In this section we present some examples which illustrate our results. Now, we present an example to illustrate the obtained result given by the previous theorems.
Example . Let X = [, +∞). We define an order on X as x y if and only if x ≥ y for all x, y ∈ X. We take the usual metric d(x, y) = |x -y| for x, y ∈ X. It is easy to see that (X, d, ) is a partially ordered complete metric space. Let f , g, S, T : X → X be defined by Define ψ, α, β : [, +∞) → [, +∞) by ψ(t) = t, Then f , g, S, T, ψ , α and β satisfy all the hypotheses of Theorems . and .. http://www.journalofinequalitiesandapplications.com/content/2014/1/517 Proof The proof of (i) and (ii) is clear. To prove (iii), let {x n } be any sequence in X such that lim n→∞ fx n = lim n→∞ Sx n = t for some t ∈ X. Since fx n = ln( + x n ) and Sx n = e x n -, we have x n → e t - and x n →   ln( + t). By the uniqueness of limit, we get that e t - =   ln( + t) and hence t = . Thus, x n →  as n → ∞. Since f and S are continuous, we have fx n → f  =  and Sx n → S =  as n → ∞. Therefore, Thus, the pair (f , S) is compatible. Similarly, one can show that the pair (g, T) is compatible.
To prove that (f , g) is partially weakly increasing with respect to T, let x, y ∈ X be such that y ∈ T - (fx). Then Ty = fx. By the definition of f and T, we have ln( + x) = e y -. So, we have y = ln( + ln( + x)). Now, since e x - ≥ x ≥ x ≥ ln( + x), we have Therefore, fx gy. Thus, we have proved that (f , g) is partially weakly increasing with respect to T. Similarly, one can show that (g, f ) is partially weakly increasing with respect to S. Now, we prove that ψ , α and β do satisfy the inequality of (). If t > , then In order to show that f , g, S, T, ψ , α and β do satisfy the contractive condition () in Theorem ., using a mean value theorem, we have, for x, y ∈ X,  N(x, y)) =   . Therefore, we have If N(x, y) ≤ , then α(M(x, y)) =   M(x, y) +   and β (N(x, y) N(x, y)) = . So, we obtain N(x, y) .
Therefore in this case () is satisfied. Thus, f , g, S, T, ψ and ϕ satisfy all the hypotheses of Theorems .. Therefore, f , g, S and T have a coincidence point. Moreover, since f , g, S and T satisfy all the hypotheses of Theorem ., we obtain that f , g, S and T have a unique common fixed point. In fact,  is the unique common fixed point of f , g, S and T. Let {x n } be a non-decreasing sequence in X with respect to such that x n → x. By the definition of metric d, there exists k ∈ N such that x n = x for all n ≥ k. So (X, d, ) is regular.
Let ψ, α, β : [, ∞) → [, ∞) be defined by ψ(t) = t, and self-maps f , g, S and T on X be given by It is easy to see that f , g, S and T satisfy all the conditions given in Theorem .. Thus ,  and  are coincidence points of the pairs (f , S) and (g, T). Since S =  and T =  are comparable,  is a coincidence point f , g, S and T. Moreover, since S =  and SS =  are not comparable, so Theorem . is not applicable for this example. It is observed that  is not a common fixed point f , g, S and T.

Application: existence of a common solution to integral equations
Consider the integral equations: Then (X, ) is a partially ordered set and regular. Also (X, · ) is a complete metric space. Define f , g : X → X by Put ψ(t) = t  , α(t) =   t  and β(t) = t log( + t  ), we get ψ d(fx, gy) ≤ α M(x, y)β N(x, y) , and ψ(t)α(t) + β(t) >  for each t > . By taking S = T = I X (the identity mapping on X), all the required hypotheses of Theorem . (or Theorem .) are satisfied. Then there exists x ∈ X, a common fixed point of f and g, that is, x is a solution to ().