Abstract
In this paper, the existence, uniqueness and iterative approximations of fixed points for contractive mappings of integral type in complete metric spaces are established. As applications, the existence, uniqueness and iterative approximations of solutions for a class of functional equations arising in dynamic programming are discussed. The results presented in this paper extend and improve essentially the results of Branciari (A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29, 531536, 2002), Kannan (Some results on fixed points. Bull. Calcutta Math. Soc. 60, 7176, 1968) and several known results. Four concrete examples involving the contractive mappings of integral type with uncountably many points are constructed.
2010 Mathematics Subject Classfication: 54H25, 47H10, 49L20, 49L99, 90C39
Keywords:
contractive mappings of integral type; complete metric space; fixed point theorem; functional equation; dynamic programming; bounded solution1. Introduction
Throughout this paper, we assume that ℝ = (∞, + ∞), ℝ^{+ }= [0, + ∞), ℕ denotes the set of all positive integers, opt stands for sup or inf, Z and Y are Banach spaces, S ⊆ Z is the state space, D ⊆ Y is the decision space, B(S) denotes the Banach space of all bounded realvalued functions on S with norm
and
The famous Banach contraction principle is as follows.
Theorem 1.1. ([1]) Let f be a mapping from a complete metric space (X,d) into itself satisfying
where c ∈ (0, 1) is a constant. Then f has a unique fixed point a ∈ X such that lim_{n→∞ }f^{n }x = a for each x ∈ X.
It is well known that the Banach contraction principle has a lot of generalizations and various applications in many directions, see, for example, [230] and the references cited therein. In 1962, Rakotch [29] extended the Banach contraction principle with replacing the contraction constant c in (1.1) by a contraction function γ and established the result later.
Theorem 1.2. ([29]) Let f be a mapping from a complete metric space (X, d) into itself satisfying
where γ : ℝ^{+ }→ [0,1) is monotonically decreasing. Then f has a unique fixed point a ∈ X such that lim_{n→∞ }f^{n}x = a for each x ∈ X.
In 1968, Kannan [12] generalized the Banach contraction principle from continuous mappings to noncontinuous mappings and proved the following fixed point theorem.
Theorem 1.3. ([12]) Let f be a mapping from a complete metric space (X, d) into itself satisfying
where is a constant. Then f has a unique fixed point in X.
In 2002, Branciari [8] gave an integral version of the Banach contraction principles and showed the following fixed point theorem.
Theorem 1.4. ([8]) Let f be a mapping from a complete metric space (X,d) into itself satisfying
where c ∈ (0, 1) is a constant and φ ∈ Φ. Then f has a unique fixed point a ∈ X such that lim_{n→∞}f^{n}x = a for each x ∈ X.
In recent years, there has been increasing interest in the study of fixed points and common fixed points of mappings satisfying contractive conditions of integral type. The authors [2,3,911,28,30] and others continued the study of Branciari. In 2006, Aliouche [2] proved a fixed point theorem using a general contractive condition of integral type in symmetric spaces. In 2007, Djoudi and Aliouche [9] obtained common fixed point theorems of Gregus type for two pairs of weakly compatible mappings satisfying contractive conditions of integral type, and Suzuki [30] proved that Theorem 1.4 previously is a corollary of the MeirKeeler fixed point theorem and that the MeirKeeler contractions of integral type are still MeirKeeler contractions. In 2009, Pathak [28] bore out a general common fixed point theorem of integral φtype for two pairs of weakly compatible mappings satisfying certain integral type implicit relations in symmetric spaces, and Jachymski [10] testified that most contractive conditions of integral type given recently by many authors coincide with classical ones and got a new contractive condition of integral type which is independent of classical ones. However, to the best of our knowledge, the concrete examples constructed in [8,10], which guarantee the existence of fixed points for the contractive mappings of integral type in complete metric spaces, include at most countably many points.
On the other hand, by using various fixed point theorems, the authors [47,1326] studied the existence, uniqueness and iterative approximations of solutions, coincidence solutions and nonnegative solutions for the functional equations arising in dynamic programming below
where x and y signify the state and decision vectors, respectively, T represents the transformation of the process, and f(x) denotes the optimal return function with the initial state x.
The purposes of this paper are both to study the existence, uniqueness and iterative approximations of fixed points for three classes of contractive mappings of integral type, respectively, under different from or weaker than the conditions in [13,811,28,30], to construct four examples with uncountably many points to show the superiority of the results presented in this paper and to show solvability of the functional Equation (1.7) in B(S). Our results improve essentially Theorems 1.11.4.
2. Lemmas
The following lemmas play important roles in this paper.
Lemma 2.1. Let φ ∈ Φ and {r_{n}}_{n∈ℕ }be a nonnegative sequence with lim_{n→∞}r_{n }= a. Then
The proof of Lemma 2.1 follows from Remark 2.1 in [27].
Lemma 2.2. Let φ ∈ Φ and {r_{n}}_{n∈ℕ }be a nonnegative sequence. Then if and only if lim_{n→∞}r_{n }= 0.
The proof of Lemma 2.2 follows by Lemma 2.1 in [27].
Lemma 2.3. ([18]) Let E be a set, p and q :E → ℝ be mappings. If opt_{y∈E }p(y) and opt_{y∈E }q(y) are bounded, then
3. Fixed point theorems for contractive mappings of integral type
In this section, we show the existence, uniqueness and iterative approximations of fixed points for three classes of contractive mappings of integral type. For each x ∈ X and n ≥ 0, put d_{n }= d(f^{n}x, f^{n+}^{1}x).
Theorem 3.1. Let f be a mapping from a complete metric space (X,d) into itself satisfying
where φ ∈ Φ and α : ℝ^{+ }→ [0, 1) is a function with
Then f has a unique fixed point a ∈ X such that for each x ∈ X, lim_{n→∞ }f^{n}x = a.
Proof. Let x be an arbitrary point in X. It follows from (3.1) and (3.2) that
Now, we show that
Suppose that (3.4) does not hold. That is, there exists some n_{0 }∈ ℕ satisfying
Since φ ∈ Φ, it follows from (3.2), (3.3) and (3.5) that
which means that
which is a contradiction and hence (3.4) holds. Note that (3.4) yields that the sequence {d_{n}}_{n∈ℕ }is nonincreasing, which implies that there exists a constant c with lim_{n→∞ }d_{n }= c ≥ 0.
Next, we show that c = 0. Otherwise c > 0. Taking upper limit in (3.3) and using (3.2), Lemma 2.1 and φ ∈ Φ, we conclude that
which is absurd. Therefore, c = 0, that is,
Now, we claim that {f^{n}x}_{n∈ℕ }is a Cauchy sequence. Suppose that {f^{n}x}_{n∈ℕ }is not a Cauchy sequence, which means that there is a constant ε > 0 such that for each positive integer k, there are positive integers m(k) and n(k) with m(k) > n(k) > k such that
For each positive integer k, let m(k) denote the least integer exceeding n(k) and satisfying the above inequality. It follows that
Note that ∀k ∈ ℕ
In light of (3.6)(3.8), we conclude that
In view of (3.1), we deduce that
Taking upper limit in (3.10) and by virtue of (3.2), (3.9), Lemma 2.1 and φ ∈ Φ, we get that
which is a contradiction. Thus, {f^{n}x}_{n∈ℕ }is a Cauchy sequence. Since (X, d) is a complete metric space, there exists a point a ∈ X such that lim_{n→∞ }f^{n}x = a. By (3.1), (3.2) and Lemma 2.2, we arrive at
which yields that
which together with Lemma 2.2 gives that lim_{n→∞ }d(f^{n+1}x, fa) = 0. Consequently, we conclude immediately that
which means that a = fa.
Finally, we prove that f has a unique fixed point in X. Suppose that f has another fixed point b ∈ X\{a}. It follows from φ ∈ Φ, (3.2) and (3.3) that
which is a contradiction. This completes the proof.
Theorem 3.2. Let f be a mapping from a complete metric space (X, d) into itself satisfying
where φ ∈ Φ and α, β : ℝ^{+ }→ [0, 1) are two functions with
Then f has a unique fixed point a ∈ X such that for each x ∈ X, lim_{n→∞ }f^{n}x = a.
Proof. Let x be an arbitrary point in X. By (3.11), we obtain that
which together with (3.12) yields that
As in the proof of Theorem 3.1, we conclude similarly that the sequence {d_{n}}_{n∈ℕ }is nonincreasing and converges to 0.
Next, we show that {f^{n}x}_{n∈ℕ }is a Cauchy sequence. Suppose that {f^{n}x}_{n∈ℕ }is not a Cauchy sequence. It follows that there is a constant ε > 0 such that for each positive integer k, there are positive integers m(k) and n(k) with m(k) > n(k) > k with
For each positive integer k, let m(k) denote the least integer exceeding n(k) and satisfying the above inequality. It is easy to verify that (3.7)(3.9) hold. By means of (3.9), (3.11), (3.12), Lemma 2.1 and φ ∈ Φ, we get that
which is a contradiction. Hence, {f^{n}x}_{n∈ℕ }is a Cauchy sequence. Since (X, d) is a complete metric space, there exists a point a ∈ X such that lim_{n→∞ }f^{n}x = a, which means that lim_{n→∞ }d(f^{n+1}x, fa) = d(a, fa). If d(a, fa) ≠ 0, by (3.11), (3.12) and Lemma 2.1, we infer that
which is impossible. Thus, d(a, fa) = 0. That is, a = fa.
Finally, we prove that f has a unique fixed point in X. Suppose that f has another fixed point b ∈ X\{a}. It follows from φ ∈ Φ and (3.12) that
which is a contradiction. This completes the proof.
As in the proof of Theorem 3.2, we get similarly the below result.
Theorem 3.3. Let f be a mapping from a complete metric space (X, d) into itself satisfying
where φ ∈ Φ and is a function with
Then f has a unique fixed point a ∈ X such that for each x ∈ X, lim_{n→∞ }f^{n}x = a.
4. Remarks and illustrative examples
In this section, by constructing four nontrivial examples with uncountably many points, we discuss and compare the fixed point theorems obtained in Section 3 with the known results in Section 1.
Remark 4.1. If α(t) = c for all t ∈ ℝ^{+}, where c ∈ (0,1) is a constant, then Theorem 3.1 changes into Theorem 1.4; furthermore, if φ(t) = 1 for all t ∈ ℝ^{+}, then Theorem 3.1 brings Theorem 1.1. The following example manifests that Theorem 3.1 extends substantially Theorems 1.1 and 1.4.
Example 4.1. Let X = ℝ^{+ }be endowed with the Euclidean metric d =  · , f: X → X, α: ℝ^{+ }→ [0,1) and φ ∈ Φ be defined by
and
It is obvious that (3.2) holds and
That is, the conditions of Theorem 3.1 are fulfilled. It follows from Theorem 3.1 that f has a unique fixed point 0 ∈ X. But, we can neither invoke Theorem 1.1 nor Theorem 1.4 to show the existence of a fixed point of f in X because (1.1) and (1.4) do not hold.
Suppose that (1.1) holds. It follows that there exists a constant c ∈ (0,1) satisfying
which gives that
which yields that c ≥ 1, which is absurd.
Suppose that (1.4) holds. It follows that there exists some constant c ∈ (0,1) satisfying
which yields that
which means that
which is a contradiction.
Remark 4.2. In case φ(t) = 1 for all t ∈ ℝ^{+}, then Theorem 3.1 reduces to a result, which generalizes Theorem 1.2. The following example reveals that Theorem 3.1 is a proper generalization Theorem 1.2.
Example 4.2. Let X = ℝ^{+ }be endowed with the Euclidean metric d =  · , f: X → X, α: ℝ^{+}→ [0,1) and φ ∈ Φ be defined by
and
It is easy to see that (3.2) holds. In order to verify (3.1), we have to consider three possible cases as follows:
Case 1. x, y ∈ X with x = y. It is clear that
Case 2. x, y ∈ X with 0 < x  y ≤ 1. Note that
Case 3. x,y ∈ X with x  y > 1. It follows that
Hence, (3.1) holds. Consequently, the conditions of Theorem 3.1 are satisfied.
It follows from Theorem 3.1 that f has a unique fixed point .
However, Theorem 1.2 is useless in guaranteeing the existence of a fixed point of f in X. Otherwise, suppose that the conditions of Theorem 1.2 are fulfilled. Notice that γ: ℝ^{+ }→ [0,1) is monotonically decreasing. It follows that lim_{t→∞+ }γ(t) exists and belongs to [γ(1), γ(0)] ⊂ [0,1). Using (1.2), we infer that
which implies that
which yields that
which is impossible.
Remark 4.3. In case φ(t) = 1 and γ(t) = h for all t ∈ ℝ^{+}, then Theorem 3.3 comes into being Theorem 1.3. The below example demonstrates that Theorem 3.3 is indeed a proper extension of Theorem 1.3.
Example 4.3. Let X = [0, 4] be endowed with the Euclidean metric and φ ∈ Φ be defined by
and
respectively. It is obvious that (3.14) holds and (3.13) is equivalent to
Note that x and y in (4.1) are symmetric, (4.1) holds for all x = y ∈ X and
In order to verify (3.13), by (4.1) and (4.2) we need only to show that
Now, we have to consider the below six possible cases:
Case 1. x, y ∈ X with 4 ≥ x > y ≥ 2. It follows that
Case 2. x, y ∈ X with 4 ≥ x ≥ 2 > y ≥ 0 and y ≠ 1. It is clear that
Case 3. x; y ∈ X with 4 ≥ x ≥ 2 and y = 1. It follows that
Case 4. x, y ∈ X with 2 > x > y ≥ 0, x ≠ 1 and y ≠ 1. Notice that
Case 5. x, y ∈ X with x = 1 > y ≥ 0. Obviously
Case 6. x, y ∈ X with 2 > x > 1 = y. Notice that
Hence, (3.13) holds. That is, the conditions of Theorem 3.3 are satisfied. It follows from Theorem 3.3 that f has a unique fixed point in X. However, it is easy to verify that for x_{0 }= 1 and y_{0 }= 0
which yields that (1.3) in Theorem 1.3 does not hold.
Next, we construct an example with uncountably many points to explain Theorem 3.2.
Example 4.4. Let X = [1,3] be endowed with the Euclidean metric d = ; · , f: X → X, α, β: ℝ^{+ }→ [0,1) and φ ∈ Φ be defined by
and
It is easy to see that (3.12) holds. In order to verify (3.11), we have to consider the below five possible cases:
Case 1. x, y ∈ X with 3 ≥ x ≥ y ≥ 2. Note that
Case 2. x, y ∈ X with x ∈ [2,3] and y ∈ [1, 2). It follows that
Case 3. x, y ∈ X with x, y ∈ [1, 2). Notice that fx = fy = 1. It follows that
Case 4. x, y ∈ X with 3 ≥ y > x ≥ 2. Note that
Case 5. x, y ∈ X with x ∈ [1, 2) and y ∈ [2,3]. Note that
that is, (3.11) holds. Thus, all the conditions of Theorem 3.2 are satisfied. It follows from Theorem 3.2 that f has a unique fixed point 1 ∈ X.
5. Applications
In this section, by using the fixed point theorems obtained in Section 3, we study solvability of the functional Equation (1.7) in B(S).
Theorem 5.1. Let u: S × D → ℝ, T : S × D → S, H : S × D × ℝ → ℝ, φ ∈ Φ and α : ℝ^{+ }→ [0, 1) satisfy (3.2),
and
Then the functional Equation (1.7) has a unique solution w ∈ B(S) and {A^{n}z}_{n∈ℕ }converges to w for each z ∈ B(S), where the mapping A is defined by
Proof. It follows from (5.1) that there exists M > 0 satisfying
It is easy to see that A is a selfmappings in B(S) by (5.3), (5.4) and Lemma 2.3.
Using Theorem 12.34 in [31] and φ ∈ Φ, we conclude that for each ε > 0, there exists δ > 0 satisfying
where m(C) denotes the Lebesgue measure of C.
Let x ∈ S,h,g ∈ B(S). Suppose that opt_{y∈D }= inf_{y∈D}. Clearly, (5.3) implies that there exist y, z ∈ D satisfying
Put
It is easy to verify that
and
which yield that
Similarly, we infer that (5.6) holds also for opt_{y∈D }= sup_{y∈D}. Combining (5.2), (5.5) and (5.6), we arrive at
which means that
letting ε → 0^{+ }in the above inequality, we deduce that
Thus, Theorem 5.1 follows from Theorem 3.1. This completes the proof.
Remark 5.1. Theorem 5.1 extends and unifies Theorem 2.1 in [7], Theorem 3.1 in [18] and Theorem 3.2 in [25].
Theorem 5.2. Let u : S × D → ℝ, T: S × D → S, H : S × D × ℝ → ℝ, φ ∈ Φ and α, β : ℝ^{+ }→ [0, 1) satisfy (3.12), (5.1) and
Then the functional Equation (1.7) has a unique solution w ∈ B(S) and {A^{n}z}_{n∈ℕ }converges to w for each z ∈ B(S), where the mapping A is defined by (5.3).
Proof. As in the proof of Theorem 5.1, by (3.12), (5.1), (5.3) and (5.7), we conclude that (5.4)(5.6) hold and
which yields that
letting ε→ 0^{+ }in the above inequality, we infer that
Thus, Theorem 5.2 follows from Theorem 3.2. This completes the proof.
Theorem 5.3. Let u : S × D → ℝ, T : S × D → S, H:S × D × ℝ → ℝ, φ ∈ Φ and satisfy (3.14), (5.1) and
Then the functional Equation (1.7) has a unique solution w ∈ B(S) and {A^{n}z}_{n∈ℕ }converges to w for each z ∈ B(S), where the mapping A is defined by (5.3).
Proof. As in the proof of Theorem 5.1, by (3.14), (5.1), (5.3) and (5.8), we conclude that (5.4)(5.6) hold and
which yields that
letting ε → 0^{+ }in the above inequality, we get that
Thus, Theorem 5.3 follows from Theorem 3.3. This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are indebted to the referees for their helpful comments.
References

Banach, A: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math. 3, 133–181 (1922)

Aliouche, A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type. J Math Anal Appl. 322, 796–802 (2006). Publisher Full Text

Altun, I, Türkoğlu, D: Some fixed point theorems for weakly compatible mappings satisfying an implicit relation. Taiwanese J Math. 13, 1291–1304 (2009)

Bellman, R: Dynamic Programming. Princeton University Press, Princeton (1957)

Bellman, R, Lee, ES: Functional equations arising in dynamic programming. Aequationes Math. 17, 1–18 (1978). Publisher Full Text

Bhakta, PC, Choudhury, SR: Some existence theorems for functional equations arising in dynamic programming II. J Math Anal Appl. 131, 217–231 (1988). Publisher Full Text

Bhakta, PC, Mitra, S: Some existence theorems for functional equations arising in dynamic programming. J Math Anal Appl. 98, 348–362 (1984). Publisher Full Text

Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int J Math Math Sci. 29, 531–536 (2002). Publisher Full Text

Djoudi, A, Aliouche, A: Common fixed point theorems of Greguš type for weakly compatible mappings satisfying contractive conditions of integral type. J Math Anal Appl. 329, 31–45 (2007). Publisher Full Text

Jachymski, J: Remarks on contractive conditions of integral type. Nonlinear Anal. 71, 1073–1081 (2009). Publisher Full Text

Kamran, T: MizoguchiTakahashi's type fixed point theorem. Comput Math Appl. 57, 507–511 (2009). Publisher Full Text

Kannan, R: Some results on fixed points. Bull Calcutta Math Soc. 60, 71–76 (1968)

Liu, Z: Coincidence theorems for expansion mappings with applications to the solutions of functional equations arising in dynamic programming. Acta Sci Math (Szeged). 65, 359–369 (1999)

Liu, Z: Compatible mappings and fixed points. Acta Sci Math (Szeged). 65, 371–383 (1999)

Liu, Z: Existence theorems of solutions for certain classes of functional equations arising in dynamic programming. J Math Anal Appl. 262, 529–553 (2001). Publisher Full Text

Liu, Z, Agarwal, RP, Kang, SM: On solvability of functional equations and system of functional equations arising in dynamic programming. J Math Anal Appl. 297, 111–130 (2004). Publisher Full Text

Liu, Z, Kang, SM: Properties of solutions for certain functional equations arising in dynamic programming. J Global Optim. 34, 273–292 (2006). Publisher Full Text

Liu, Z, Kang, SM: Existence and uniqueness of solutions for two classes of functional equations arising in dynamic programming. Acta Math Appl Sini (English Series). 23, 195–208 (2007). Publisher Full Text

Liu, Z, Kang, SM, Ume, JS: Solvability and convergence of iterative algorithms for certain functional equations arising in dynamic programming. Optimization. 59, 887–916 (2010). Publisher Full Text

Liu, Z, Ume, JS: On properties of solutions for a class of functional equations arising in dynamic programming. J Optim Theory Appl. 117, 533–551 (2003). Publisher Full Text

Liu, Z, Ume, JS, Kang, SM: Some existence theorems for functional equations arising in dynamic programming. J Korean Math Soc. 34, 11–28 (2006)

Liu, Z, Ume, JS, Kang, SM: On properties of solutions for two functional equations arising in dynamic programming. Fixed Point Theory Appl. 19, (2010) Article ID 905858

Liu, Z, Ume, JS, Kang, SM: Some existence theorems for functional equations and system of functional equations arising in dynamic programming. Taiwanese J Math. 14, 1517–1536 (2010)

Liu, Z, Wang, LL, Kim, HG, Kang, SM: Common fixed point theorems for contractive type mappings and their applications in dynamic programming. Bull Korean Math Soc. 45, 573–585 (2008). Publisher Full Text

Liu, Z, Xu, YG, Ume, JS, Kang, SM: Solutions to two functional equations arising in dynamic programming. J Comput Appl Math. 192, 251–269 (2006). Publisher Full Text

Liu, Z, Zhao, LS, Kang, SM, Ume, JS: On solvability of a functional equation. Optimization. 60, 365–375 (2011). Publisher Full Text

Mocanu, M, Popa, V: Some fixed point theorems for mappings satisfying implicit relations in symmetric spaces. Libertas Math. 28, 1–13 (2008)

Pathak, HK: Integral Φtype contractions and existence of continuous solutions for nonlinear integral inclusions. Nonlinear Anal. 71, e2577–e2591 (2009). Publisher Full Text

Rakotch, E: A note on contractive mappings. Proc Amer Math Soc. 13, 459–465 (1962). Publisher Full Text

Suzuki, T: MeirKeeler contractions of integral type are still MeirKeeler contractions. Int J Math Math Sci. 6, (2007) Article ID 39281

Hewitt, E, Stromberg, K: Real and Abstract Analysis. Springer, Berlin (1978)