Mediterr. J. Math. 9 (2012), 225–238 DOI 10.1007/s00009-011-0120-1 1660-5446/12/010120-14, published online February 19, 2011 © 2011 Springer Basel AG
Mediterranean Journal of Mathematics
´ c’s Fixed Point An Integral Version of Ciri´ Theorem Bessem Samet and Calogero Vetro∗ Abstract. We establish a new fixed point theorem for mappings satisfying a general contractive condition of integral type. The presented theorem gen´ c’s fixed point theorem [Lj. B. Ciri´ ´ c, Generalized eralizes the well known Ciri´ contractions and fixed point theorems, Publ. Inst. Math. 12 (26) (1971) 19-26]. Some examples and applications are given. Mathematics Subject Classification (2010). 47H10; 54H25. Keywords. Complete metric space, λ-generalized contraction, fixed point, contractive condition of integral type.
1. Introduction One of the most fundamental fixed point theorems is the Banach’s contraction principle [2]. Many authors have extended this well known fixed point theorem in various ways (cf. [1], [3-19]). ´ c introduced the notion of λ-generalized contraction. To be In 1971, Ciri´ precise, we have the following: ´ c [5]) Let (X, d) be a metric space and T : X → X be Definition 1.1. (Ciri´ a given mapping. The mapping T is said to be a λ-generalized contraction iff for every x, y ∈ X there exist non-negative numbers q(x, y), r(x, y), s(x, y) and t(x, y) such that sup {q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} = λ < 1
(1.1)
x,y∈X
and d(T x, T y)
≤
q(x, y)d(x, y) + r(x, y)d(x, T x) + s(x, y)d(y, T y) (1.2) +t(x, y)[d(x, T y) + d(y, T x)]
The second author is supported by University of Palermo, Local University Project R. S. ex 60 %. ∗ Corresponding author.
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holds for every x, y ∈ X. ´ c proved the following important fixed point theorem. Thus, Ciri´ ´ c [5]) Let (X, d) be a complete metric space, T : X → X Theorem 1.1. (Ciri´ be a λ-generalized contraction. Then, T admits a unique fixed point u ∈ X and for all x ∈ X, the sequence {T n x} converges to u. It is clear that Theorem 1.1 generalizes the Banach contraction principle. In recent years, Branciari [4] initiated a study of contractive conditions of integral type, giving an integral version of the Banach contraction principle. More precisely, Branciari established the following theorem. Theorem 1.2. (Branciari [4]) Let (X, d) be a complete metric space, c ∈ (0, 1) and T : X → X be a given mapping such that for each x, y ∈ X,
d(T x,T y) 0
ϕ(s) ds ≤ c
d(x,y) 0
ϕ(s) ds,
(1.3)
where ϕ : [0, +∞) → [0, +∞) is a Lebesgue integrable mapping on each compact subset of [0, +∞) and such that for all ε > 0,
ε 0
ϕ(s) ds > 0.
Then, T admits a unique fixed point u ∈ X and for each x ∈ X, the sequence {T n x} converges to u. Taking ϕ ≡ 1 in Theorem 1.2, we retrieve immediately the Banach fixed point theorem. Later on, fixed point theorems involving more general contractive conditions of integral type were obtained (cf. [1], [15]-[17], [19]-[20]). Suzuki [18] showed that Meir-Keeler contractions of integral type are still Meir-Keeler contractions and so proved that Theorem 1.2 of Branciari is a particular case of the Meir-Keeler fixed point theorem [14]. ´ c’s fixed point In this paper, we establish an integral version of Ciri´ theorem (Theorem 1.1). Some examples and applications are given.
2. Main result We introduce the following notations. Let N ∈ N∗ = {1, 2, . . .} be fixed. Let ϕi : [0, +∞) → [0, +∞) (i = 1, . . . , N ) be a Lebesgue integrable mapping on each compact subset of [0, +∞). For all t ≥ 0, we denote:
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I1 (t)
t
=
ϕ1 (s) ds,
0
I2 (t)
=
.. .
=
IN (t)
=
227
.. .
t
ϕ1 (s) ds
0
ϕ2 (s) ds =
0
I1 (t)
ϕ2 (s) ds
0
IN −1 (t) 0
ϕN (s) ds.
Now, we state and prove our main result. Theorem 2.1. Let (X, d) be a complete metric space, T : X → X be a given mapping and N ∈ N∗ be fixed. Let ϕi : [0, +∞) → [0, +∞) (i = 1, . . . , N ) be a Lebesgue integrable mapping on each compact subset of [0, +∞) such that for all ε > 0, ε
0
ϕi (s) ds > 0 f or all i = 1, . . . , N.
(2.1)
We assume that for every x, y ∈ X there exist non-negative numbers q(x, y), r(x, y), s(x, y) and t(x, y) such that sup {q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} = λ < 1
(2.2)
x,y∈X
and IN (d(T x, T y))
≤
q(x, y)IN (d(x, y)) + r(x, y)IN (d(x, T x)) +s(x, y)IN (d(y, T y))
(2.3)
+2t(x, y)IN ([d(x, T y) + d(y, T x)]/2) holds for every x, y ∈ X. Then, T admits a unique fixed point u ∈ X and for each x ∈ X, the sequence {T n x} converges to u. Proof. Without restriction to the generality, we can assume that N = 2. The general case can be treated in the same manner. Then, (2.3) becomes 0d(T x,T y) ϕ1 (s) ds 0d(x,y) ϕ1 (s) ds ϕ2 (s) ds ≤ q(x, y) ϕ2 (s) ds (2.4) 0
0
d(x,T x)
+r(x, y) +s(x, y)
0
0
ϕ2 (s) ds d(y,T y) 0
ϕ1 (s) ds
ϕ2 (s) ds
0
+2t(x, y)
ϕ1 (s) ds
d(x,T y)+d(y,T x) 2
0
0
ϕ1 (s) ds
ϕ2 (s) ds.
Let x ∈ X be arbitrary. Define the sequence {xn } by x0 = x, x1 = T x0 , x2 = T x1 = T 2 x0 , . . . , xn+1 = T xn = T n+1 x0 , . . .
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If there exists k ∈ N such that xk = xk+1 , then xk is a fixed point of T and T n x = xk for all n ≥ k. Hence, the result is immediate in this case. Now, we consider the case xn = xn+1 for all n ∈ N. For x = xn−1 and y = xn , using the triangular inequality, condition (2.4) becomes d(T xn−1 ,T xn )
0
ϕ2 (s)ds
0
d(xn−1 ,xn )
ϕ1 (s)ds
≤
q(xn−1 , xn )
0
ϕ2 (s)ds
0
d(xn−1 ,xn )
+r(xn−1 , xn ) +s(xn−1 , xn )
0
0
ϕ1 (s)ds
ϕ2 (s)ds d(xn ,xn+1 ) 0
0
+2t(xn−1 , xn )
ϕ1 (s)ds
ϕ1 (s)ds
ϕ2 (s)ds Θ
0
ϕ2 (s)ds,
max{d(xn−1 ,xn ),d(xn ,xn+1 )} ϕ1 (s) ds. where Θ := 0 Suppose now that max{d(xn−1 , xn ), d(xn , xn+1 )} = d(xn , xn+1 ). We get: d(xn ,xn+1 )
[1 − (q + r + s + 2t)(xn−1 , xn )]
0
ϕ1 (s) ds
0
ϕ2 (s) ds ≤ 0,
which is a contradiction with (2.1). Then, max{d(xn−1 , xn ), d(xn , xn+1 )} = d(xn−1 , xn ) and we have: d(xn ,xn+1 )
0
ϕ1 (s) ds
0
d(xn−1 ,xn )
ϕ2 (s) ds ≤ F (xn−1 , xn )
0
ϕ1 (s) ds
ϕ2 (s) ds,
0
(2.5)
where F (x, y) =
q(x, y) + r(x, y) + 2t(x, y) . 1 − s(x, y)
Since λ < 1, from q(x, y) + r(x, y) + λs(x, y) + 2t(x, y) ≤ q(x, y) + r(x, y) + s(x, y) + 2t(x, y) ≤ λ, it follows F (x, y) ≤ λ, for all x, y ∈ X. Thus, from (2.5), we get: d(xn ,xn+1 )
0
ϕ1 (s) ds
0
d(xn−1 ,xn )
ϕ2 (s) ds ≤ λ
0
ϕ1 (s) ds
ϕ2 (s) ds.
0
Repeating this argument n times, we conclude that d(xn ,xn+1 )
0
0
ϕ1 (s) ds
d(x0 ,x1 )
ϕ2 (s) ds ≤ λ
0
n 0
ϕ1 (s) ds
ϕ2 (s) ds,
(2.6)
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for all n ∈ N∗ . Letting n → +∞ in (2.6), we get: 0d(xn ,xn+1 ) ϕ1 (s) ds lim ϕ2 (s) ds = 0. n→+∞
(2.7)
0
Let us prove now that lim d(xn , xn+1 ) = 0.
(2.8)
n→+∞
Suppose that (2.8) does not hold. Then, there exists ε > 0 such that for each integer p there exists an integer n(p) ≥ p such that d(xn(p) , xn(p)+1 ) ≥ ε.
Let δ :=
ε 0
ϕ1 (s) ds > 0,
then for every integer p we have: d(xn(p) ,xn(p)+1 ) δ ϕ1 (s) ds 0 ϕ2 (s) ds ≤ ϕ2 (s) ds. 0
0
Using (2.7) and letting p → +∞, we get: δ ϕ2 (s) ds ≤ 0. 0
Hence, we obtain a contradiction with (2.1). Consequently, (2.8) holds. Now, let us prove that {xn } is a Cauchy sequence. Suppose that it is not the case. Then, there exists ε > 0 such that for each integer p there exist integers m(p), n(p) with m(p) > n(p) > p, such that d(xm(p) , xn(p) ) ≥ ε.
(2.9)
For every integer p, let m(p) be the least positive integer exceeding n(p) satisfying condition (2.9) such that d(xm(p)−1 , xn(p) ) < ε. Hence, ε ≤ d(xm(p) , xn(p) )
≤
d(xm(p) , xm(p)−1 ) + d(xm(p)−1 , xn(p) )
<
d(xm(p) , xm(p)−1 ) + ε → ε+ as p → +∞.
Then, lim d(xm(p) , xn(p) ) = ε.
n→+∞
(2.10)
Moreover, by the triangular inequality, we have: | d(xm(p)−1 , xn(p) ) − d(xm(p) , xn(p) ) |≤ d(xm(p)−1 , xm(p) )
(2.11)
and so d(xm(p)−1 , xn(p) )−d(xm(p) , xn(p) ) → 0 as p → +∞. Then, from (2.10), we have: (2.12) lim d(xm(p)−1 , xn(p) ) = ε. p→+∞
From d(xm(p) , xn(p) )
≤
d(xm(p) , xn(p)+1 ) + d(xn(p)+1 , xn(p) )
=
d(T xm(p)−1 , T xn(p) ) + d(xn(p)+1 , xn(p) ),
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d(T xm(p)−1 ,T xn(p) )+d(xn(p)+1 ,xn(p) )
lim sup 0
p→+∞
ϕ1 (s)ds
d(T xm(p)−1 ,T xn(p) )
ϕ1 (s)ds,
= lim sup p→+∞
0
and (2.10), we obtain:
δ 0
≤ lim sup p→+∞
p→+∞
ϕ1 (s) ds
0
ϕ2 (s) ds = lim
d(xm(p) ,xn(p) )
ϕ2 (s) ds
0
(2.13)
d(T xm(p)−1 ,T xn(p) )
ϕ1 (s)ds
0
ϕ2 (s) ds
0
d(xm(p)−1 ,xn(p) )
≤ lim sup q(xm(p) , xn(p) )
+s(xm(p) , xn(p) )
0
0
ϕ1 (s) ds
ϕ2 (s) ds d(xn(p) ,xn(p)+1 ) 0
0
+2t(xm(p) , xn(p) )
ϕ2 (s) ds
d(xm(p)−1 ,xm(p) )
ϕ1 (s) ds
0
p→+∞
+r(xm(p) , xn(p) )
0
0
ϕ1 (s) ds
ϕ2 (s) ds Π
ϕ2 (s) ds ,
d(xm(p)−1 ,xn(p) )+d(xn(p)2,xn(p)+1 )+d(xn(p) ,xm(p) ) ϕ1 (s) ds. where Π := 0 Since q(·, ·), r(·, ·), s(·, ·) and t(·, ·) are bounded, as p → +∞, we can assume, without loss, that q(xm(p) , xn(p) ) → q, r(xm(p) , xn(p) ) → r, s(xm(p) , xn(p) ) → s, t(xm(p) , xn(p) ) → t. Combining (2.8), (2.10), (2.12) and (2.13), we get: δ δ ϕ2 (s) ds ≤ (q + 2t) ϕ2 (s) ds 0
0
≤
λ
δ 0
ϕ2 (s) ds.
As λ ∈ [0, 1), we obtain a contradiction and so, we can conclude that {xn } is a Cauchy sequence. Now, since (X, d) is a complete metric space, there exists u ∈ X such that lim xn = u.
n→+∞
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Let us prove now that u is a fixed point of T . From (2.4), we have: 0d(T xn ,T u) ϕ1 (s) ds 0d(xn ,u) ϕ1 (s) ds ϕ2 (s) ds ≤ q(xn , u) ϕ2 (s) ds 0
0
d(xn ,xn+1 )
+r(xn , u) +s(xn , u)
0
0
ϕ2 (s) ds d(u,T u)
ϕ1 (s) ds
0
ϕ2 (s) ds
0
+2t(xn , u)
ϕ1 (s) ds
Υ
0
ϕ2 (s) ds,
d(xn ,xn+1 )+d(T x2n ,T u)+d(u,xn+1 ) ϕ1 (s) ds. where Υ := 0 Since q(·, ·), r(·, ·), s(·, ·) and t(·, ·) are bounded, as n → +∞, we can assume, without loss, that q(xn , u) → q,
r(xn , u) → r,
Now, letting n → +∞, we obtain: 0d(u,T u) ϕ1 (s) ds ϕ2 (s) ds ≤
s(xn , u) → s, 0
ϕ1 (s) ds
(s + 2t)
0
λ
ϕ2 (s) ds
0
d(u,T u)
≤
d(u,T u)
t(xn , u) → t.
0
ϕ1 (s) ds
0
ϕ2 (s) ds.
Since λ ∈ [0, 1), this implies that 0d(u,T u) ϕ1 (s) ds 0
ϕ2 (s) ds = 0,
which gives us from (2.1) that d(u, T u) = 0, i.e., u is a fixed point of T . To finish the proof, we have to prove that u is the unique fixed point of T . Suppose that v is another fixed point of T , i.e., T v = v. By (2.4), we get:
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B. Samet and C. Vetro d(u,v)
0
0
d(T u,T v)
ϕ1 (s) ds
ϕ2 (s) ds
0
ϕ1 (s) ds
=
ϕ2 (s) ds
0
≤
d(u,v)
q(u, v)
0
+r(u, v) +s(u, v)
d(u,u) 0
ϕ2 (s) ds d(v,v) 0
d(u,v) 0
ϕ1 (s) ds
ϕ2 (s) ds
0
d(u,v)
≤
λ
0
0
ϕ1 (s) ds
0
0
0
ϕ1 (s) ds
ϕ2 (s) ds
ϕ1 (s) ds
0
d(u,v)
d(u,v)
(q(u, v) + 2t(u, v))
(1 − λ)
ϕ1 (s) ds
ϕ2 (s) ds
0
+2t(u, v)
ϕ1 (s) ds
0
Hence
ϕ1 (s) ds
ϕ2 (s) ds
0
≤
Mediterr. J. Math.
ϕ2 (s) ds.
ϕ2 (s) ds ≤ 0.
Since λ ∈ [0, 1), from (2.1), we get d(u, v) = 0, i.e., u = v. Then, the uniqueness of the fixed point of T is proved. This makes end to the proof. Remark 2.1. Taking ϕi ≡ 1 in Theorem 2.1, we retrieve immediately the ´ c’s fixed point theorem. Ciri´ Remark 2.2. By choosing the non-negative numbers q(x, y), r(x, y), s(x, y) and t(x, y) suitably, one can derive results involving different (and independent) contractive conditions.
3. Examples and Applications The following example shows that condition (2.3) in Theorem 2.1 is indeed a proper extension of the same condition with ϕi ≡ 1. Example 3.1. Consider X = {1/n | n ∈ N∗ } ∪ {0} equipped with the usual metric d(x, y) =| x − y | for every x, y ∈ X. Clearly (X, d) is a complete metric space. Define T : X → X as 1 if x = n1 , T x = n+1 0 if x = 0. Now the reader, following the lines of Example 3.6 in [4], can easily verify that the condition I1 (d(T x, T y)) ≤ q(x, y)I1 (d(x, y))
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is satisfied assuming ϕ1 (s) = max{0, s(1/s)−2 (1 − log s)} for s > 0, ϕ1 (0) = 0 and q(x, y) = 1/2 for every x, y ∈ X. Clearly, on the same assumptions outlined above, also the condition I1 (d(T x, T y))
≤
q(x, y)I1 (d(x, y)) + r(x, y)I1 (d(x, T x)) +s(x, y)I1 (d(y, T y)) +2t(x, y)I1 ([d(x, T y) + d(y, T x)]/2)
is satisfied with 0 ≤ supx,y∈X {r(x, y) + s(x, y) + 2t(x, y)} = μ < 1/2. However, for x = 1/n, y = 0 and r(x, y) = t(x, y), condition (2.3) with ϕ1 ≡ 1 becomes 1 1 1 1 1 1 ≤ q(x, y) + r(x, y) − + s(x, y) · 0 + t(x, y) + n+1 n n n+1 n n+1 that is 1 1 1 ≤ (q(x, y) + 2t(x, y)) ≤ λ for all n ∈ N∗ , n+1 n n which implies λ → 1+ as n → +∞, a contradiction. In the following example (see Example 1 in [5]) we show that Theorem 1.2 cannot be applied, since the condition in the theorem is not satisfied. Instead condition (2.3) holds and so, we can apply Theorem 2.1 to this example. Example 3.2. Consider X = [0, 2] equipped with the usual metric d(x, y) = | x − y | for every x, y ∈ X. Clearly (X, d) is a complete metric space. Define T : X → X as x if x ∈ [0, 1], T x = 10 x if x ∈ (1, 2]. 11 The reader can easily verify, for x = 999/1000 and y = 1001/1000, that the condition (1.3) with ϕ(s) = 2s (s ≥ 0), is not satisfied for every c ∈ (0, 1). However, on the same assumptions outlined above, the condition (2.3) is satisfied with N = 1, q(x, y) = 1/10 and r(x, y) = s(x, y) = t(x, y) = 1/5 for every x, y ∈ X. ´ c investigated a family of mappings satisfying a generalized In [8], Ciri´ contractive condition. Now, as application of our Theorem 2.1, we give a common fixed point theorem for a family of mappings which satisfy a common condition of type (2.3). We start with the following lemma, which is an essential part of the proof of the next theorem. Lemma 3.1. Let T0 , T : X → X be two mappings on a metric space (X, d). Let α : [0, +∞) → [0, +∞) be a nondecreasing and continuous function such that α(t) = 0 iff t = 0. Define: Fix (T0 ) = {x ∈ X | x = T0 x}
and
Fix (T ) = {x ∈ X | x = T x}.
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We assume that, for every x, y ∈ X there exist non-negative numbers q(x, y), r(x, y), s(x, y) and t(x, y) such that sup {q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} = λ < 1
(3.1)
x,y∈X
and α(d(T0 x, T y))
≤
q(x, y)α(d(x, y)) + r(x, y)α(d(x, T0 x)) +s(x, y)α(d(y, T y)) +2t(x, y)α([d(x, T y) + d(y, T0 x)]/2)
(3.2)
holds for every x, y ∈ X. If Fix (T0 ) is a non-empty set, then Fix (T0 ) is a singleton and Fix (T0 ) = Fix (T ). Proof. Let u ∈ Fix (T0 ) ⊂ X any fixed point of T0 . Then by (3.2), we have: α(d(T0 u, T u))
≤
q(u, u)α(d(u, u)) + r(u, u)α(d(u, T0 u)) +s(u, u)α(d(u, T u)) +2t(u, u)α([d(u, T u) + d(u, T0 u)]/2)
that is ≤ ≤
α(d(u, T u))
s(u, u)α(d(u, T u)) + 2t(u, u)α(d(u, T u)/2) (s + 2t)(u, u)α(d(u, T u))
which implies d(u, T u) = 0. Therefore u ∈ Fix (T ). Let now v ∈ Fix (T0 ) be arbitrary. Then v ∈ Fix (T ) and by (3.2), we have: α(d(T0 u, T v))
≤
q(u, v)α(d(u, v)) + r(u, v)α(d(u, T0 u)) +s(u, v)α(d(v, T v)) + 2t(u, v)α([d(u, T v) + d(v, T0 u)]/2)
that is α(d(u, v)) ≤ (q(u, v) + 2t(u, v)) α(d(u, v)) which implies d(u, v) = 0. Then, u = v and hence Fix (T0 ) = Fix (T ) = {u}. Theorem 3.1. Let (X, d) be a complete metric space and {Tn } be a sequence of self mappings on X. Let N ∈ N∗ be fixed and ϕi : [0, +∞) → [0, +∞) (i = 1, . . . , N ) be a Lebesgue integrable mapping on each compact subset of [0, +∞) such that for all ε > 0, ε ϕi (s) ds > 0 f or all i = 1, . . . , N. (3.3) 0
We assume that for every x, y ∈ X there exist non-negative numbers q(x, y), r(x, y), s(x, y) and t(x, y) such that sup {q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} = λ < 1 x,y∈X
(3.4)
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and IN (d(T0 x, Tn y))
≤
q(x, y)IN (d(x, y)) + r(x, y)IN (d(x, T0 x)) (3.5) +s(x, y)IN (d(y, Tn y)) +2t(x, y)IN ([d(x, Tn y) + d(y, T0 x)]/2)
holds for every x, y ∈ X and n ∈ N∗ . Then, there exists a unique point u ∈ X such that Tn u = u for each n ∈ N and for arbitrary x0 ∈ X the sequence x 0 , x 1 = T0 x 0 , x 2 = T1 x 1 , x 3 = T0 x 2 , . . . , x2n−1 = T0 x2n−2 , x2n = Tn x2n−1 , . . . converges to u. Proof. Proceeding as in the proof of Theorem 2.1, for N = 2, x = x2n−2 and y = x2n−1 , we have: 0d(T0 x2n−2 ,Tn x2n−1 ) ϕ1 (s) ds ϕ2 (s) ds 0
d(x2n−2 ,x2n−1 )
≤ q(x2n−2 , x2n−1 ) +r(x2n−2 , x2n−1 ) +s(x2n−2 , x2n−1 )
0
0 d(x2n−2 ,x2n−1 ) 0
0
ϕ2 (s) ds ϕ1 (s) ds
ϕ2 (s) ds d(x2n−1 ,x2n ) 0
0
+2t(x2n−2 , x2n−1 )
ϕ1 (s) ds
ϕ1 (s) ds
ϕ2 (s) ds max{d(x2n−2 ,x2n−1 ),d(x2n−1 ,x2n }) 0
ϕ1 (s) ds
ϕ2 (s) ds.
0
Suppose now that max{d(x2n−2 , x2n−1 ), d(x2n−1 , x2n )} = d(x2n−1 , x2n ). In this case, we get: 0d(x2n−1 ,x2n ) ϕ1 (s) ds [1 − (q + r + s + 2t)(x2n−2 , x2n−1 )] ϕ2 (s) ds ≤ 0. 0
From (3.3)-(3.4), this implies that d(x2n−1 , x2n ) = 0. Suppose now that max{d(x2n−2 , x2n−1 ), d(x2n−1 , x2n )} = d(x2n−2 , x2n−1 ). In this case, we get: 0d(x2n−1 ,x2n ) ϕ1 (s) ds 0d(x2n−2 ,x2n−1 ) ϕ1 (s) ds ϕ2 (s) ds ≤ λ ϕ2 (s) ds. (3.6) 0
0
Hence (3.6) holds in all cases. By the same reason, for x = x2n−2 and y = x2n−3 , we obtain: 0d(x2n−1 ,x2n−2 ) ϕ1 (s) ds 0d(x2n−2 ,x2n−3 ) ϕ1 (s) ds ϕ2 (s) ds ≤ λ ϕ2 (s) ds. 0
0
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Consequently, it can be concluded that 0d(xn ,xn+1 ) ϕ1 (s) ds n ϕ2 (s) ds ≤ λ 0
d(x0 ,x1 ) 0
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ϕ1 (s) ds
ϕ2 (s) ds,
0
(3.7)
for all n ∈ N∗ . Letting n → +∞ in (3.7), we get: d(xn ,xn+1 )
0
n→+∞
ϕ1 (s) ds
ϕ2 (s) ds = 0.
lim
0
(3.8)
Using the same arguments in the proof of Theorem 2.1, we conclude that lim d(xn , xn+1 ) = 0.
n→+∞
Now, we are ready to show that {xn } is a Cauchy sequence in (X, d). To this end, it is sufficient to verify that {x2n } is a Cauchy sequence. Suppose that it is not the case. Then, there exists ε > 0 such that for each even integer 2p there exist even integers 2m(p), 2n(p) with 2m(p) > 2n(p) > 2p, such that d(x2m(p) , x2n(p) ) ≥ ε. Thus, using the same arguments in the proof of Theorem 2.1 we can conclude that {x2n } is a Cauchy sequence. Then {xn } is also a Cauchy sequence. As (X, d) is a complete metric space, there exists u ∈ X such that lim xn = u.
n→+∞
Again, using the same arguments in the proof of Theorem 2.1, one can easily shows that u = T0 u, i.e., u is a fixed point of T0 . Now, we observe that condition (3.5) implies condition (3.2) with α(t) = I (t) IN (t) = 0 N −1 ϕN (s) ds. Then, applying Lemma 3.1, we have that there is a unique fixed point u = Tn u for each n = 0, 1, 2, . . .. This makes end to the proof.
References [1] I. Altun, D. T¨ urko˘ glu and B. E. Rhoades, Fixed points of weakly compatible maps satisfying a general contractive condition of integral type. Fixed Point Theory Appl. 2007 (2007), article ID 17301, 9 pages. [2] S. Banach, Sur les op´erations dans les ensembles absraites et leurs applications. Fund. Math. 3 (1922), 133-181. [3] D. W. Boyd and J. S. Wong, On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458-464. [4] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29 (2002), 531-536. ´ c, Generalized contractions and fixed point theorems. Publ. Inst. [5] Lj. B. Ciri´ Math. 12 (1971), 19-26. ´ c, Fixed points for generalized multi-valued mappings. Mat. Vesnik. [6] Lj. B. Ciri´ 9 (1972), 265-272.
Vol. 9 (2012)
´ iric´’s Fixed Point Theorem Integral Version of C
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´ c, A generalization of Banach’s contraction principle. Proc. Amer. [7] Lj. B. Ciri´ Math. Soc. 45 (1974), 267-273. ´ c, On a family of contractive maps and fixed points. Publ. Inst. Math. [8] Lj. B. Ciri´ 17 (1974), 45-51. ´ c, Coincidence and fixed points for maps on topological spaces. Topol. [9] Lj. B. Ciri´ Appl. 154 (2007), 3100-3106. ´ c, Fixed point theorems for multi-valued contractions in complete [10] Lj. B. Ciri´ metric spaces. J. Math. Anal. Appl. 348 (2008), 499-507. ´ c, Non-self mappings satisfying nonlinear contractive condition with [11] Lj. B. Ciri´ applications. Nonlinear Anal. 71 (2009), 2927-2935. [12] G. Jungck, Commuting mappings and fixed points. Amer. Math. Monthly 73 (1976), 261-263. [13] G. Jungck and B. E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory 7 (2006), 286-296. [14] A. Meir and E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl. 28 (1969), 326-329. [15] B. E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 63 (2003), 40074013. [16] B. Samet, A fixed point theorem in a generalized metric space for mappings satisfying a contractive condition of integral type. Int. Journal of Math. Anal. 3 (2009), 1265-1271. [17] B. Samet and H. Yazidi, An extension of Banach fixed point theorem for mappings satisfying a contractive condition of integral type. J. Nonlinear Sci. Appl., in press. [18] T. Suzuki, Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Int. J. Math. Math. Sci. 2007 (2007), article ID 39281, 6 pages. [19] C. Vetro, On Branciari’s theorem for weakly compatible mappings. Appl. Math. Lett. 23 (2010), 700-705. [20] P. Vijayaraju, B. E. Rhoades and R. Mohanraj, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 15 (2005), 2359-2364.
Bessem Samet Ecole sup´erieure des Sciences et Techniques de Tunis D´epartement de Math´ematiques 5, avenue taha hussein-tunis B.P. 56, bab menara-1008 Tunisie e-mail:
[email protected]
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Calogero Vetro Department of Mathematics and Informatics University of Palermo Via archirafi 34 90123 Palermo Italy e-mail:
[email protected] Received: April 16, 2010. Revised: July 28, 2010. Accepted: September 13, 2010.
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