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Metric or partial metric spaces endowed with a finite number of graphs: A tool to obtain fixed point results
Topology and its Applications 164 (2014) 125–137
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Topology and its Applications www.elsevier.com/locate/topol
Metric or partial metric spaces endowed with a finite number of graphs: A tool to obtain fixed point results ✩ Calogero Vetro a,∗ , Francesca Vetro b a Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy b Università degli Studi di Palermo, Dipartimento Energia, Ingegneria dell’Informazione e Modelli Matematici (DEIM), Viale delle Scienze, 90128 Palermo, Italy
a r t i c l e
i n f o
Article history: Received 16 July 2013 Received in revised form 12 December 2013 Accepted 14 December 2013 MSC: 54H25 47H10
a b s t r a c t We give some fixed point theorems in the setting of metric spaces or partial metric spaces endowed with a finite number of graphs. The presented results extend and improve several well-known results in the literature. In particular, we discuss a Caristi type fixed point theorem in the setting of partial metric spaces, which has a close relation to Ekeland’s principle. © 2013 Elsevier B.V. All rights reserved.
Keywords: Caristi’s fixed point theorem Ekeland’s principle Graph Metric space Partial metric space
1. Introduction The notion of fixed point is of great interest in mathematics as well as in many fields of applied sciences. Formally, this notion can be introduced as follows: Definition 1. Let X be a nonempty set and T : X → X be a self-mapping. A point x ∈ X is said to be fixed under T if T x = x. Moreover, the set of all these points is denoted by Fix(T ). Metric fixed point theory is the branch of mathematical analysis which focuses on the existence and uniqueness of fixed points under metric conditions on both the domain of the mapping and the mapping itself. We give now the fundamental result of this theory. ✩
The authors are supported by Università degli Studi di Palermo, R. S. ex 60%.
* Corresponding author. E-mail addresses: [email protected] (C. Vetro), [email protected] (F. Vetro). 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.12.008
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Theorem 1 (Banach’s contraction principle). Let (X, d) be a complete metric space and T : X → X be such that d(T x, T y) kd(x, y), for all x, y ∈ X, and some k ∈ (0, 1). Then Fix(T ) is singleton, that is, there exists a unique x ∈ X such that T x = x. In the last decades, this principle was generalized and extended by many authors in several directions (see, for example, [13,16,22–24,27,28]). The reason of our study is to establish fixed point theorems for self-mappings defined on a metric space or a partial metric space endowed with a finite number of graphs. We are motivated by the following two observations: Firstly, we recall that in the mathematical field of domain theory, attempts were made in order to equip semantic domain with a notion of distance. In particular, Matthews [12] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that Banach’s contraction principle can be generalized to the partial metric context for applications in program verification (see also [5,6,17,26]). Secondly, graph theory provides useful tools for many disciplines in which there is a connectedness of elements or components that seem to be related in a system-type. Such a case occurs in the constructive proof of many fixed point theorems, which leads to an algorithm for finding the fixed point. In particular, Espinola and Kirk [8] provided useful results by combining fixed point theory and graph theory. Also, Jachymski [9] continued this idea from a different perspective and his work was followed by several authors [1,3]. After Introduction and Preliminaries we discuss fixed point results in metric spaces in Section 3. Our results generalize and improve many existing fixed point theorems in the literature including Theorem 2.2 of [21]; also an illustrative example is given. In Section 4, we deduce several fixed point theorems in partial metric spaces, by using the results in Section 3. In particular, we state a Caristi type fixed point theorem in partial metric spaces and show how it can be derived from the corresponding result in metric spaces. We recall that Caristi’s fixed point theorem [4] is a variation of Ekeland’s principle, which provides a characterization of complete metric spaces [7]. Finally, we give some fixed point theorems for cyclic mappings by using the results in the previous sections. 2. Preliminaries In this section, we introduce some concepts which are useful for the correct understanding of the paper. 2.1. Graph theory Definition 2. A graph G is an ordered pair (V, E), where V is a set and E ⊆ V × V is a binary relation. We say that V is the vertex set and E is the edge set. For a more detailed background on this topic, we refer the reader to [29]. Definition 3. Let G = (V, E) be a graph and D be a subset of V . We say that D is G-directed if for every x, y ∈ D, there exists z ∈ V such that (x, z), (y, z) ∈ E. Example 1. Let V = F([0, 1], R) be the set of functions f : [0, 1] → R, and define E ⊆ V × V by: (u, v) ∈ E
⇐⇒
u(t) v(t),
for all t ∈ [0, 1].
Then G = (V, E) is a graph. Let D = M ([0, 1], R) be the set of measurable functions f : [0, 1] → R. Then D is G-directed. Indeed, for every u, v ∈ D, the function z = max{u, v} satisfies (u, z), (v, z) ∈ E.
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Let (V, d) be a metric space. We consider a family G = {Gi : 1 i q} of q 1 graphs such that Gi = (V, Ei ), Ei ⊆ V × V , i = 1, 2, . . . , q. Definition 4. Let T : V → V be a given mapping. We say that T is G-monotone if for all i = 1, 2, . . . , q, we have that (x, y) ∈ Ei implies (T x, T y) ∈ Ei+1 , with Eq+1 = E1 . Consequently, (T kq x, T kq y) ∈ Ei for each nonnegative integer number k if T is G-monotone. Remark 1. If q = 1 (G = G1 ), we say that T preserves edges of G (see [9]). Example 2. Consider the sets V = [0, 1] ∪ [2, 3], E1 = [0, 1] × [2, 3] and E2 = [2, 3] × [0, 1]. Let G = {Gi : i = 1, 2} be the family of graphs Gi = (V, Ei ), i = 1, 2. Define the mapping T : V → V by: T x = x + 2 if x ∈ [0, 1], T x = x − 2 if x ∈ [2, 3]. If (x, y) ∈ E1 , we have (T x, T y) ∈ E2 . If (x, y) ∈ E2 , we have (T x, T y) ∈ E1 . Then T is G-monotone. Definition 5. We say that the pair (G, d) is regular if the following condition holds: if {xn } is a sequence in V and x is a point in V such that ∀ i ∈ {1, 2, . . . , q}, there exists a subsequence {xmi,k } of {xn } with (xmi,k , xmi,k +1 ) ∈ Ei , ∀k; d(xn , x) → 0
as n → ∞,
(1) (2)
then there exist a subsequence {xnk } of {xn } and a rank j ∈ {1, 2, . . . , q} such that (xnk , x) ∈ Ej for all k. Example 3. Let V = B([0, 1], R) be the set of bounded functions f : [0, 1] → R. Define E ⊆ V × V by: (u, v) ∈ E
⇐⇒
u(t) v(t),
for all t ∈ [0, 1].
Consider the graph G = (V, E). We endow V with the metric d given by: d(u, v) = sup u(t) − v(t), 0t1
for all u, v ∈ V.
Let {xn } be a sequence in V and x be a point in V satisfying conditions (1) and (2), that is, (i) there exists a subsequence {xmk } of {xn } such that for all k ∈ N and t ∈ [0, 1]: xmk (t) xmk +1 (t), (ii) d(xn , x) → 0 as n → ∞. Then, (xmk , x) ∈ E for all k ∈ N; hence, the pair (G, d) is regular. 2.2. Partial metric spaces Here, we recall some definitions and some properties of partial metric spaces that can be found in [12, 14,15,19,25]. A partial metric on a nonempty set X is a function p : X × X → [0, ∞) such that for all x, y, z ∈ X: (p1 ) (p2 ) (p3 ) (p4 )
x = y ⇔ p(x, x) = p(x, y) = p(y, y), p(x, x) p(x, y), p(x, y) = p(y, x), p(x, y) p(x, z) + p(z, y) − p(z, z).
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A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X. It is clear that, if p(x, y) = 0, then from (p1 ) and (p2 ) it follows that x = y. But if x = y, p(x, y) may not be 0. A basic example of a partial metric space is the pair ([0, ∞), p), where p(x, y) = max{x, y} for all x, y ∈ [0, ∞). Other examples of partial metric spaces which are interesting from a computational point of view can be found in [12]. Each partial metric p on X generates a T0 topology τp on X which has as a base the family of open p-balls {Bp (x, ε): x ∈ X, ε > 0}, where Bp (x, ε) = y ∈ X: p(x, y) < p(x, x) + ε for all x ∈ X and ε > 0. If p is a partial metric on X, then the function ps : X × X → [0, ∞) given by ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) is a metric on X. Let (X, p) be a partial metric space. A sequence {xn } in (X, p) converges to a point x ∈ X if and only if p(x, x) = limn→∞ p(x, xn ). A sequence {xn } in (X, p) is called a Cauchy sequence if there exists (and is finite) limn,m→∞ p(xn , xm ). A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn } in X converges, with respect to τp , to a point x ∈ X such that p(x, x) = limn,m→∞ p(xn , xm ). It is easy to see that, every closed subset of a complete partial metric space is complete. Lemma 1. [12,15] Let (X, p) be a partial metric space. Then (a) {xn } is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metric space (X, ps ). (b) A partial metric space (X, p) is complete if and only if the metric space (X, ps ) is complete. Furthermore, limn→∞ ps (xn , x) = 0 if and only if p(x, x) = lim p(xn , x) = n→∞
lim p(xn , xm ).
n,m→∞
3. Fixed point results in metric spaces In this section, we also consider the family G = {Gi : 1 i q} of q 1 graphs such that Gi = (V, Ei ), Ei ⊆ V × V , i = 1, 2, . . . , q. Let Ψ be the family of nondecreasing functions ψ : [0, ∞) → [0, ∞) satisfying: lim ψ n (t) = 0,
n→∞
for all t > 0.
The following auxiliary fact is immediate; so, we omit its proof. Lemma 2. If ψ ∈ Ψ , then ψ(t) < t for all t > 0. Thus, we state and prove our main result in metric spaces. Theorem 2. Let (V, d) be a complete metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold: (a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, d) is regular;
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(c) there exist ψ ∈ Ψ and a lower semi-continuous function φ : V → [0, ∞) such that d(T x, T y) + φ(T x) + φ(T y) ψ d(x, y) + φ(x) + φ(y) ,
129
(3)
for all (x, y) ∈ Ei , i = 1, 2, . . . , q; (d) there exist i ∈ {1, 2, . . . , q}, and Ai , Bi ⊆ V such that Ei = Ai × Bi . Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Proof. From (a), there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 . Next, define the sequence {xn } in V by: xn+1 = T xn ,
n = 0, 1, 2, . . . .
If xn+1 = xn for some n, then xn is a fixed point of T and the existence of a fixed point is proved. Now, suppose that xn+1 = xn ,
n = 0, 1, 2, . . . .
(4)
Since T is G-monotone, for all n 0, there exists i = i(n) ∈ {1, 2, . . . , q} such that (xn , xn+1 ) ∈ Ei . For all n 1, applying the inequality (3) with x = xn−1 and y = xn , we obtain d(T xn−1 , T xn ) + φ(T xn−1 ) + φ(T xn ) ψ d(xn−1 , xn ) + φ(xn−1 ) + φ(xn ) . (5) Using (5), we get d(xn , xn+1 ) + φ(xn ) + φ(xn+1 ) ψ n d(x0 , x1 ) + φ(x0 ) + φ(x1 ) .
(6)
Since ψ ∈ Ψ and letting n → ∞, we get that d(xn , xn+1 ) + φ(xn ) + φ(xn+1 ) → 0. This implies that d(xn , xn+1 ) → 0 and φ(xn ) → 0 as n → ∞. Fix ε > 0 and let n(ε) be a positive integer such that d(xm , xm+1 ) + φ(xm ) + φ(xm+1 ) <
ε − ψ(ε) q
(7)
for all m n(ε). Now, let i be as in condition (d) and we choose m n(ε) such that (xm , xm+1 ) ∈ Ei . We prove that d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) < ε
(8)
for all n m such that n − m = kq with k a nonnegative integer number. If k = 0, then (8) holds by (7). Now, suppose that (8) holds for some n such that n − m = kq with k > 0. Since T is G-monotone, we deduce easily that (xn , xn+1 ) = (xm+kq , xm+kq+1 ) ∈ Ei . Consequently, from (d), we have (xm , xn+1 ) ∈ Ei . Next, using (3) and (7), we have d(xm , xn+q+1 ) + φ(xm ) + φ(xn+q+1 ) d(xm , xm+1 ) + d(xm+1 , xn+2 ) +
q
d(xn+k , xn+k+1 ) + φ(xm ) + φ(xn+q+1 )
k=2
ε − ψ(ε) + ψ d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) q
ε − ψ(ε) + ψ(ε) = ε. Thus (8) holds for all n m such that n − m = kq with k 0 and (xm , xm+1 ) ∈ Ei .
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Clearly, (8) holds for all m n(ε) and n m with n − m q. If n − m > q and (xm , xm+1 ) ∈ Ej with j = i, we choose h ∈ {1, . . . , q} and a nonnegative integer k such that (xm+h , xm+h+1 ) ∈ Ei and s = m + h + kq n < s + q. Then for all n m, we have d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) d(xm , xm+h ) + d(xm+h , xs+1 ) + d(xs+1 , xn+1 ) + φ(xm ) + φ(xn+1 ). From the above, we get easily d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) < 3ε for all m n(ε) and n m and hence {xn } is a Cauchy sequence. Now, since (V, d) is complete, there exists u ∈ V such that xn → u. Clearly the sequence {xn } satisfies conditions (1) and (2) with x = u. Since (G, d) is regular, there exist a subsequence {xnk } of {xn } and j ∈ {1, 2, . . . , q} such that (xnk , u) ∈ Ej for all k. Since φ is lower semi-continuous, we get φ(u) lim inf φ(xn ) = 0, n→∞
that is, φ(u) = 0. Applying (3) with x = xnk and y = u, we obtain d(xnk+1 , T u) d(T xnk , T u) + φ(xnk +1 ) + φ(T u) ψ d(xnk , u) + φ(xnk ) + φ(u) < d(xnk , u) + φ(xnk ) + φ(u)
(9)
for all k. Letting k → ∞, we get d(u, T u) = 0, hence u ∈ V is a fixed point of T . Now, suppose that there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed. We will prove that u is the unique fixed point of T . Suppose that y ∈ V is another fixed point of T . Then, there is z ∈ V such that (u, z), (y, z) ∈ Ej . Define the sequence {zn } in V by z0 = z and zn+1 = T zn for all n 0. Since T is G-monotone, for all n 0, there exists i = i(n) ∈ {1, 2, . . . , q} such that (u, zn ) ∈ Ei . Applying (3), for all n 0, we have d(u, zn+1 ) d(T u, T zn ) + φ(T u) + φ(T zn ) ψ d(u, zn ) + φ(u) + φ(zn ) ψ n d(u, z0 ) + φ(u) + φ(z0 ) .
(10)
Letting n → ∞ in the above inequality yields d(u, zn ) → 0. Similarly, we can prove that d(y, zn ) → 0 and hence y = u. 2 Remark 2. One can prove Theorem 2 without assuming hypothesis (d), provided that the function ψ ∈ Ψ ∞ satisfies the condition n=1 ψ n (t) < ∞ for all t > 0. Taking ψ(t) = kt with k ∈ (0, 1) in Theorem 2, by Remark 2, we obtain the following corollary which is a generalization of Theorem 2.2 of [21]. Corollary 1. Let (V, d) be a complete metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold:
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(a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, d) is regular; (c) there exist k ∈ (0, 1) and a lower semi-continuous function φ : V → [0, ∞) such that
d(T x, T y) + φ(T x) + φ(T y) k d(x, y) + φ(x) + φ(y) ,
(11)
for all (x, y) ∈ Ei , i = 1, 2, . . . , q. Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Remark 3. Taking V = X, q = 1 and E1 = X × X in Corollary 1, we obtain the following generalization of Banach’s contraction principle. Corollary 2. Let (X, d) be a complete metric space and let T : X → X be a mapping. Suppose that there exist k ∈ (0, 1) and a lower semi-continuous function φ : X → [0, ∞) such that
d(T x, T y) + φ(T x) + φ(T y) k d(x, y) + φ(x) + φ(y) ,
(12)
for all x, y ∈ X. Then T has a unique fixed point. Remark 4. From Corollary 2, we deduce Banach’s contraction principle, whenever we assume φ(x) = 0 for every x ∈ X. To support our results, we give an illustrative example. Precisely, we show that our Corollary 2 can be used to cover this example while Banach’s contraction principle cannot be applied. Example 4. Let X = [0, 1] endowed with the usual metric d(x, y) = |x − y| for all x, y ∈ X. Obviously (X, d) is a complete metric space. Also, fix r ∈ [0, 1), and define T : X → X by ⎧ 0 if x = 0, ⎪ ⎨ r 2n−1 1 1 T x = 2n − r 2n (2nx − 1) if 2n x 2n−1 , ⎪ ⎩ r + r 2n+1 (2nx − 1) if 1 x 1 . 2n 2n 2n+1 2n Firstly, we prove that T is not a contraction. In fact, if for odd n > 1 we choose x = we have d(T x, T y) =
1 2n−1
and y =
1 n−1 ,
n 3 r d(x, y) = , n−1 (n − 1)(2n − 1) 5(n − 1)
which is not satisfied for r > 3/5. Therefore, Banach’s contraction principle cannot be applied. On the other hand, if we consider the function φ : X → [0, ∞) defined by φ(x) = x, then we obtain d(T x, T y) + φ(T x) + φ(T y) = 2 max{T x, T y} 2 max{rx, ry} = r2 max{x, y} = r d(x, y) + φ(x) + φ(y) , for all x, y ∈ X. Thus, all the conditions of Corollary 2 are satisfied. Therefore, T has a unique fixed point in X.
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Another interesting result is given in the following theorem. Theorem 3. Let (V, d) be a complete metric space endowed with a graph G = (V, E). Let T : V → V be a mapping such that the following conditions hold: (a) (x, T x) ∈ E for all x ∈ V ; (b) there exists k ∈ (0, 1) such that d(T x, T y) kd(x, y), for all (x, y) ∈ E; (c) if (x, y) ∈ / E, then d(x, T x) > (1 + k)d(x, y), for all (x, y) ∈ X × X. Then T has a unique fixed point. Proof. From (a) and (b) we deduce that d T x, T 2 x kd(x, T x),
(13)
for all x ∈ X. Let x0 ∈ X and consider the sequence {xn } defined by xn = T xn−1 for all n ∈ N. If xn+1 = xn for some n, then xn is a fixed point of T and the existence of a fixed point is proved. Now, suppose that xn+1 = xn ,
for all n ∈ N.
(14)
From (a), it follows that (xn , xn+1 ) ∈ E for all n ∈ N. Now, from (b) with x = xn−1 and y = xn , we get d(xn , xn+1 ) kd(xn−1 , xn ) and hence d(xn , xn+1 ) kn d(x0 , x1 ),
for all n ∈ N.
(15)
This ensures that {xn } is a Cauchy sequence. Since V is a complete metric space there exists x ∈ V such that xn → x. Next, we prove that for all (u, v) ∈ X × X, we have either (u, v) ∈ E or (T u, v) ∈ E. Assume that (u, v) ∈ / E and (T u, v) ∈ / E, then by (c), we have d(u, T u) > (1 + k)d(u, v)
and d T u, T 2 u > (1 + k)d(T u, v).
(16)
By (13) and (16), we get d(u, T u) d(u, v) + d(T u, v) 1 d(u, T u) + d T u, T 2 u < 1+k 1 d(u, T u) + kd(u, T u) 1+k d(u, T u) a contradiction and so (u, v) ∈ E or (T u, v) ∈ E. This implies that either (x2n , x) ∈ E or (x2n+1 , x) ∈ E. Then, from (b) we obtain either d(x2n+1 , T x) kd(x2n , x) or d(x2n+2 , T x) kd(x2n+1 , x). Thus, there exists a subsequence {xnk } of {xn } such that d(xnj +1 , T x) kd(xnj , x),
for all j ∈ N.
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Letting j → ∞ in the previous inequality, we obtain xnj +1 → T x and since xnj +1 → x, we get x = T x. We will prove that x is the unique fixed point of T . Suppose that y ∈ V is another fixed point of T . If (x, y) ∈ E, then from (b), we deduce d(x, y) = d(T x, T y) kd(x, y), / E, then (T x, y) ∈ E and again from (b), we obtain which implies x = y. If (x, y) ∈ d(x, y) = d T 2 x, T y kd(T x, y) = kd(x, y), which implies x = y. This completes the proof. 2 4. Fixed point in partial metric spaces In this section, we deduce easily several fixed point theorems in partial metric spaces, by using the results in Section 3. We recall the following definition. Definition 6. Let (V, p) be a partial metric space and let φ : V → [0, ∞) be a function on V . Then, the function φ is called p-lower semi-continuous on V whenever lim p(xn , x) = p(x, x)
n→∞
=⇒
φ(x) lim inf φ(xn ) = sup inf φ(xm ). n→∞
n1 mn
Clearly, the above definition reduces to the classical definition of lower semi-continuity, whenever we consider a classical metric instead of a partial metric. In particular, referring to the metric ps , we observe that every p-lower semi-continuous function on (V, p) is a ps -lower semi-continuous function. The following auxiliary fact is useful in the proof of the next theorem. Lemma 3. Let (V, p) be a complete partial metric space and φ : V → [0, ∞) be the function defined by φ(x) = p(x, x), for every x ∈ V . Then, φ is a ps -lower semi-continuous function. Proof. Let {xn } be a sequence in V such that limn→∞ ps (xn , x) = 0, where x ∈ V . Then by Lemma 1, we get lim p(xn , xn ) = p(x, x),
n→∞
that is, lim φ(xn ) = φ(x).
n→∞
Thus, φ is continuous and therefore lower semi-continuous on (V, ps ), that is, ps -lower semi-continuous. 2 Finally, we state and prove our result. Theorem 4. Let (V, p) be a complete partial metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold: (a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, p) is regular;
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(c) there exists ψ ∈ Ψ such that p(T x, T y) ψ(p(x, y)), for all (x, y) ∈ Ei , i = 1, 2, . . . , q; (d) there exist i ∈ {1, 2, . . . , q} and Ai , Bi ⊆ V such that Ei = Ai × Bi . Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Proof. Note that since (V, p) is complete, from Lemma 1, (V, ps ) is a complete metric space. Now, for all x, y ∈ V , we have p(x, y) =
ps (x, y) + p(x, x) + p(y, y) 2
and hence the result follows by Theorem 2 if we choose d(x, y) = x, y ∈ V . 2
1 s 2 p (x, y)
and φ(x) =
1 2 p(x, x)
for all
From Corollary 1, we obtain the following corollary which is a generalization of Matthews fixed point theorem. Corollary 3. Let (V, p) be a complete partial metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold: (a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, p) is regular; (c) there exists k ∈ (0, 1) such that p(T x, T y) kd(x, y), for all (x, y) ∈ Ei , i = 1, 2, . . . , q. Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Remark 5. Taking V = X, q = 1 and E1 = X × X in Corollary 3, we recover the Matthews fixed point theorem [12]. Recently, several authors worked on Caristi’s fixed point theorem in the setting of partial metric spaces [2,10,19], also using the well-known Ekeland’s variational principle [7]. Following this direction of research, we present some results which can be easily derived from the corresponding results in metric spaces. Theorem 5. Let (V, p) be a complete partial metric space and let F : V → R be a ps -lower semi-continuous function bounded from below. Let ε > 0 be given, and u ∈ V be such that F (u) inf x∈V F (x) + ε. Then there exists some point v ∈ V such that F (v) F (u), ps (u, v) 1 and F (w) > F (v) − εps (v, w), for every w ∈ V with w = v. Proof. Note that since (V, p) is complete, from Lemma 1, (V, ps ) is a complete metric space. Thus the thesis follows from Theorem 1 of [7]. 2 Since every p-lower semi-continuous function on (V, p) is a ps -lower semi-continuous function, we get the following corollary. Corollary 4. Let (V, p) be a complete partial metric space and let F : V → R be a p-lower semi-continuous function bounded from below. Let ε > 0 be given, and u ∈ V be such that F (u) inf x∈V F (x) + ε. Then there exists some point v ∈ V such that F (v) F (u), ps (u, v) 1 and F (w) > F (v) − εps (v, w), for every w ∈ V with w = v.
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Now, we deduce Caristi type fixed point theorem in the setting of partial metric spaces as consequence of the corresponding result in metric spaces. Theorem 6. Let (V, p) be a complete partial metric space and let T : V → V satisfying p(u, T u) φ(u) − φ(T u),
for all u ∈ V,
(17)
where φ : V → R is a ps -lower semi-continuous function. Then T has a fixed point. Proof. Since (V, p) is a complete partial metric space, then also (V, ps ) is a complete metric space. From ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) we deduce p(x, y) =
1 s p (x, y) + p(x, x) + p(y, y) 2
and so
1 1 s p (u, T u) ps (u, T u) + p(u, u) + p(T u, T u) φ(u) − φ(T u), 2 2 that is ps (u, T u) 2φ(u) − 2φ(T u). Thus, the result follows from Caristi’s fixed point theorem in complete metric spaces.
2
Remark 6. Theorem 6 holds also if φ is a p-lower semi-continuous function. 5. Cyclic results In this section, we deduce some fixed point results for cyclic mappings on metric spaces, by using the results in Section 3. In 2003, Kirk et al. [11] introduced the following definition. Definition 7. [11] Let X be a nonempty set, q be a positive integer and T : X → X be a mapping. q X = i=1 Ai is said to be a cyclic representation of X with respect to T if (i) Ai , i = 1, 2, . . . , q, are nonempty closed sets; (ii) T (A1 ) ⊂ A2 , . . . , T (Aq−1 ) ⊂ Aq , T (Aq ) ⊂ A1 . Recently, fixed point theorems involving a cyclic representation of X with respect to a self-mapping T appeared in many articles (see, for example, [18,20]). Following this direction of research, we introduce some definitions. Definition 8. Let (X, d) be a metric space, q be a positive integer, A1 , A2 , . . . , Aq be nonempty closed subsets q of X and Y = i=1 Ai . An operator T : Y → Y is said to be a cyclic weak (φ, ψ)-contraction if q (i) i=1 Ai is a cyclic representation of Y with respect to T ; (ii) there exist ψ ∈ Ψ and a lower semi-continuous function φ : V → [0, ∞) such that d(T x, T y) + φ(T x) + φ(T y) ψ(d(x, y) + φ(x) + φ(y)), for every x ∈ Ai , y ∈ Ai+1 , i = 1, 2, . . . , q, where Aq+1 = A1 .
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Taking ψ(t) = kt with k ∈ (0, 1) in Definition 8, we obtain the following notion. Definition 9. Let (X, d) be a metric space, q be a positive integer, A1 , A2 , . . . , Aq be nonempty closed subsets q of X and Y = i=1 Ai . An operator T : Y → Y is said to be a cyclic weak φk -contraction if q (i) i=1 Ai is a cyclic representation of Y with respect to T ; (ii) there exist k ∈ (0, 1) and a lower semi-continuous function φ : V → [0, ∞) such that d(T x, T y) + φ(T x) + φ(T y) k[d(x, y) + φ(x) + φ(y)], for every x ∈ Ai , y ∈ Ai+1 , i = 1, 2, . . . , q, where Aq+1 = A1 . Now, we state and prove the following theorem, which is an extension of Kirk et al. cyclic fixed point theorem [11]. Theorem 7. Let (X, d) be a complete metric space, q a positive integer, A1 , . . . , Aq ∈ Pcl (X) (where Pcl (X) q is the collection of nonempty closed subsets of X), Y = i=1 Ai and T : Y → Y be a cyclic weak (φ, ψ)-contraction. Then, T has a unique fixed point in Y . Proof. We take V = Y and consider the family of graphs G = {Gi : 1 i q} of q 1 graphs such that Gi = (V, Ei ), Ei = Ai × Ai+1 , for all i = 1, 2, . . . , q + 1 with Aq+1 = A1 . Since A1 , . . . , Aq ∈ Pcl (X) and (X, d) is complete, then (V, d) is a complete metric space. Let (x, y) ∈ Ei for some i. This implies that x ∈ Ai and y ∈ Ai+1 . From condition (i) of Definition 8, we have T (Ai ) ⊆ Ai+1
and T (Ai+1 ) ⊆ Ai+2 .
Then we get (T x, T y) ∈ Ai+1 × Ai+2 ,
that is,
(T x, T y) ∈ Ei+1 .
Thus we proved that T is G-monotone. Let x0 ∈ A1 (such a point exists since A1 = ∅). Since T (A1 ) ⊆ A2 , we have T x0 ∈ A2 . This implies that (x0 , T x0 ) ∈ E1 . Now, we shall prove that (G, d) is regular. Let {xn } be a sequence in V and x be a point in V satisfying (1) and (2) of Definition 5. For i ∈ {1, . . . , q}, there exists a subsequence {xmi,k } of {xn } such that (xmi,k , xmi,k +1 ) ∈ Ei for all k. This implies that xmi,k ∈ Ai for all k. Since Ai is closed and d(xn , x) → 0 as n → ∞, we have x ∈ Ai ; so, as i ∈ {1, . . . , q} was q arbitrarily chosen, we get that x ∈ i=1 Ai . This yields, by the choice of our subsequences, (xmi,k , x) ∈ Ei for all i ∈ {1, . . . , q} and all k. Thus we proved that (G, d) is regular. Finally, we shall prove that Fix(T ) is G1 -directed. Let x, y ∈ Fix(T ). q From (i) of Definition 8, we have x, y ∈ i=1 Ai , which implies that (x, y) ∈ E1 and (y, y) ∈ E1 , and our claim is proved. Now, from the definition of Ei , it follows that all the hypotheses of Theorem 2 are satisfied, then we deduce that T has a unique fixed point in V . 2 In view of Definition 9, we give the following result. Corollary 5. Let (X, d) be a complete metric space, q a positive integer, A1 , . . . , Aq ∈ Pcl (X), Y = and T : Y → Y be a cyclic weak φk -contraction. Then, T has a unique fixed point in Y .
q i=1
Ai
Finally, reasoning as in the proof of Theorem 4 in Section 4, one can prove the following result in the setting of partial metric spaces. Theorem 8. Let (X, p) be a complete partial metric space, q a positive integer, A1 , . . . , Aq ∈ Pclp (X) (where q Pclp (X) is the collection of nonempty closed subsets of X), Y = i=1 Ai and T : Y → Y an operator. Assume that
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q (i) i=1 Ai is a cyclic representation of Y with respect to T ; (ii) there exists ψ ∈ Ψ such that p(T x, T y) ψ(p(x, y)), for every x ∈ Ai , y ∈ Ai+1 , i = 1, 2, . . . , q, where Aq+1 = A1 . Then, T has a unique fixed point in Y . References [1] S.M.A. Aleomraninejad, Sh. Rezapour, N. Shahzad, Some fixed point results on a metric space with a graph, Topol. Appl. 159 (2012) 659–663. [2] H. Aydi, E. Karapinar, C. Vetro, On Ekeland’s variational principle in partial metric spaces, submitted for publication. [3] I. Beg, A.R. Butt, S. Radojevic, The contraction principle for set valued mappings on a metric space with a graph, Comput. Math. Appl. 60 (2010) 1214–1219. [4] J. Caristi, Fixed point theorems for mapping satisfying inwardness conditions, Trans. Am. Math. Soc. 215 (1976) 241–251. [5] C. Di Bari, P. Vetro, Common fixed points for ψ-contractions on partial metric spaces, Hacet. J. Math. Stat. 42 (2013) 1–8. [6] C. Di Bari, P. Vetro, Fixed points for weak ϕ-contractions on partial metric spaces, Int. J. Eng. Contemp. Math. Sci. 1 (2011) 5–13. [7] I. Ekeland, Nonconvex minimization problems, Bull. Am. Math. Soc. 1 (1979) 443–474. [8] R. Espinola, W.A. Kirk, Fixed point theorems in R-trees with applications to graph theory, Topol. Appl. 153 (2006) 1046–1055. [9] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Am. Math. Soc. 136 (2008) 1359–1373. [10] E. Karapınar, Generalizations of Caristi Kirk’s theorem on partial metric spaces, Fixed Point Theory Appl. 2011 (4) (2011). [11] W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical weak contractive conditions, Fixed Point Theory 4 (2003) 79–89. [12] S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. N.Y. Acad. Sci., vol. 728, 1994, pp. 183–197. [13] S.B. Nadler, Multivalued contraction mappings, Pac. J. Math. 30 (1969) 475–488. [14] S.J. O’Neill, Partial metrics, valuations and domain theory, in: Proc. 11th Summer Conference on General Topology and Applications, in: Ann. N.Y. Acad. Sci., vol. 806, 1996, pp. 304–315. [15] S. Oltra, O. Valero, Banach’s fixed point theorem for partial metric spaces, Rend. Ist. Mat. Univ. Trieste 36 (2004) 17–26. [16] D. O’Regan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241–1252. [17] D. Paesano, P. Vetro, Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topol. Appl. 159 (2012) 911–920. [18] M. Pacurar, I.A. Rus, Fixed point theory for cyclic ϕ-contractions, Nonlinear Anal. 72 (2010) 1181–1187. [19] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. 2010 (2010), Article ID 493298, 6 pp. [20] I.A. Rus, Cyclic representations and fixed points, Ann. “Tiberiu Popoviciu” Sem. Funct. Equ. Approx. Convexity 3 (2005) 171–178. [21] B. Samet, C. Vetro, F. Vetro, From metric spaces to partial metric spaces, Fixed Point Theory Appl. 2013 (5) (2013). [22] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012) 2154–2165. [23] M. Sgroi, C. Vetro, Multi-valued F -contractions and the solution of certain functional and integral equations, Filomat 27 (2013) 1259–1268. [24] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Am. Math. Soc. 136 (2008) 1861–1869. [25] O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol. 6 (2005) 229–240. [26] F. Vetro, S. Radenović, Nonlinear ψ-quasi-contractions of Ćirić-type in partial metric spaces, Appl. Math. Comput. 219 (2012) 1594–1600. [27] P. Vetro, Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 56 (2007) 464–468. [28] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012 (94) (2012). [29] D.B. West, Introduction to Graph Theory, second edition, Prentice Hall, 1996.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Metric or partial metric spaces endowed with a finite number of graphs: A tool to obtain fixed point results ✩ Calogero Vetro a,∗ , Francesca Vetro b a Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy b Università degli Studi di Palermo, Dipartimento Energia, Ingegneria dell’Informazione e Modelli Matematici (DEIM), Viale delle Scienze, 90128 Palermo, Italy
a r t i c l e
i n f o
Article history: Received 16 July 2013 Received in revised form 12 December 2013 Accepted 14 December 2013 MSC: 54H25 47H10
a b s t r a c t We give some fixed point theorems in the setting of metric spaces or partial metric spaces endowed with a finite number of graphs. The presented results extend and improve several well-known results in the literature. In particular, we discuss a Caristi type fixed point theorem in the setting of partial metric spaces, which has a close relation to Ekeland’s principle. © 2013 Elsevier B.V. All rights reserved.
Keywords: Caristi’s fixed point theorem Ekeland’s principle Graph Metric space Partial metric space
1. Introduction The notion of fixed point is of great interest in mathematics as well as in many fields of applied sciences. Formally, this notion can be introduced as follows: Definition 1. Let X be a nonempty set and T : X → X be a self-mapping. A point x ∈ X is said to be fixed under T if T x = x. Moreover, the set of all these points is denoted by Fix(T ). Metric fixed point theory is the branch of mathematical analysis which focuses on the existence and uniqueness of fixed points under metric conditions on both the domain of the mapping and the mapping itself. We give now the fundamental result of this theory. ✩
The authors are supported by Università degli Studi di Palermo, R. S. ex 60%.
* Corresponding author. E-mail addresses: [email protected] (C. Vetro), [email protected] (F. Vetro). 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.12.008
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Theorem 1 (Banach’s contraction principle). Let (X, d) be a complete metric space and T : X → X be such that d(T x, T y) kd(x, y), for all x, y ∈ X, and some k ∈ (0, 1). Then Fix(T ) is singleton, that is, there exists a unique x ∈ X such that T x = x. In the last decades, this principle was generalized and extended by many authors in several directions (see, for example, [13,16,22–24,27,28]). The reason of our study is to establish fixed point theorems for self-mappings defined on a metric space or a partial metric space endowed with a finite number of graphs. We are motivated by the following two observations: Firstly, we recall that in the mathematical field of domain theory, attempts were made in order to equip semantic domain with a notion of distance. In particular, Matthews [12] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that Banach’s contraction principle can be generalized to the partial metric context for applications in program verification (see also [5,6,17,26]). Secondly, graph theory provides useful tools for many disciplines in which there is a connectedness of elements or components that seem to be related in a system-type. Such a case occurs in the constructive proof of many fixed point theorems, which leads to an algorithm for finding the fixed point. In particular, Espinola and Kirk [8] provided useful results by combining fixed point theory and graph theory. Also, Jachymski [9] continued this idea from a different perspective and his work was followed by several authors [1,3]. After Introduction and Preliminaries we discuss fixed point results in metric spaces in Section 3. Our results generalize and improve many existing fixed point theorems in the literature including Theorem 2.2 of [21]; also an illustrative example is given. In Section 4, we deduce several fixed point theorems in partial metric spaces, by using the results in Section 3. In particular, we state a Caristi type fixed point theorem in partial metric spaces and show how it can be derived from the corresponding result in metric spaces. We recall that Caristi’s fixed point theorem [4] is a variation of Ekeland’s principle, which provides a characterization of complete metric spaces [7]. Finally, we give some fixed point theorems for cyclic mappings by using the results in the previous sections. 2. Preliminaries In this section, we introduce some concepts which are useful for the correct understanding of the paper. 2.1. Graph theory Definition 2. A graph G is an ordered pair (V, E), where V is a set and E ⊆ V × V is a binary relation. We say that V is the vertex set and E is the edge set. For a more detailed background on this topic, we refer the reader to [29]. Definition 3. Let G = (V, E) be a graph and D be a subset of V . We say that D is G-directed if for every x, y ∈ D, there exists z ∈ V such that (x, z), (y, z) ∈ E. Example 1. Let V = F([0, 1], R) be the set of functions f : [0, 1] → R, and define E ⊆ V × V by: (u, v) ∈ E
⇐⇒
u(t) v(t),
for all t ∈ [0, 1].
Then G = (V, E) is a graph. Let D = M ([0, 1], R) be the set of measurable functions f : [0, 1] → R. Then D is G-directed. Indeed, for every u, v ∈ D, the function z = max{u, v} satisfies (u, z), (v, z) ∈ E.
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Let (V, d) be a metric space. We consider a family G = {Gi : 1 i q} of q 1 graphs such that Gi = (V, Ei ), Ei ⊆ V × V , i = 1, 2, . . . , q. Definition 4. Let T : V → V be a given mapping. We say that T is G-monotone if for all i = 1, 2, . . . , q, we have that (x, y) ∈ Ei implies (T x, T y) ∈ Ei+1 , with Eq+1 = E1 . Consequently, (T kq x, T kq y) ∈ Ei for each nonnegative integer number k if T is G-monotone. Remark 1. If q = 1 (G = G1 ), we say that T preserves edges of G (see [9]). Example 2. Consider the sets V = [0, 1] ∪ [2, 3], E1 = [0, 1] × [2, 3] and E2 = [2, 3] × [0, 1]. Let G = {Gi : i = 1, 2} be the family of graphs Gi = (V, Ei ), i = 1, 2. Define the mapping T : V → V by: T x = x + 2 if x ∈ [0, 1], T x = x − 2 if x ∈ [2, 3]. If (x, y) ∈ E1 , we have (T x, T y) ∈ E2 . If (x, y) ∈ E2 , we have (T x, T y) ∈ E1 . Then T is G-monotone. Definition 5. We say that the pair (G, d) is regular if the following condition holds: if {xn } is a sequence in V and x is a point in V such that ∀ i ∈ {1, 2, . . . , q}, there exists a subsequence {xmi,k } of {xn } with (xmi,k , xmi,k +1 ) ∈ Ei , ∀k; d(xn , x) → 0
as n → ∞,
(1) (2)
then there exist a subsequence {xnk } of {xn } and a rank j ∈ {1, 2, . . . , q} such that (xnk , x) ∈ Ej for all k. Example 3. Let V = B([0, 1], R) be the set of bounded functions f : [0, 1] → R. Define E ⊆ V × V by: (u, v) ∈ E
⇐⇒
u(t) v(t),
for all t ∈ [0, 1].
Consider the graph G = (V, E). We endow V with the metric d given by: d(u, v) = sup u(t) − v(t), 0t1
for all u, v ∈ V.
Let {xn } be a sequence in V and x be a point in V satisfying conditions (1) and (2), that is, (i) there exists a subsequence {xmk } of {xn } such that for all k ∈ N and t ∈ [0, 1]: xmk (t) xmk +1 (t), (ii) d(xn , x) → 0 as n → ∞. Then, (xmk , x) ∈ E for all k ∈ N; hence, the pair (G, d) is regular. 2.2. Partial metric spaces Here, we recall some definitions and some properties of partial metric spaces that can be found in [12, 14,15,19,25]. A partial metric on a nonempty set X is a function p : X × X → [0, ∞) such that for all x, y, z ∈ X: (p1 ) (p2 ) (p3 ) (p4 )
x = y ⇔ p(x, x) = p(x, y) = p(y, y), p(x, x) p(x, y), p(x, y) = p(y, x), p(x, y) p(x, z) + p(z, y) − p(z, z).
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A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X. It is clear that, if p(x, y) = 0, then from (p1 ) and (p2 ) it follows that x = y. But if x = y, p(x, y) may not be 0. A basic example of a partial metric space is the pair ([0, ∞), p), where p(x, y) = max{x, y} for all x, y ∈ [0, ∞). Other examples of partial metric spaces which are interesting from a computational point of view can be found in [12]. Each partial metric p on X generates a T0 topology τp on X which has as a base the family of open p-balls {Bp (x, ε): x ∈ X, ε > 0}, where Bp (x, ε) = y ∈ X: p(x, y) < p(x, x) + ε for all x ∈ X and ε > 0. If p is a partial metric on X, then the function ps : X × X → [0, ∞) given by ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) is a metric on X. Let (X, p) be a partial metric space. A sequence {xn } in (X, p) converges to a point x ∈ X if and only if p(x, x) = limn→∞ p(x, xn ). A sequence {xn } in (X, p) is called a Cauchy sequence if there exists (and is finite) limn,m→∞ p(xn , xm ). A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn } in X converges, with respect to τp , to a point x ∈ X such that p(x, x) = limn,m→∞ p(xn , xm ). It is easy to see that, every closed subset of a complete partial metric space is complete. Lemma 1. [12,15] Let (X, p) be a partial metric space. Then (a) {xn } is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metric space (X, ps ). (b) A partial metric space (X, p) is complete if and only if the metric space (X, ps ) is complete. Furthermore, limn→∞ ps (xn , x) = 0 if and only if p(x, x) = lim p(xn , x) = n→∞
lim p(xn , xm ).
n,m→∞
3. Fixed point results in metric spaces In this section, we also consider the family G = {Gi : 1 i q} of q 1 graphs such that Gi = (V, Ei ), Ei ⊆ V × V , i = 1, 2, . . . , q. Let Ψ be the family of nondecreasing functions ψ : [0, ∞) → [0, ∞) satisfying: lim ψ n (t) = 0,
n→∞
for all t > 0.
The following auxiliary fact is immediate; so, we omit its proof. Lemma 2. If ψ ∈ Ψ , then ψ(t) < t for all t > 0. Thus, we state and prove our main result in metric spaces. Theorem 2. Let (V, d) be a complete metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold: (a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, d) is regular;
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(c) there exist ψ ∈ Ψ and a lower semi-continuous function φ : V → [0, ∞) such that d(T x, T y) + φ(T x) + φ(T y) ψ d(x, y) + φ(x) + φ(y) ,
129
(3)
for all (x, y) ∈ Ei , i = 1, 2, . . . , q; (d) there exist i ∈ {1, 2, . . . , q}, and Ai , Bi ⊆ V such that Ei = Ai × Bi . Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Proof. From (a), there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 . Next, define the sequence {xn } in V by: xn+1 = T xn ,
n = 0, 1, 2, . . . .
If xn+1 = xn for some n, then xn is a fixed point of T and the existence of a fixed point is proved. Now, suppose that xn+1 = xn ,
n = 0, 1, 2, . . . .
(4)
Since T is G-monotone, for all n 0, there exists i = i(n) ∈ {1, 2, . . . , q} such that (xn , xn+1 ) ∈ Ei . For all n 1, applying the inequality (3) with x = xn−1 and y = xn , we obtain d(T xn−1 , T xn ) + φ(T xn−1 ) + φ(T xn ) ψ d(xn−1 , xn ) + φ(xn−1 ) + φ(xn ) . (5) Using (5), we get d(xn , xn+1 ) + φ(xn ) + φ(xn+1 ) ψ n d(x0 , x1 ) + φ(x0 ) + φ(x1 ) .
(6)
Since ψ ∈ Ψ and letting n → ∞, we get that d(xn , xn+1 ) + φ(xn ) + φ(xn+1 ) → 0. This implies that d(xn , xn+1 ) → 0 and φ(xn ) → 0 as n → ∞. Fix ε > 0 and let n(ε) be a positive integer such that d(xm , xm+1 ) + φ(xm ) + φ(xm+1 ) <
ε − ψ(ε) q
(7)
for all m n(ε). Now, let i be as in condition (d) and we choose m n(ε) such that (xm , xm+1 ) ∈ Ei . We prove that d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) < ε
(8)
for all n m such that n − m = kq with k a nonnegative integer number. If k = 0, then (8) holds by (7). Now, suppose that (8) holds for some n such that n − m = kq with k > 0. Since T is G-monotone, we deduce easily that (xn , xn+1 ) = (xm+kq , xm+kq+1 ) ∈ Ei . Consequently, from (d), we have (xm , xn+1 ) ∈ Ei . Next, using (3) and (7), we have d(xm , xn+q+1 ) + φ(xm ) + φ(xn+q+1 ) d(xm , xm+1 ) + d(xm+1 , xn+2 ) +
q
d(xn+k , xn+k+1 ) + φ(xm ) + φ(xn+q+1 )
k=2
ε − ψ(ε) + ψ d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) q
ε − ψ(ε) + ψ(ε) = ε. Thus (8) holds for all n m such that n − m = kq with k 0 and (xm , xm+1 ) ∈ Ei .
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Clearly, (8) holds for all m n(ε) and n m with n − m q. If n − m > q and (xm , xm+1 ) ∈ Ej with j = i, we choose h ∈ {1, . . . , q} and a nonnegative integer k such that (xm+h , xm+h+1 ) ∈ Ei and s = m + h + kq n < s + q. Then for all n m, we have d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) d(xm , xm+h ) + d(xm+h , xs+1 ) + d(xs+1 , xn+1 ) + φ(xm ) + φ(xn+1 ). From the above, we get easily d(xm , xn+1 ) + φ(xm ) + φ(xn+1 ) < 3ε for all m n(ε) and n m and hence {xn } is a Cauchy sequence. Now, since (V, d) is complete, there exists u ∈ V such that xn → u. Clearly the sequence {xn } satisfies conditions (1) and (2) with x = u. Since (G, d) is regular, there exist a subsequence {xnk } of {xn } and j ∈ {1, 2, . . . , q} such that (xnk , u) ∈ Ej for all k. Since φ is lower semi-continuous, we get φ(u) lim inf φ(xn ) = 0, n→∞
that is, φ(u) = 0. Applying (3) with x = xnk and y = u, we obtain d(xnk+1 , T u) d(T xnk , T u) + φ(xnk +1 ) + φ(T u) ψ d(xnk , u) + φ(xnk ) + φ(u) < d(xnk , u) + φ(xnk ) + φ(u)
(9)
for all k. Letting k → ∞, we get d(u, T u) = 0, hence u ∈ V is a fixed point of T . Now, suppose that there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed. We will prove that u is the unique fixed point of T . Suppose that y ∈ V is another fixed point of T . Then, there is z ∈ V such that (u, z), (y, z) ∈ Ej . Define the sequence {zn } in V by z0 = z and zn+1 = T zn for all n 0. Since T is G-monotone, for all n 0, there exists i = i(n) ∈ {1, 2, . . . , q} such that (u, zn ) ∈ Ei . Applying (3), for all n 0, we have d(u, zn+1 ) d(T u, T zn ) + φ(T u) + φ(T zn ) ψ d(u, zn ) + φ(u) + φ(zn ) ψ n d(u, z0 ) + φ(u) + φ(z0 ) .
(10)
Letting n → ∞ in the above inequality yields d(u, zn ) → 0. Similarly, we can prove that d(y, zn ) → 0 and hence y = u. 2 Remark 2. One can prove Theorem 2 without assuming hypothesis (d), provided that the function ψ ∈ Ψ ∞ satisfies the condition n=1 ψ n (t) < ∞ for all t > 0. Taking ψ(t) = kt with k ∈ (0, 1) in Theorem 2, by Remark 2, we obtain the following corollary which is a generalization of Theorem 2.2 of [21]. Corollary 1. Let (V, d) be a complete metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold:
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(a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, d) is regular; (c) there exist k ∈ (0, 1) and a lower semi-continuous function φ : V → [0, ∞) such that
d(T x, T y) + φ(T x) + φ(T y) k d(x, y) + φ(x) + φ(y) ,
(11)
for all (x, y) ∈ Ei , i = 1, 2, . . . , q. Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Remark 3. Taking V = X, q = 1 and E1 = X × X in Corollary 1, we obtain the following generalization of Banach’s contraction principle. Corollary 2. Let (X, d) be a complete metric space and let T : X → X be a mapping. Suppose that there exist k ∈ (0, 1) and a lower semi-continuous function φ : X → [0, ∞) such that
d(T x, T y) + φ(T x) + φ(T y) k d(x, y) + φ(x) + φ(y) ,
(12)
for all x, y ∈ X. Then T has a unique fixed point. Remark 4. From Corollary 2, we deduce Banach’s contraction principle, whenever we assume φ(x) = 0 for every x ∈ X. To support our results, we give an illustrative example. Precisely, we show that our Corollary 2 can be used to cover this example while Banach’s contraction principle cannot be applied. Example 4. Let X = [0, 1] endowed with the usual metric d(x, y) = |x − y| for all x, y ∈ X. Obviously (X, d) is a complete metric space. Also, fix r ∈ [0, 1), and define T : X → X by ⎧ 0 if x = 0, ⎪ ⎨ r 2n−1 1 1 T x = 2n − r 2n (2nx − 1) if 2n x 2n−1 , ⎪ ⎩ r + r 2n+1 (2nx − 1) if 1 x 1 . 2n 2n 2n+1 2n Firstly, we prove that T is not a contraction. In fact, if for odd n > 1 we choose x = we have d(T x, T y) =
1 2n−1
and y =
1 n−1 ,
n 3 r d(x, y) = , n−1 (n − 1)(2n − 1) 5(n − 1)
which is not satisfied for r > 3/5. Therefore, Banach’s contraction principle cannot be applied. On the other hand, if we consider the function φ : X → [0, ∞) defined by φ(x) = x, then we obtain d(T x, T y) + φ(T x) + φ(T y) = 2 max{T x, T y} 2 max{rx, ry} = r2 max{x, y} = r d(x, y) + φ(x) + φ(y) , for all x, y ∈ X. Thus, all the conditions of Corollary 2 are satisfied. Therefore, T has a unique fixed point in X.
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Another interesting result is given in the following theorem. Theorem 3. Let (V, d) be a complete metric space endowed with a graph G = (V, E). Let T : V → V be a mapping such that the following conditions hold: (a) (x, T x) ∈ E for all x ∈ V ; (b) there exists k ∈ (0, 1) such that d(T x, T y) kd(x, y), for all (x, y) ∈ E; (c) if (x, y) ∈ / E, then d(x, T x) > (1 + k)d(x, y), for all (x, y) ∈ X × X. Then T has a unique fixed point. Proof. From (a) and (b) we deduce that d T x, T 2 x kd(x, T x),
(13)
for all x ∈ X. Let x0 ∈ X and consider the sequence {xn } defined by xn = T xn−1 for all n ∈ N. If xn+1 = xn for some n, then xn is a fixed point of T and the existence of a fixed point is proved. Now, suppose that xn+1 = xn ,
for all n ∈ N.
(14)
From (a), it follows that (xn , xn+1 ) ∈ E for all n ∈ N. Now, from (b) with x = xn−1 and y = xn , we get d(xn , xn+1 ) kd(xn−1 , xn ) and hence d(xn , xn+1 ) kn d(x0 , x1 ),
for all n ∈ N.
(15)
This ensures that {xn } is a Cauchy sequence. Since V is a complete metric space there exists x ∈ V such that xn → x. Next, we prove that for all (u, v) ∈ X × X, we have either (u, v) ∈ E or (T u, v) ∈ E. Assume that (u, v) ∈ / E and (T u, v) ∈ / E, then by (c), we have d(u, T u) > (1 + k)d(u, v)
and d T u, T 2 u > (1 + k)d(T u, v).
(16)
By (13) and (16), we get d(u, T u) d(u, v) + d(T u, v) 1 d(u, T u) + d T u, T 2 u < 1+k 1 d(u, T u) + kd(u, T u) 1+k d(u, T u) a contradiction and so (u, v) ∈ E or (T u, v) ∈ E. This implies that either (x2n , x) ∈ E or (x2n+1 , x) ∈ E. Then, from (b) we obtain either d(x2n+1 , T x) kd(x2n , x) or d(x2n+2 , T x) kd(x2n+1 , x). Thus, there exists a subsequence {xnk } of {xn } such that d(xnj +1 , T x) kd(xnj , x),
for all j ∈ N.
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Letting j → ∞ in the previous inequality, we obtain xnj +1 → T x and since xnj +1 → x, we get x = T x. We will prove that x is the unique fixed point of T . Suppose that y ∈ V is another fixed point of T . If (x, y) ∈ E, then from (b), we deduce d(x, y) = d(T x, T y) kd(x, y), / E, then (T x, y) ∈ E and again from (b), we obtain which implies x = y. If (x, y) ∈ d(x, y) = d T 2 x, T y kd(T x, y) = kd(x, y), which implies x = y. This completes the proof. 2 4. Fixed point in partial metric spaces In this section, we deduce easily several fixed point theorems in partial metric spaces, by using the results in Section 3. We recall the following definition. Definition 6. Let (V, p) be a partial metric space and let φ : V → [0, ∞) be a function on V . Then, the function φ is called p-lower semi-continuous on V whenever lim p(xn , x) = p(x, x)
n→∞
=⇒
φ(x) lim inf φ(xn ) = sup inf φ(xm ). n→∞
n1 mn
Clearly, the above definition reduces to the classical definition of lower semi-continuity, whenever we consider a classical metric instead of a partial metric. In particular, referring to the metric ps , we observe that every p-lower semi-continuous function on (V, p) is a ps -lower semi-continuous function. The following auxiliary fact is useful in the proof of the next theorem. Lemma 3. Let (V, p) be a complete partial metric space and φ : V → [0, ∞) be the function defined by φ(x) = p(x, x), for every x ∈ V . Then, φ is a ps -lower semi-continuous function. Proof. Let {xn } be a sequence in V such that limn→∞ ps (xn , x) = 0, where x ∈ V . Then by Lemma 1, we get lim p(xn , xn ) = p(x, x),
n→∞
that is, lim φ(xn ) = φ(x).
n→∞
Thus, φ is continuous and therefore lower semi-continuous on (V, ps ), that is, ps -lower semi-continuous. 2 Finally, we state and prove our result. Theorem 4. Let (V, p) be a complete partial metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold: (a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, p) is regular;
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(c) there exists ψ ∈ Ψ such that p(T x, T y) ψ(p(x, y)), for all (x, y) ∈ Ei , i = 1, 2, . . . , q; (d) there exist i ∈ {1, 2, . . . , q} and Ai , Bi ⊆ V such that Ei = Ai × Bi . Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Proof. Note that since (V, p) is complete, from Lemma 1, (V, ps ) is a complete metric space. Now, for all x, y ∈ V , we have p(x, y) =
ps (x, y) + p(x, x) + p(y, y) 2
and hence the result follows by Theorem 2 if we choose d(x, y) = x, y ∈ V . 2
1 s 2 p (x, y)
and φ(x) =
1 2 p(x, x)
for all
From Corollary 1, we obtain the following corollary which is a generalization of Matthews fixed point theorem. Corollary 3. Let (V, p) be a complete partial metric space and let T : V → V be a G-monotone mapping. Also suppose that the following conditions hold: (a) there exists x0 ∈ V such that (x0 , T x0 ) ∈ E1 ; (b) (G, p) is regular; (c) there exists k ∈ (0, 1) such that p(T x, T y) kd(x, y), for all (x, y) ∈ Ei , i = 1, 2, . . . , q. Then T has a fixed point. Moreover, if there exists j ∈ {1, 2, . . . , q} such that Fix(T ) is Gj -directed, we obtain uniqueness of the fixed point. Remark 5. Taking V = X, q = 1 and E1 = X × X in Corollary 3, we recover the Matthews fixed point theorem [12]. Recently, several authors worked on Caristi’s fixed point theorem in the setting of partial metric spaces [2,10,19], also using the well-known Ekeland’s variational principle [7]. Following this direction of research, we present some results which can be easily derived from the corresponding results in metric spaces. Theorem 5. Let (V, p) be a complete partial metric space and let F : V → R be a ps -lower semi-continuous function bounded from below. Let ε > 0 be given, and u ∈ V be such that F (u) inf x∈V F (x) + ε. Then there exists some point v ∈ V such that F (v) F (u), ps (u, v) 1 and F (w) > F (v) − εps (v, w), for every w ∈ V with w = v. Proof. Note that since (V, p) is complete, from Lemma 1, (V, ps ) is a complete metric space. Thus the thesis follows from Theorem 1 of [7]. 2 Since every p-lower semi-continuous function on (V, p) is a ps -lower semi-continuous function, we get the following corollary. Corollary 4. Let (V, p) be a complete partial metric space and let F : V → R be a p-lower semi-continuous function bounded from below. Let ε > 0 be given, and u ∈ V be such that F (u) inf x∈V F (x) + ε. Then there exists some point v ∈ V such that F (v) F (u), ps (u, v) 1 and F (w) > F (v) − εps (v, w), for every w ∈ V with w = v.
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Now, we deduce Caristi type fixed point theorem in the setting of partial metric spaces as consequence of the corresponding result in metric spaces. Theorem 6. Let (V, p) be a complete partial metric space and let T : V → V satisfying p(u, T u) φ(u) − φ(T u),
for all u ∈ V,
(17)
where φ : V → R is a ps -lower semi-continuous function. Then T has a fixed point. Proof. Since (V, p) is a complete partial metric space, then also (V, ps ) is a complete metric space. From ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) we deduce p(x, y) =
1 s p (x, y) + p(x, x) + p(y, y) 2
and so
1 1 s p (u, T u) ps (u, T u) + p(u, u) + p(T u, T u) φ(u) − φ(T u), 2 2 that is ps (u, T u) 2φ(u) − 2φ(T u). Thus, the result follows from Caristi’s fixed point theorem in complete metric spaces.
2
Remark 6. Theorem 6 holds also if φ is a p-lower semi-continuous function. 5. Cyclic results In this section, we deduce some fixed point results for cyclic mappings on metric spaces, by using the results in Section 3. In 2003, Kirk et al. [11] introduced the following definition. Definition 7. [11] Let X be a nonempty set, q be a positive integer and T : X → X be a mapping. q X = i=1 Ai is said to be a cyclic representation of X with respect to T if (i) Ai , i = 1, 2, . . . , q, are nonempty closed sets; (ii) T (A1 ) ⊂ A2 , . . . , T (Aq−1 ) ⊂ Aq , T (Aq ) ⊂ A1 . Recently, fixed point theorems involving a cyclic representation of X with respect to a self-mapping T appeared in many articles (see, for example, [18,20]). Following this direction of research, we introduce some definitions. Definition 8. Let (X, d) be a metric space, q be a positive integer, A1 , A2 , . . . , Aq be nonempty closed subsets q of X and Y = i=1 Ai . An operator T : Y → Y is said to be a cyclic weak (φ, ψ)-contraction if q (i) i=1 Ai is a cyclic representation of Y with respect to T ; (ii) there exist ψ ∈ Ψ and a lower semi-continuous function φ : V → [0, ∞) such that d(T x, T y) + φ(T x) + φ(T y) ψ(d(x, y) + φ(x) + φ(y)), for every x ∈ Ai , y ∈ Ai+1 , i = 1, 2, . . . , q, where Aq+1 = A1 .
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Taking ψ(t) = kt with k ∈ (0, 1) in Definition 8, we obtain the following notion. Definition 9. Let (X, d) be a metric space, q be a positive integer, A1 , A2 , . . . , Aq be nonempty closed subsets q of X and Y = i=1 Ai . An operator T : Y → Y is said to be a cyclic weak φk -contraction if q (i) i=1 Ai is a cyclic representation of Y with respect to T ; (ii) there exist k ∈ (0, 1) and a lower semi-continuous function φ : V → [0, ∞) such that d(T x, T y) + φ(T x) + φ(T y) k[d(x, y) + φ(x) + φ(y)], for every x ∈ Ai , y ∈ Ai+1 , i = 1, 2, . . . , q, where Aq+1 = A1 . Now, we state and prove the following theorem, which is an extension of Kirk et al. cyclic fixed point theorem [11]. Theorem 7. Let (X, d) be a complete metric space, q a positive integer, A1 , . . . , Aq ∈ Pcl (X) (where Pcl (X) q is the collection of nonempty closed subsets of X), Y = i=1 Ai and T : Y → Y be a cyclic weak (φ, ψ)-contraction. Then, T has a unique fixed point in Y . Proof. We take V = Y and consider the family of graphs G = {Gi : 1 i q} of q 1 graphs such that Gi = (V, Ei ), Ei = Ai × Ai+1 , for all i = 1, 2, . . . , q + 1 with Aq+1 = A1 . Since A1 , . . . , Aq ∈ Pcl (X) and (X, d) is complete, then (V, d) is a complete metric space. Let (x, y) ∈ Ei for some i. This implies that x ∈ Ai and y ∈ Ai+1 . From condition (i) of Definition 8, we have T (Ai ) ⊆ Ai+1
and T (Ai+1 ) ⊆ Ai+2 .
Then we get (T x, T y) ∈ Ai+1 × Ai+2 ,
that is,
(T x, T y) ∈ Ei+1 .
Thus we proved that T is G-monotone. Let x0 ∈ A1 (such a point exists since A1 = ∅). Since T (A1 ) ⊆ A2 , we have T x0 ∈ A2 . This implies that (x0 , T x0 ) ∈ E1 . Now, we shall prove that (G, d) is regular. Let {xn } be a sequence in V and x be a point in V satisfying (1) and (2) of Definition 5. For i ∈ {1, . . . , q}, there exists a subsequence {xmi,k } of {xn } such that (xmi,k , xmi,k +1 ) ∈ Ei for all k. This implies that xmi,k ∈ Ai for all k. Since Ai is closed and d(xn , x) → 0 as n → ∞, we have x ∈ Ai ; so, as i ∈ {1, . . . , q} was q arbitrarily chosen, we get that x ∈ i=1 Ai . This yields, by the choice of our subsequences, (xmi,k , x) ∈ Ei for all i ∈ {1, . . . , q} and all k. Thus we proved that (G, d) is regular. Finally, we shall prove that Fix(T ) is G1 -directed. Let x, y ∈ Fix(T ). q From (i) of Definition 8, we have x, y ∈ i=1 Ai , which implies that (x, y) ∈ E1 and (y, y) ∈ E1 , and our claim is proved. Now, from the definition of Ei , it follows that all the hypotheses of Theorem 2 are satisfied, then we deduce that T has a unique fixed point in V . 2 In view of Definition 9, we give the following result. Corollary 5. Let (X, d) be a complete metric space, q a positive integer, A1 , . . . , Aq ∈ Pcl (X), Y = and T : Y → Y be a cyclic weak φk -contraction. Then, T has a unique fixed point in Y .
q i=1
Ai
Finally, reasoning as in the proof of Theorem 4 in Section 4, one can prove the following result in the setting of partial metric spaces. Theorem 8. Let (X, p) be a complete partial metric space, q a positive integer, A1 , . . . , Aq ∈ Pclp (X) (where q Pclp (X) is the collection of nonempty closed subsets of X), Y = i=1 Ai and T : Y → Y an operator. Assume that
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