() @ Appl. Gen. Topol. 18, no. 2 (2017), 277-287doi:10.4995/agt.2017.6776 c© AGT, UPV, 2017 Existence of common fixed points of improved F-contraction on partial metric spaces Muhammad Nazam a, Muhammad Arshad a and Mujahid Abbas b a Department of Mathematics and Statistics, International Islamic University, Islamabad, Pak- istan (nazim.phdma47@iiu.edu.pk, marshadzia@iiu.edu.pk) b Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa. (abbas.mujahid@gmail.com) Communicated by S. Romaguera Abstract Following the approach of F-contraction introduced by Wardowski [13], in this paper, we introduce improved F-contraction of rational type in the framework of partial metric spaces and used it to obtain a common fixed point theorem for a pair of self mappings. We show, through ex- ample, that improved F-contraction is more general than F- contrac- tion and guarantees fixed points in those cases where F-contraction fails to provide. Moreover, we apply this fixed point result to show the existence of common solution of the system of integral equations. 2010 MSC: 47H10; 47H04; 54H25. Keywords: fixed point; improved F-contraction; integral equations; partial metric. 1. Introduction Matthews [11] introduced the concept of partial metric spaces and proved an analogue of Banach’s fixed point theorem in partial metric spaces. In fact, a partial metric space is a generalization of metric space in which the self dis- tances p(r1, r1) of elements of a space may not be zero and follows the inequal- ity p(r1, r1) ≤ p(r1, r2). After this remarkable contribution, many authors took interest in partial metric spaces and its topological properties and presented Received 29 October 2016 – Accepted 18 March 2017 http://dx.doi.org/10.4995/agt.2017.6776 M. Nazam, M. Arshad and M. Abbas several well known fixed point results in the framework of partial metric spaces (see [1, 2, 3, 4, 12] and references therein). In 1922, Banach presented a landmark fixed point result (Banach Contraction Principle). This result proved a gateway for the fixed point researchers and opened a new door in metric fixed point theory. A number of efforts have been made to enrich and generalize Banach Contraction Principle (see [6, 7] and references therein). Following Banach, in 2012, Wardowski [13] presented a new contraction (known as F - contraction). Since 2012, a number of fixed point results have been established by using F-contraction (see [5, 9, 10] and references therein). Recently, Vetro et al.[8] proved some fixed point theorems for Hardy-Rogers-type self-mappings in complete metric spaces and complete ordered metric spaces for F- contractions. In this article, following Wardowski [13] and Vetro et al.[8], we prove a com- mon fixed point theorem for a pair of self mappings satisfying improved F- contraction of rational type in complete partial metric spaces. An example is constructed to illustrate this result. We apply the mentioned theorem to show the existence of solution of system of Volterra type integral equations. 2. preliminaries Throughout this paper, we denote (0, ∞) by R+, [0, ∞) by R+0 , (−∞, +∞) by R and set of natural numbers by N. Following concepts and results will be required for the proofs of main results. Definition 2.1 ([13]). A mapping T : M → M, is said to be F-contraction if it satisfies following condition (2.1) (d(T (r1), T (r2)) > 0 ⇒ τ + F(d(T (r1), T (r2)) ≤ F(d(r1, r2))), for all r1, r2 ∈ M and some τ > 0. Where F : R + → R is a function satisfying following properties. (F1) : F is strictly increasing. (F2) : For each sequence {rn} of positive numbers limn→∞ rn = 0 if and only if limn→∞ F(rn) = −∞. (F3) : There exists θ ∈ (0, 1) such that limα→0+(α) θF(α) = 0. Wardowski [13] established the following result using F-contraction. Theorem 2.2 ([13]). Let (M, d) be a complete metric space and T : M → M be a F-contraction. Then T has a unique fixed point υ ∈ M and for every r0 ∈ M the sequence {T n(r0)} for all n ∈ N is convergent to υ. We denote by ∆F , the set of all functions satisfying the conditions (F1)− (F3). Example 2.3 ([13]). Let F : R+ → R be given by the formula F(α) = ln(α). It is clear that F satisfies (F1)−(F3) for any κ ∈ (0, 1). Each mapping T : M → M satisfying (2.1) is a F-contraction such that d(T (r1), T (r2)) ≤ e −τ d(r1, r2), for all r1, r2 ∈ M, T (r1) 6= T (r2). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 278 Improved F -contraction on partial metric spaces Obviously, for all r1, r2 ∈ M such that T (r1) = T (r2), the inequality d(T (r1), T (r2)) ≤ e −τ d(r1, r2) holds, that is T is a Banach contraction. Remark 2.4. From (F1) and (2.1) it is easy to conclude that every F-contraction is necessarily continuous. Definition 2.5 ([11]). Let M be a nonempty set and if the function p : M × M → R+0 satisfies following properties, (p1) r1 = r2 ⇔ p (r1, r1) = p (r1, r2) = p (r2, r2) , (p2) p (r1, r1) ≤ p (r1, r2) , (p3) p (r1, r2) = p (r2, r1) , (p4) p (r1, r3) ≤ p (r1, r2) + p (r2, r3) − p (r2, r2) . for all r1, r2, r3 ∈ M. Then p is called a partial metric on M and the pair (M, p) is known as partial metric space. In [11], Matthews proved that every partial metric p on M induces a metric dp : M × M → R + 0 defined by (2.2) dp (r1, r2) = 2p (r1, r2) − p (r1, r1) − p (r2, r2) ; for all r1, r2 ∈ M. Notice that a metric on a set M is a partial metric p such that p(r, r) = 0 for all r ∈ M and p(r1, r2) = 0 implies r1 = r2 ( using (p1) and (p2)). Matthews [11] established that each partial metric p on M generates a T0 topology τ(p) on M. The base of topology τ(p) is the family of open p-balls {Bp (r, ǫ) : r ∈ M, ǫ > 0}, where Bp (r, ǫ) = {r1 ∈ M : p (r, r1) < p (r, r) + ǫ} for all r ∈ M and ǫ > 0. A sequence {rn}n∈N in (M, p) converges to a point r ∈ M if and only if p(r, r) = limn→∞ p(r, rn). Definition 2.6 ([11]). Let (M, p) be a partial metric space. (1) A sequence {rn}n∈N in (M, p) is called a Cauchy sequence if limn,m→∞ p(rn, rm) exists and is finite. (2) A partial metric space (M, p) is said to be complete if every Cauchy sequence {rn}n∈N in M converges, with respect to τ(p), to a point r ∈ X such that p(r, r) = limn,m→∞ p(rn, rm). The following lemma will be helpful in the sequel. Lemma 2.7 ([11]). (1) A sequence rn is a Cauchy sequence in a partial metric space (M, p) if and only if it is a Cauchy sequence in metric space (M, dp) (2) A partial metric space (M, p) is complete if and only if the metric space (M, dp) is complete. (3) A sequence {rn}n∈N in M converges to a point r ∈ M, with respect to τ(dp) if and only if limn→∞ p(r, rn) = p(r, r) = limn,m→∞ p(rn, rm). (4) If limn→∞ rn = υ such that p(υ, υ) = 0 then limn→∞ p(rn, r) = p(υ, r) for every r ∈ M. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 279 M. Nazam, M. Arshad and M. Abbas In the following example, we shall show that there are mappings which are not F-contractions in metric spaces, nevertheless, such mappings follow the conditions of F-contraction in partial metric spaces. Example 2.8. Let M = [0, 1] and define partial metric by p(r1, r2) = max {r1, r2} for all r1, r2 ∈ M. The metric d induced by partial metric p is given by d(r1, r2) = |r1 − r2| for all r1, r2 ∈ M. Define the mappings F : R + → R by F(r) = ln(r) and T by T (r) =      r 5 if r ∈ [0, 1); 0 if r = 1 Then T is not a F- contraction in a metric space (M, d). Indeed, for r1 = 1 and r2 = 5 6 , d(T (r1), T (r2)) > 0 and we have τ + F (d(T (r1), T (r2))) ≤ F (d(r1, r2)) , τ + F ( d(T (1), T ( 5 6 )) ) ≤ F ( d(1, 5 6 ) ) , τ + F ( d(0, 1 6 ) ) ≤ F ( 1 6 ) , 1 6 < 1 6 , which is a contradiction for all possible values of τ. Now if we work in partial metric space (M, p), we get a positive answer that is τ + F (p(T (r1), T (r2))) ≤ F (p(r1, r2)) implies τ + F ( 1 6 ) ≤ F (1) , which is true. Similarly, for all other points in M our claim proves true. 3. main result We begin with following definitions. Definition 3.1. Let (M, p) be a partial metric space. The mapping T : M → M is called a improved F-contraction of rational type, if for all m1, m2 ∈ M, we have (3.1) τ + F(p(T (m1), T (m2))) ≤ F (N(m1, m2)) , for some F ∈ ∆F , τ > 0 and N(m1, m2) = max        p(m1, m2), p(m1, T (m1))p(m2, T (m2)) 1 + p(m1, m2) , p(m1, T (m1))p(m2, T (m2)) 1 + p(T (m1), T (m2))        . c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 280 Improved F -contraction on partial metric spaces Definition 3.2. Let (M, p) be a partial metric space. The mappings S, T : M → M are called a pair of improved F-contraction of rational type, if for all m1, m2 ∈ M , we have (3.2) (p(S(m1), T (m2)) > 0 implies τ + F(p(S(m1), T (m2))) ≤ F (M(m1, m2))) , for some F ∈ ∆F , τ > 0 and M(m1, m2) = max        p(m1, m2), p(m1, S(m1))p(m2, T (m2)) 1 + p(m1, m2) , p(m1, S(m1))p(m2, T (m2)) 1 + p(S(m1), T (m2))        . The following theorem is one of our main results. Theorem 3.3. Let (M, p) be a complete partial metric space and S, T : M → M be a pair of mappings such that (1) S or T is a continuous mapping, (2) (S, T ) is a pair of improved F-contraction of rational type. Then there exists a common fixed point υ of the pair (S, T ) in M such that p(υ, υ) = 0. Proof. We begin with the following observation: M(m1, m2) = 0 if and only if m1 = m2 is a common fixed point of (S, T ). Indeed, if m1 = m2 is a common fixed point of (S, T ), then T (m2) = T (m1) = m1 = m2 = S(m2) = S(m1) and M(m1, m2) = max        p(m1, m1), p(m1, S(m1))p(m2, T (m2)) 1 + p(m1, m2) , p(m1, S(m1))p(m2, T (m2)) 1 + p(S(m1), T (m2))        , = p(m1, m1). From contractive condition (3.2), we get τ + F(p(m1, m1)) = τ + F(p(S(m1), T (m2))) ≤ F (p(m1, m1)) . This is only possible if p(m1, m1) = 0, which entails M(m1, m1) = 0. Con- versely, if M(m1, m2) = 0, it is easy to check that m1 = m2 is a fixed point of S and T . In order to find common fixed points of S and T for the situation when M(r1, r2) > 0 for all r1, r2 ∈ M with r1 6= r2, we construct an iterative sequence {rn} of points in M such a way that, r2i+1 = S(r2i) and r2i+2 = T (r2i+1) where i = 0, 1, 2, . . . . Assume that p(S(r2i), T (r2i+1)) > 0, then from contractive con- dition (3.2), we get F (p(r2i+1, r2i+2)) = F (p(S(r2i), T (r2i+1))) ≤ F (M(r2i, r2i+1)) − τ, c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 281 M. Nazam, M. Arshad and M. Abbas for all i ∈ N ∪ {0}, where M(r2i, r2i+1) = max        p(r2i, r2i+1), p(r2i, S(r2i))p(r2i+1, T (r2i+1)) 1 + p(r2i, r2i+1) , p(r2i, S(r2i))p(r2i+1, T (r2i+1)) 1 + p(S(r2i), T (r2i+1))        = max        p(r2i, r2i+1), p(r2i, r2i+1)p(r2i+1, r2i+2) 1 + p(r2i, r2i+1) , p(r2i, r2i+1)p(r2i+1, r2i+2) 1 + p(r2i+1, r2i+2)        ≤ max {p(r2i, r2i+1), p(r2i+1, r2i+2)} . For if M(r2i, r2i+1) ≤ p(r2i+1, r2i+2), then F (p(r2i+1, r2i+2)) ≤ F (p(r2i+1, r2i+2)) − τ, which is a contradiction due to F1. Therefore, F (p(r2i+1, r2i+2)) ≤ F (p(r2i, r2i+1)) − τ, for all i ∈ N ∪ {0}. Hence, (3.3) F (p(rn+1, rn+2)) ≤ F (p(rn, rn+1)) − τ, for all n ∈ N ∪ {0}. Following (3.3), we obtain F (p(rn, rn+1)) ≤ F (p(rn−2, rn−1)) − 2τ. Repeating these steps we get, (3.4) F (p(rn, rn+1)) ≤ F (p(r0, r1)) − nτ. From (3.4), we obtain limn→∞ F (p(rn, rn+1)) = −∞. Since F ∈ ∆F , (3.5) lim n→∞ p(rn, rn+1) = 0. From the property (F3) of F-contraction, there exists κ ∈ (0, 1) such that (3.6) lim n→∞ ((p(rn, rn+1)) κ F (p(rn, rn+1))) = 0. Following (3.4), for all n ∈ N, we obtain (3.7) (p(rn, rn+1)) κ (F (p(rn, rn+1)) − F (p(r0, x1))) ≤ − (p(rn, rn+1)) κ nτ ≤ 0. Considering (3.5), (3.6) and letting n → ∞ in (3.7), we have (3.8) lim n→∞ (n (p(rn, rn+1)) κ ) = 0. Since (3.8) holds, there exists n1 ∈ N, such that n (p(rn, rn+1)) κ ≤ 1 for all n ≥ n1 or, (3.9) p(rn, rn+1) ≤ 1 n 1 κ for all n ≥ n1. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 282 Improved F -contraction on partial metric spaces Using (3.9), we get for m > n ≥ n1, p(rn, rm) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) − m−1 ∑ j=n+1 p(rj, rj) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) = m−1 ∑ i=n p(ri, ri+1) ≤ ∞ ∑ i=n p(ri, ri+1) ≤ ∞ ∑ i=n 1 i 1 k . The convergence of the series ∑ ∞ i=n 1 i 1 κ entails limn,m→∞ p(rn, rm) = 0. Hence {rn} is a Cauchy sequence in (M, p). Due to Lemma 2.7, {rn} is a Cauchy sequence in (M, dp). Since (M, p) is a complete partial metric space, so (M, dp) is a complete metric space and as a result there exists υ ∈ M such that limn→∞ dp(rn, υ) = 0, moreover, by Lemma 2.7 (3.10) lim n→∞ p(υ, rn) = p(υ, υ) = lim n,m→∞ p(rn, rm). Since limn,m→∞ p(rn, rm) = 0, from (3.10) we deduce that (3.11) p(υ, υ) = 0 = lim n→∞ p(υ, rn). Now from (3.11) it follows that r2n+1 → υ and r2n+2 → υ as n → ∞ with respect to τ(p). The continuity of T implies υ = lim n→∞ rn = lim n→∞ r2n+1 = lim n→∞ r2n+2 = lim n→∞ T (r2n+1) = T ( lim n→∞ r2n+1) = T (υ), and from contractive (3.2), we have τ + F(p(υ, S(υ))) = τ + F(p(S(υ), T (υ))) ≤ F(M(υ, υ)) = F(p(υ, υ)). This implies that p(υ, S(υ)) = 0 and due to (p1), (p2) we conclude that υ = S(υ). Thus we have S(υ) = T (υ) = υ. Hence (S, T ) has a common fixed point υ. Now we show that υ is the unique common fixed point of S and T . Assume the contrary, that is, there exists ω ∈ M such that υ 6= ω and ω = T (ω). From the contractive condition (3.2), we have (3.12) τ + F(p(S(υ), T (ω))) ≤ F (M(υ, ω)) , c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 283 M. Nazam, M. Arshad and M. Abbas where M(υ, ω) = max        p(υ, ω), p(υ, S(υ))p(ω, T (ω)) 1 + p(υ, y) , p(υ, S(υ))p(ω, T (ω)) 1 + p(S(υ), T (ω))        . From (3.12), we have (3.13) τ + F(p(υ, ω)) ≤ F (p(υ, ω)) , The inequality (3.13), leads to p(υ, ω) < p(υ, ω), which is a contradiction. Hence, υ = ω and υ is a unique common fixed point of a pair (S, T ). � The following example illustrates Theorem 3.3 and shows that condition (3.2) is more general than contractivity condition given by Wardowski ([13]). Example 3.4. Let M = [0, 1] and define p(r1, r2) = max {r1, r2}, then (M, p) is a complete partial metric space. Moreover, define d (r1, r2) = |r1 − r2|, so, (M, d) is a complete metric space. Define the mappings S, T : M → M as follows: T (r) =      r 5 if r ∈ [0, 1); 0 if r = 1 and S(r) = 3r 7 for all r ∈ M Clearly, S, T are self mappings. Define the function F : R+ → R by F(r) = ln(r), for all r ∈ R+ > 0. Let r1, r2 ∈ M such that p(S(r1), T (r2)) > 0 and suppose that r1 ≤ r2. Then M(r1, r2) = max { r2, r1r2 1 + r1 , r1r2 1 + max { 3r1 7 , r2 5 } } . Since r1 1+r1 < 1 and r1 1+max{ 3r17 , r2 5 } < 1, we have that M(r1, r2) = r2. In a similar way, if r1 ≥ r2, we obtain that M(r1, r2) = r1, i.e., M(r1, r2) = p(r1, r2). Let τ ≤ ln(7 3 ). Then τ + (p(S(r1), T (r2))) = τ + ln ( max { 3r1 7 , r2 5 }) ≤ ln( 7 3 ) + ln ( max { 3p(r1, r2) 7 , p(r1, r2) 5 }) = ln( 7 3 ) + ln ( 3p(r1, r2) 7 ) = ln (p(r1, r2)) = F (M(r1, r2)) . Thus, the contractive condition (3.2) is satisfied for all r1, r2 ∈ M. Hence, all the hypotheses of the Theorem 3.3 are satisfied, note that (S, T ) have a unique c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 284 Improved F -contraction on partial metric spaces common fixed point r = 0. As we have seen in Example 2.8, T is not a F- contraction in (M, d) and consequently we can not apply Theorem 2.2. Corollary 3.5. Let (M, p) be a complete partial metric space and T : M → M be a mapping such that (1) T is a continuous mapping, (2) T is an improved F-contraction of rational type. Then T has a unique fixed point υ in M such that p(υ, υ) = 0. Proof. Setting S = T in Theorem 3.3, we obtain required result. � Remark 3.6. If we set N(r1, r2) = max {p(r1, r2), p(r1, T (r1)), p(r2, T (r2))} in inequality (3.1), Corollary 3.5 remains true. Similarly, by setting M(r1, r2) = max {p(r1, r2), p(r1, T (r1)), p(r2, S(r2))} in inequality (3.2), Theorem 3.3 remains true. 4. Application of Theorem 3.3 In this part of paper, we shall apply Theorem 3.3 to show the existence of common solution of the system of Volterra type integral equations. Such system is given by the following equations. u(t) = f(t) + t ∫ 0 Kn(t, s, u(s))ds,(4.1) w(t) = f(t) + t ∫ 0 Jn(t, s, w(s))ds,(4.2) for all t ∈ [0, a], and a > 0. We shall show, by using Theorem 3.3, that the solution of integral equations (4.1) and (4.2) exists. Let C([0, a], R) be the space of all continuous functions defined on [0, a]. For u ∈ C([0, a], R), define sup norm as: ‖u‖τ = sup t∈[0,a] {u(t)e−τt}, where τ > 0. Let C([0, a], R) be endowed with the partial metric (4.3) pτ(u, v) = dτ (u, v) + cn = sup t∈[0,a] ‖ |u(t) − v(t)| e−τt‖τ + cn for all u, v ∈ C([0, a], R) and {cn} is a sequence of positive real numbers such that limn→∞ cn = 0. Obviously, C([0, a], R, ‖ · ‖τ) is a Banach space. Now we prove the following theorem to ensure the existence of solution of system of integral equations. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 2 285 M. Nazam, M. Arshad and M. Abbas Theorem 4.1. Assume the following conditions are satisfied. (i) Kn, Jn : [0, a] × [0, a] × R → R and f, g : [0, a] → R are continuous. (ii) Define Su(t) = f(t) + t ∫ 0 Kn(t, s, u(s))ds, T u(t) = f(t) + t ∫ 0 Jn(t, s, u(s))ds, and when n → ∞ there exists τ ≥ 1 such that |Kn(t, s, u) − Jn(t, s, v)| ≤ τe −τ [M(u, v)] for all t, s ∈ [0, a] and u, v ∈ C([0, a], R), where M(u, v) = max        p(u(t), v(t)), p(u(t), Su(t))p(v(t), T v(t)) 1 + p(u(t), v(t)) , p(u(t), Su(t))p(v(t), T v(t)) 1 + p(Su(t), T v(t))        . Then the system of integral equations given in (4.1) and (4.2) has a solution. Proof. Following assumption (ii), we have p(Su(t), T v(t)) = d(Su(t), T v(t)) + cn = t ∫ 0 |Kn(t, s, u(s) − Jn(t, s, v(s)))| ds + cn ≤ t ∫ 0 τe −τ ([M(u, v)]e−τs)eτsds (by taking limit n → ∞) ≤ t ∫ 0 τe−τ ‖M(u, v)‖τe τsds ≤ τe−τ ‖M(u, v)‖τ t ∫ 0 e τs ds ≤ τe−2τ ‖M(u, v)‖τ 1 τ e τt ≤ e−τ‖M(u, v)‖τe τt . This implies p(Su(t), T v(t))e−τt ≤ e−τ‖M(u, v)‖τ, That is ‖p(Su(t), T v(t))‖ τ ≤ e−τ‖M(u, v)‖τ, c© AGT, UPV, 2017 Appl. Gen. 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