Fuzzy relation-theoretic contraction principle

1 Introduction

The definition of a fuzzy set was first formulated by Zadeh [27]. This notion has been used extensively in topology as well as modern analysis by many researchers. The idea of fuzzy metric spaces was first given by Kramosil and Michalek [16] in 1975. Thereafter, in order to have Hausdorffness property, George and Veeramani [7] adjusted the definition of fuzzy metric spaces given in [16]. The fuzzy fixed point theory was started by Grabiec [9] in 1988 wherein he introduced the concepts of G-Cauchy sequences, G-complete fuzzy metric spaces and provide a fuzzy metric version of Banach contraction. After that several fixed point results have been proven in the setting of fuzzy metric spaces. The above mentioned concept of G-completeness is not a very natural notion as even ℝ is not complete in this sense. In this quest, in 1994, George and Veeramani [7] slightly modified the concepts of fuzzy metric spaces and Cauchy sequences therein, namely M-Cauchy sequences. Also, the authors in [7] found a Hausdorff topology in their new defined fuzzy metric spaces. In 2002, Gregori and Sapena [10] defined fuzzy contractive mappings and proved a very natural extension of the Banach theorem for such mappings in G-complete as well as M-complete fuzzy metric spaces. Relation-theoretic fixed point theory is a relatively new direction of fixed point theory. This direction was initiated by Turinici [26] and it becomes very active area after the existence of the well-known results due to Ran and Reurings [23] and Nieto and Lopez [21, 22] with their interesting applications to boundary value problems and matrix equations. Recently, there are several researchers working in this direction (e.g. Samet and Turinici [25], Ben-El-Mechaiekh [5], Bhaskar and Lakshmikantham [6], Imdad et al. [1–3, 24] and several others).

The aim of this manuscript can be described as follows:

to prove fuzzy relation-theoretic contraction principle;

2 Preliminaries

Now, we recall some relative notions and basic results as under. From now on, E stands for a non-empty set, f a self mapping on E and Fix (f) = {c ∈ E : c = f (c)}.

Definition 2.1. [16] A continuous t-norm * is a continuous binary operation * : [0, 1] × [0, 1] → [0, 1] which is commutative, associative and satisfies:

t * 1 = t ∀ t ∈ [0, 1];

Kramosil and Michalek [16] defined fuzzy metric spaces as under.

Definition 2.2. [16] Let ℵ be a fuzzy set on E² × [0, ∞) and * a continuous t-norm. Assume that (∀ c, d, g ∈ E and t, s > 0):

(KM-i)  ℵ (c, d, 0) =0;

(KM-ii)  ℵ (c, d, t) =1 iff c = d;

(KM-iii)  ℵ (c, d, t) = ℵ (d, c, t);

(KM-iv)  ℵ (c, d, t) * ℵ (d, g, s) ≤ ℵ (c, g, t + s);

(KM-v)  ℵ (c, d, .) : [0, ∞) → [0, 1] is left continuous.

Then (E, ℵ , *) is called a fuzzy metric space (Kramosil and Michalek’s sense). Definition 2.3. If we replace the axiom (KM-iv) by:

(KM-iv)^′   ℵ (c, d, t) * ℵ (d, g, s) ≤ ℵ (c, g, max {t, s}) ∀ c, d, g ∈ E and t, s > 0, then (E, ℵ , *) is known as a non-Archimedean fuzzy metric space. Remark 2.1. [16] For all c, d ∈ E, ℵ (c, d, .) is a non-decreasing mapping. The topology of a fuzzy metric space (Kramosil and Michalek’s sense) is not Hausdorff in general. In order to have Hausdorffness property, George and Veeramani [7, 8] slightly modified the definition of fuzzy metric spaces such that the topology of the newly defined fuzzy metric space becomes Hausdorff.

Definition 2.4. [7, 8] Let ℵ be a fuzzy set on E² × (0, ∞) and * a continuous t-norm. Assume that (∀ c, d, g ∈ E and t, s > 0):

(GV-i)  ℵ (c, d, t) >0;

(GV-ii)  ℵ (c, d, t) =1 iff c = d;

(GV-iii)  ℵ (c, d, t) = ℵ (d, c, t);

(GV-iv)  ℵ (c, d, t) * ℵ (d, g, s) ≤ ℵ (c, g, t + s);

(GV-v)  ℵ (c, d, .) : (0, ∞) → [0, 1] is continuous.

Then (E, ℵ , *) is called a fuzzy metric space (George and Veeramani’s sense). Remark 2.2. [7] The topology of a fuzzy metric space in the sense of Definition 2.4 is Hausdorff.

Remark 2.3. [7] Every fuzzy metric space in the sense of Definition 2.4 is a fuzzy metric space in the sense of Definition 2.2, the converse is not true in general.

From now on, (E, ℵ , *) stands for a fuzzy metric space in the sense of Definition 2.4.

Definition 2.5. [7–9] A sequence {c _n } ⊆ E is said to be

(a) convergent to c if ∀ ε > 0 and t > 0, ∃ N ∈ ℕ satisfying ℵ ( c n , c , t ) > 1 - ε ∀ n ≥ N . (b) M-Cauchy if ∀ ε > 0 and t > 0, ∃ N ∈ ℕ satisfying ℵ ( c n , c n + p , t ) > 1 - ε ∀ n ≥ N and p ∈ ℕ 0 . (c) G-Cauchy if lim n → ∞ ℵ ( c n + p , c n , t ) = 1 , ∀ t > 0 and p ∈ ℕ . Let (E, ℵ , *) be a fuzzy metric space. If every M-Cauchy (G-Cauchy) sequence in E is convergent in E, then E is said to be M-complete (G-complete).

George and Sapena [10] provided a fuzzy version of Banach contraction mappings as under.

Definition 2.6. [10] [Fuzzy Banach contraction mappings] A self-mapping f on (E, ℵ , *) is called a fuzzy contractive mapping if ∃λ ∈ (0, 1) satisfying 1 ℵ ( fc , fd , t ) - 1 ≤ λ ( 1 ℵ ( c , d , t ) - 1 ) ∀ c , d ∈ E . The authors in [10] proved a fuzzy version of Banach contraction principle as under.

Theorem 2.1. [10] Every fuzzy contractive mapping defined on a G-complete fuzzy metric space possesses a unique fixed point.

Now, we recall some relation theoretic notions as under.

Definition 2.7. [17] A subset R of E² is called a binary relation on E. If ( c , d ) ∈ R (sometimes we write c R d ), then we say that “c is related to d under R ". If either c R d or d R c , then we write [ c , d ] ∈ R . Observe that E² is a binary relation on E called the universal relation. In this presentation, R refers for a non-empty binary relation on E.

Definition 2.8. (see [1, 19]) A binary relation R on a non-empty set E is said to be:

(i) reflexive if c R c ∀ c ∈ E;

(ii) transitive if c R d and d R g imply c R g ∀ c, d, g ∈ E;

(iii) antisymmetric if c R d and d R c imply c = d ∀ c, d ∈ E;

(iv) partial order if it is reflexive, antisymmetric and transitive;

(v) complete if [ c , d ] ∈ R ∀ c, d ∈ E;

(vi) f-closed if ( c , d ) ∈ R ⇒ ( fc , fd ) ∈ R ∀ c , d ∈ E . Definition 2.9. [1] A sequence {c _n } ⊆ E is called R -preserving if ( c n , c n + 1 ) ∈ R ∀ n ∈ ℕ . Now, we define R -preserving fuzzy contractive sequences as under.

Definition 2.10. A sequence {c _n } ⊆ E is said to be an R -preserving fuzzy contractive sequence if there exists λ ∈ (0, 1) such that (∀t > 0) 1 ℵ ( fc n + 1 , fc n + 2 , t ) - 1 ≤ λ ( 1 ℵ ( c n , c n + 1 , t ) - 1 ) ∀ n ∈ ℕ .

Example 2.1. Let E = (-1, ∞) and let * be the product t-norm given by t * s = ts ∀t, s ∈ [0, 1]. Define ℵ by (∀c, d ∈ E and t > 0) ℵ ( c , d , t ) = { 0 if t = 0 ; e - | c - d | t if t ≠ 0 . Define R on E by c R d ⇔ c , d ∈ [ 0 , 1 ] and c ≥ d . Then { 1 2 n } is an R -preserving fuzzy contractive sequence.

In what follows, we provide relation-theoretic versions of the fuzzy metrical notions: ℵ-self-closedness, convergence and completeness.

Definition 2.11. A binary relation R on E is said to be an ℵ-self-closed if given any convergent R -preserving sequence {c _n } ⊆ E which converges (in fuzzy sense) to some c ∈ E, ∃ {c _{n k} } ⊆ {c _n } with ( c n k , c ) ∈ R . Example 2.2. Let E = (0, 4] and * be the product t-norm given by t * s = ts ∀t, s ∈ [0, 1]. Define ℵ by (∀c, d ∈ E and t > 0) ℵ ( c , d , t ) = { 0 if t = 0 2 t 2 t + | c - d | if t ≠ 0 . Define R on E by R = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 1 , 4 ) , ( 2 , 4 ) } . Observe that if {c _n } is an R -preserving sequence which converges to some c ∈ E, then ∃ n 0 ∈ ℕ such that either c _n  = 1  ∀  n ≥ n₀ or c _n  = 2  ∀  n ≥ n₀. Therefore, { c n 0 + i } i ∈ ℕ 0 is a subsequence of {c _n } such that c n 0 + i R c for each i ∈ ℕ 0 . Hence, R is ℵ-self closed.

Definition 2.12. A sequence {c _n } ⊆ E is called

(i) R -M-Cauchy if c n R c n + 1 ∀ n ∈ ℕ 0 and ∀ ε > 0 and t > 0 ∃ n 0 ∈ ℕ satisfying ℵ ( c n , c n + p , t ) > 1 - ε ∀ n ≥ N and p ∈ ℕ 0 . (ii) R -G-Cauchy if c n R c n + 1 ∀ n ∈ ℕ 0 and lim n → ∞ ℵ ( c n , c n + p , t ) = 1 ∀ p ∈ ℕ , t > 0 . Remark 2.4. Every M-Cauchy (G-Cauchy) sequence is an R -M-Cauchy ( R -G-Cauchy) sequence, for any arbitrary binary relation R . R -M-Cauchyness ( R -G-Cauchyness) coincides with M-Cauchyness (G-Cauchyness) if R is taken to be the universal relation.

Definition 2.13. A fuzzy metric space (E, ℵ , *) which is endowed with a binary relation R is said to be R -M-complete ( R -G-complete) if every R -M-Cauchy ( R -G-Cauchy) sequence is convergent in E. Remark 2.5. Every M-complete (G-complete) fuzzy metric space is R -M-complete ( R -G-complete) fuzzy metric space, for any arbitrary binary relation R . R -M-completeness ( R -G-completeness) coincides with M-completeness (G-completeness) if R is taken to be the universal relation.

3 Main results

Firstly, we define R -fuzzy contractive mappings as under.

Definition 3.1. A self mapping f on (E, ℵ , *) is called an R -fuzzy contractive if ∃λ ∈ (0, 1) such that 1 ℵ ( fc , fd , t ) - 1 ≤ λ ( 1 ℵ ( c , d , t ) - 1 ) ∀ c , d ∈ E with ( c , d ) ∈ R . Remark 3.1. Under the universal relation, Definition 3.1 reduces to Definition 2.6. Now, we state and prove our first main result as under.

Theorem 3.1. Let (E, ℵ , *) be a fuzzy metric space (in the sense Definition 2.4), R a binary relation and f a self-mapping on E. Suppose that:

∃c₀ ∈ E with c 0 R fc 0 ;

Then f possesses a fixed point.

Proof. Define { c n } n ∈ ℕ ⊆ E by c n = f n c 0 ∀ n ∈ ℕ . If c _n  = c_n+1 for some n ∈ ℕ , then we are done, as c _n comes out to be a fixed point of f. Now, let us assume that c n ≠ c n + 1 ∀ n ∈ ℕ . Given that c 0 R fc 0 and we have c₁ = fc₀. Hence, c 0 R c 1 . As R is f-closed, using induction, we get c n R c n + 1 ∀ n ∈ ℕ . So, {c _n } is an R -preserving sequence. Since R is transitive, we obtain c n R c n + p ∀ n , p ∈ ℕ . Now, observe that for any t > 0 and c, d ∈ E with c R d , we have (forsome λ ∈ (0, 1)) 1 ℵ ( fc , fd , t ) - 1 ≤ λ ( 1 ℵ ( c , d , t ) - 1 ) . Thus, 1 ℵ ( c n , c n + p , t ) - 1 = 1 ℵ ( fc n - 1 , fc n + p - 1 , t ) - 1 ≤ λ ( 1 ℵ ( c n - 1 , c n + p - 1 , t ) - 1 ) Hence, {c _n } is an R -preserving fuzzy contractive sequence. Now, observe that 1 ℵ ( c n , c n + p , t ) - 1 ≤ λ ( 1 ℵ ( c n - 1 , c n + p - 1 , t ) - 1 ) ≤ λ 2 ( 1 ℵ ( c n - 2 , c n + p - 2 , t ) - 1 ) ⋮ ≤ λ n ( 1 ℵ ( c 0 , c p , t ) - 1 ) → 0 as n → ∞ . Therefore, for any t > 0, lim n → ∞ ℵ ( c n , c n + p , t ) = 1 , i.e., {c _n } is an R -G-Cauchy sequence. The R -G-completeness of E guarantees the existence of some c ∈ E such that lim n → ∞ ℵ ( c n , c , t ) = 1 ∀t > 0, i.e. {c _n } converges to c. Now, we show that c ∈ Fix (f). As f is continuous and {c _n } is a sequence converging to c, we have fc _n  → fc, i.e., c_n+1 → fc. Due to the Hausdorffness property of (E, ℵ , *), we conclude that c = fc (in view of the uniqueness of the limit). Therefore, c ∈ Fix (f). This accomplishes the required. In the coming example, we exhibit the usefulness of Theorem 3.1.

Example 3.1. Let E = (-1, ∞) and let * be the product t-norm given by t * s = ts ∀t, s ∈ [0, 1]. Define ℵ by (∀c, d ∈ E and t > 0) ℵ ( c , d , t ) = { 0 if t = 0 ; e - | c - d | t if t ≠ 0 . Define f on E by f ( c ) = { c , if c ∈ ( - 1 , 0 ] ; c 2 , if c ∈ [ 0 , 1 ] ; e c - 1 - 1 2 , otherwise . Define R on E by c R d ⇔ c , d ∈ [ 0 , 1 ] and c ≥ d . Clearly, (E, ℵ , *) is R -G-complete and R is transitive on E. Notice that 1 2 ∈ E such that 1 2 R f ( 1 2 ) as f ( 1 2 ) = 1 4 . Also R is f-closed as for any c, d ∈ [0, 1] with c ≥ d, we have c 2 , d 2 ∈ [ 0 , 1 ] and c 2 ≥ d 2 , i.e., fc R fd whenever c R d . Now, for any c, d ∈ E with c R d , we have 1 ℵ ( fc , fd , t ) - 1 = e | c - d | 2 t - 1 and 1 ℵ ( c , d , t ) - 1 = e | c - d | t - 1 . We observe that the following holds for any λ ∈ ( 1 e , 1 ) : 1 ℵ ( fc , fd , t ) - 1 ≤ λ ( 1 ℵ ( c , d , t ) - 1 ) , so that f is an R -fuzzy contractive mapping. Also, clearly f is continuous. Thus, the hypotheses of Theorem 3.1 hold true. Therefore, f admits a fixed point in E (Fix (f) = (-1, 0]). Observe that Theorem 2.1 can not be applied in the context of Example 3.1 since f is not fuzzy contractive mapping and the space (E, ℵ , *) is not G-complete.

Now, we provide an analogous of Theorem 3.1 as follows:

Theorem 3.2. Theorem 3.1 remains valid if we replace the assumption (v) by:

(v′) R is ℵ-self-closed.

Proof. Using similar arguments as in Theorem 3.1, it can be proven that {c _n } is an R -preserving sequence which converges to c ∈ E. Due to (v′), ∃ {c _{n k} } ⊆ {c _n } with c n k R c ∀ k ∈ ℕ . On using the condition (iv), we have (for any t > 0) 1 ℵ ( c n k + 1 , fc , t ) - 1 ≤ λ ( 1 ℵ ( c n k , c , t ) - 1 ) → 0 as k → ∞ .

Thus, we get lim k → ∞ ℵ ( c n k + 1 , fc , t ) = 1 , i.e., c _{n k}  → fc as k→ ∞. As the space enjoys the Hausdorffness property, the limit must be unique. Therefore, we conclude that c = fc (since c _n  → c as n→ ∞, c _{n k}  → fc as k→ ∞ and {c _{n k} } ⊆ {c _n }). In the coming example, we show the usefulness of Theorem 3.2.

Example 3.2. Let E = (0, 4] and * be the product t-norm given by t * s = ts ∀t, s ∈ [0, 1]. Define ℵ by (∀c, d ∈ E and t > 0) ℵ ( c , d , t ) = { 0 if t = 0 2 t 2 t + | c - d | if t ≠ 0 . Define R on E by R = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 1 , 4 ) , ( 2 , 4 ) } . Also, define f on E as f ( c ) = { 1 , if 0 < c ≤ 2 , 2 , if 2 < c ≤ 4 . Observe that:

1 ∈ E and 1 R f 1 ;

Case I: let (c, d) = (1, 4). Then for any t > 0 1 ℵ ( fc , fd , t ) - 1 = 1 ℵ ( 1 , 2 , t ) - 1 = 2 t + | 1 - 2 | 2 t - 1 = 1 2 t

and

1 ℵ ( c , d , t ) - 1 = 1 ℵ ( 1 , 4 , t ) - 1 = 2 t + | 1 - 4 | 2 t - 1 = 3 2 t .

For λ = 2 3 , we have 1 2 t ≤ λ ( 3 2 t ) so that 1 ℵ ( fc , fd , t ) - 1 ≤ λ ( 1 ℵ ( c , d , t ) - 1 ) .

Case II: if (c, d) = (2, 4), then

1 ℵ ( fc , fd , t ) - 1 = 1 ℵ ( 1 , 2 , t ) - 1 = 2 t + | 1 - 2 | 2 t - 1 = 1 2 t

and

1 ℵ ( c , d , t ) - 1 = 1 ℵ ( 2 , 4 , t ) - 1 = 2 t + | 2 - 4 | 2 t - 1 = 1 t .

For λ = 2 3 , we have 1 2 t ≤ λ ( 1 t ) so that 1 ℵ ( fc , fd , t ) - 1 ≤ λ ( 1 ℵ ( c , d , t ) - 1 ) . Thus, we have λ ( = 2 3 ) ∈ ( 0 , 1 ) such that inequality (3.1) holds ∀c, d ∈ E such that ( c , d ) ∈ R . Hence, f is R -fuzzy contractive. Now, if {c _n } is an R -preserving sequence which converges to some c ∈ E, then ∃ n 0 ∈ ℕ such that either c _n  = 1  ∀  n ≥ n₀ or c _n  = 2  ∀  n ≥ n₀. Therefore, { c n 0 + i } i ∈ ℕ 0 is a subsequence of {c _n } such that c n 0 + i R c for each i ∈ ℕ 0 . Hence, R is ℵ-self closed.

Thus, the requirements of Theorem 3.2 are fulfilled. Therefore, f possesses a fixed point in E (namely: c = 1). Now, we provide a correspondence uniqueness result, which runs as under.

Theorem 3.3. In addition to the assumptions of Theorem 3.1 (or Theorem 3.2), if

(vi) ∀c, d ∈ Fix (f), ∃g ∈ E such that c R g and d R g , then the fixed point of f is unique.

Proof. Let c, d ∈ Fix (f). In view of the condition (vi), there is some g ∈ E such that c R g and d R g . Define fg₀ = g₁ and g_n+1 = fg _n for all n ≥ 0. Our claim is that g _n  → c and g _n  → d as n→ ∞. As c R g 0 and f is R -fuzzy contractive, we get

1 ℵ ( c , g 1 , t ) - 1 = 1 ℵ ( fc , fg 0 , t ) - 1 ≤ λ ( 1 ℵ ( c , g 0 , t ) - 1 ) .

As R is f-closed, we have fc R fg 0 , i.e., c R g 1 . Thus, we have 1 ℵ ( c , g 2 , t ) - 1 = 1 ℵ ( fc , fg 1 , t ) - 1 ≤ λ ( 1 ℵ ( c , g 1 , t ) - 1 ) ≤ λ 2 ( 1 ℵ ( c , g 0 , t ) - 1 ) . By induction, we get 1 ℵ ( c , g n , t ) - 1 ≤ λ n ( 1 ℵ ( c , g 0 , t ) - 1 ) → 0 as n → ∞ .

Therefore, lim n → ∞ ℵ ( c , g n , t ) = 1 , i.e., {g _n } converges to c. Similarly, we can show that {g _n } converges to d. Due to the Hausdorffness property of (E, ℵ , *), we conclude that c = d (in view of the uniqueness of the limit). Hence, the fixed point of f is unique. As required. Remark 3.3. Under the universal relation, Theorem 3.5 reduces to Theorem 2.1.

Example 3.3. The fixed point of the mapping f defined in Example 3.2 is unique and in fact Theorem 3.5 is applicable in the context of Example 3.2.

By considering a cyclic contractive condition, Kirk et al. [15] obtained a new extension of the celebrated Banach contraction principle as given in the following theorem:

Theorem 3.4. [15] Let H₁ and H₂ be two non-empty closed subsets of a complete metric space (E, d) and f : E → E. Assume that the following conditions are satisfied:

f (H₁) ⊆ H₂ and f (H₂) ⊆ H₁;

Inspired by Theorem 3.4, we define cyclic fuzzy contractive mappings as under.

Theorem 3.5. Let H₁ and H₂ be two non-empty subsets of a G-complete fuzzy metric space (E, ℵ , *) such that E = H₁ ∪ H₂ and f be a self mapping on E. If

f (H₁) ⊆ H₂ and f (H₂) ⊆ H₁;

Proof. Define R on E by c R d ⇔ ( c , d ) ∈ ( H 1 × H 2 ) ∪ ( H 2 × H 1 ) .(3.2) Clearly, R is transitive. Given that H₁ is non-empty, so let c₀ ∈ H₁. In view of (i) we have fc₀ ∈ H₂ and thus we conclude that c 0 R fc 0 . Now, for any ( c , d ) ∈ R , either (c, d) ∈ H₁ × H₂ or (c, d) ∈ H₂ × H₁. So (fc, fd) ∈ H₂ × H₁ or (fc, fd) ∈ H₁ × H₂. Therefore, R is f-closed. Due to (ii) and (3.2), f is R -fuzzy contractive mapping. Also, given that E is G-complete so that it is R -G-complete. Now, we show that R is ℵ-self-closed. Let {c _n } ⊆ E be an R -preserving sequence converging to some c ∈ E. Let U = { k ∈ ℕ 0 | ( c k , c k + 1 ) ∈ H 1 × H 2 } and V = { k ∈ ℕ 0 | ( c k , c k + 1 ) ∈ H 2 × H 1 } . Observe that U ∪ V = ℕ 0 , i.e., at least one of U and V is infinite. Assume that U is infinite. Then U can be written as follows: U = { k i | i ∈ ℕ } , where { k i } is a strictly increasing sequence , so it converges to infinity . Let m _i  = k _i  + 1, then {m _i } is also a strictly increasing sequence, so it is also converging to infinity.

Observe that the two subsequences {c _{k i} } and {c _{m i} } have the following properties:

both converge to c,

As E = H₁ ∪ H₂, either c ∈ H₁ or c ∈ H₂. If c ∈ H₁, then (c _{m i} , c) ∈ H₂ × H₁. On the other hand, if c ∈ H₂, then (c _{k i} , c) ∈ H₁ × H₂. So, in both the possible cases we get a subsequence of {c _n } such that each element of that subsequence is related to c. Thus, we conclude that R is ℵ-self-closed. Hence, the requirements of Theorem 3.1 are fulfilled. Thus, f possesses a fixed point in E, say c. Since c ∈ E = H₁ ∪ H₂, we must have either c ∈ H₁ or c ∈ H₂. If c ∈ H₁, then f (c) ∈ H₂ (due to the condition (i)). As f (c) = c, we have c ∈ H₂. Similarly, if we assume that c ∈ H₂, then we have c = f (c) ∈ H₁ (again due to the condition (i)). Therefore, c ∈ H₁ ∩ H₂. Finally, if c, d ∈ Fix (f), then c, d ∈ H₁ ∩ H₂. Therefore, c R d and c R c . Hence, the condition (vi) of Theorem 3.5 guarantees that the fixed point of f is unique. As required.

Remark 3.3. The mapping f in Theorem 3.5 is called cyclic fuzzy contractive mapping.

Remark 3.4. Similar results can be proved for R -M-complete fuzzy metric spaces assuming that every R -preserving fuzzy contractive sequence is an R -M-Cauchy sequence.

Now, we deduce some consequences of our main results including fuzzy metric, order-theoretic and α-admissible results.

5 Application to ordinary differential equations

In this section, as an application of our main fixed point results, we study the existence and uniqueness of a solution for the first-order periodic boundary value problem: { x ′ ( t ) = f ( t , x ( t ) ) , t ∈ I = [ 0 , T ] ; x ( 0 ) = x ( T ) ,(5.1) where T > 0 and f : I × ℝ → ℝ is a continuous function.

In what follows, E = C ( I , ℝ ) denotes the space of all real valued continuous functions defined on I.

Now, we recall the following basic definitions:

Definition 5.1. A solution of (5.1) is a function x ∈ C 1 ( I , ℝ ) satisfying (5.1).

Definition 5.2. A lower solution of (5.1) is a function α ∈ C 1 ( I , ℝ ) such that α ′ ( t ) ≤ f ( t , α ( t ) ) , t ∈ I and α ( 0 ) ≤ α ( T ) .

Definition 5.3. An upper solution of (5.1) is a function β ∈ C 1 ( I , ℝ ) such that β ′ ( t ) ≥ f ( t , β ( t ) ) , t ∈ I and β ( 0 ) ≥ β ( T ) .

In the following results, Nieto and Rodriguez-Lopez described some suitable conditions to ensure the existence of a unique solution of (5.1) in the presence of its lower (or upper) solution.

Theorem 5.1. [21] Consider the first-order periodic problem (5.1) such that f is continuous and there exist γ > 0 and δ > 0 with δ > γ such that 0 ≤ f ( t , y ) + δ y - [ f ( t , x ) + δ x ] ≤ γ ( y - x ) , for all x , y ∈ ℝ with x ≤ y .(5.2) Then the existence of a lower (or an upper) solution of (5.1) ensures the existence of a unique solution of (5.1).

Theorem 5.2. [22] Consider the first-order periodic problem (5.1) such that f is continuous and there exist γ > 0 and δ > 0 with δ > γ such that - γ ( y - x ) ≤ f ( t , y ) + δ y - [ f ( t , x ) + δ x ] ≤ 0 , for all x , y ∈ ℝ with x ≤ y .(5.3) Then the existence of a lower (or an upper) solution of (5.1) ensures the existence of a unique solution of (5.1).

In this section, we prove the existence and uniqueness of a solution for the first-order periodic problem (5.1) in the presence of a lower (or an upper) solution under a new condition which unify conditions (5.2) and (5.3).

Theorem 5.3. Consider the first-order periodic problem (5.1) such that f is continuous, non-decreasing in the second variable and there exist γ > 0 and δ > 0 with δ > γ such that | f ( t , y ) + δ y - [ f ( t , x ) + δ x ] | ≤ γ ( y - x ) , for all x , y ∈ ℝ with x ≤ y . Then the existence of a lower solution of (5.1) ensures the existence of a unique solution of (5.1).

Proof. Observe that problem (5.1) can be written in the following form: { x ′ ( t ) + δ x ( t ) = f ( t , x ( t ) ) + δ x ( t ) , t ∈ I = [ 0 , T ] ; x ( 0 ) = x ( T ) , which is equivalent to the following integral equation: x ( t ) = ∫ 0 T G ( t , s ) [ f ( s , x ( s ) ) + δ x ( s ) ] ds , where G ( t , s ) = { e δ ( T + s - t ) e δ T - 1 , 0 ≤ s < t ≤ T , e δ ( s - t ) e δ T - 1 , 0 ≤ t < s ≤ T . Define a binary relation R on E as follows: x R y ⇔ [ x ( t ) ≤ y ( t ) , for all t ∈ I ] , for all x , y ∈ E . Clearly, R is transitive. Define a mapping g : E → E by: g ( x ( t ) ) = ∫ 0 T G ( t , s ) [ f ( s , x ( s ) ) + δ x ( s ) ] ds , t ∈ I .(5.4) Observe that x ∈ E is a fixed point of g if and only if it is a solution of (5.1). Now, we prove that R is g-closed. Let x, y ∈ E be such that x R y . This amounts to saying that x (t) ≤ y (t), for all t ∈ I. Since f is non-decreasing in the second variable, we obtain (for all t ∈ I) f ( t , y ( t ) ) + δ y ( t ) ≥ f ( t , x ( t ) ) + δ x ( t ) .(5.5) Since G (t, s) >0 for all t, s ∈ I, therefore (5.5) implies that g ( x ( t ) ) = ∫ 0 T G ( t , s ) [ f ( s , x ( s ) ) + δ x ( s ) ] ds ≤ ∫ 0 T G ( t , s ) [ f ( s , y ( s ) ) + δ y ( s ) ] ds = g ( y ( t ) ) , for all t ∈ I. That is, gx R gy so that R is g-closed and hence the condition (ii) of Theorem 3.1 is satisfied. Now, let α ∈ E be a lower solution of (5.1). We show that α R g ( α ) . As α is a lower solution of (5.1), we have α ′ ( t ) + δ α ( t ) ≤ f ( t , α ( t ) ) + δ α ( t ) , t ∈ I . Multiplying both the sides of the above inequality by e ^δt , we have ( α ( t ) e δ t ) ′ ≤ [ f ( t , α ( t ) ) + δ α ( t ) ] e δ t , t ∈ I , or α ( t ) e δ t ≤ α ( 0 ) + ∫ 0 t [ f ( s , α ( s ) ) + δ α ( s ) ] e δ s ds , t ∈ I ,(5.6) yielding thereby (with α (0) ≤ α (T))

α ( 0 ) e δ T ≤ α ( T ) e δ T ≤ α ( 0 ) + ∫ 0 T [ f ( s , α ( s ) ) + δ α ( s ) ] e δ s ds ,

so that α ( 0 ) ≤ ∫ 0 T e δ s e δ T - 1 [ f ( s , α ( s ) ) + δ α ( s ) ] ds , which together with (5.6) imply that α ( t ) ≤ ∫ 0 t e δ ( T + s - t ) e δ T - 1 [ f ( s , α ( s ) ) + δ α ( s ) ] ds + ∫ 0 T e δ s - t e δ T - 1 [ f ( s , α ( s ) ) + δ α ( s ) ] ds , t ∈ I , i . e . , α ( t ) ≤ ∫ 0 T G ( t , s ) [ f ( s , α ( s ) ) + δ α ( s ) ] ds = g ( α ( t ) ) , t ∈ I ,

so that α R g ( α ) , i . e ., condition (i) of Theorem 3.1 is satisfied. Now, define a metric d on E by: d ( x , y ) = sup t ∈ I | x ( t ) - y ( t ) | , { for all x , y ∈ E . Define ℵ : E² × (0, ∞) → [0, 1] by ℵ ( x , y , t ) = t t + d ( x , y ) , ∀ x , y ∈ E , t > 0 . Then (E, ℵ , *) is an R -G-complete fuzzy metric space with a * b = a · b,   ∀  a, b ∈ [0, 1]. So, the condition (iii) of Theorem 3.1 is fulfilled. Now, let x, y ∈ E be such that x R y . Then x (t) ≤ y (t), for all t ∈ I. Observe that

d ( gx , gy ) = sup t ∈ I | g ( x ( t ) ) - g ( y ( t ) ) | ≤ sup t ∈ I ∫ 0 T G ( t , s ) | f ( s , x ( s ) ) + δ x ( s ) - f ( s , y ( s ) ) - δ y ( s ) | ds ≤ sup t ∈ I ∫ 0 T G ( t , s ) γ | x ( s ) - y ( s ) | ds ≤ γ d ( x , y ) sup t ∈ I ∫ 0 T G ( t , s ) ds = γ d ( x , y ) sup t ∈ I 1 e δ T - 1 ( 1 δ e δ ( T + s - t ) ] 0 t + 1 δ e δ ( s - t ) ] t T ) = γ δ d ( x , y ) 1 e δ T - 1 ( e δ T - 1 ) = γ δ d ( x , y ) .

Now,

1 ℵ ( gx , gy , t ) - 1 = d ( gx , gy ) t ≤ k λ d ( x , y ) t = k λ ( 1 ℵ ( x , y , t ) - 1 ) . which shows that g satisfies the condition (iv) in Theorem 3.1. Let {x _n } ⊆ E be an R -preserving sequence converging to z ∈ E. Then, for each t ∈ I, we have x 1 ( t ) ≤ x 2 ( t ) ≤ . . . ≤ x n ( t ) ≤ . . . .(5.7) Since { x n ( t ) } ⊆ ℝ is R -preserving sequence converging to z (t), therefore (5.7) implies that x _n  (t) ≤ z (t) , for all t ∈ I , n ∈ ℕ . Thus, x n R z , for all n ∈ ℕ . This shows that R is ℵ-self-closed, i . e . , the condition (v′) of Theorem 3.2 is satisfied. Hence, Theorem 3.2 ensures the existence of a solution of (5.1). Now, if x, y ∈ Fix (g), then u = max {x, y} ∈ E. As x ≤ u and y ≤ u, we have x R u and y R u so that Theorem 3.5 shows that the fixed point of g is unique. Hence, (5.1) has a unique solution.

Finally, we present the following result which proves the existence and uniqueness of a solution for the first-order periodic problem (5.1) in the presence of an upper solution.

Theorem 5.4. Consider the first-order periodic problem (5.1) such that f is continuous and non-increasing in the second variable. Assume that there exist γ > 0 and δ > 0 with δ > γ > 0 such that | f ( t , y ) + δ y - [ f ( t , x ) + δ x ] | ≤ γ ( y - x ) , for all x , y ∈ ℝ with x ≤ y . Then the existence of an upper solution of (5.1) ensures the existence of a unique solution of (5.1).

Proof. Define a binary relation R on E as follows: x R y ⇔ [ x ( t ) ≥ y ( t ) , for all t ∈ I ] , for all x , y ∈ E . Now, following steps of the proof of Theorem 5.3 with an analogous procedure, one can check that all the hypotheses of Theorem 3.2 are validated. Therefore, in this analogue case Theorem 3.2 ensures the existence of a fixed point of g which is indeed unique in view of Theorem 3.5. Thus (5.1) admits a unique solution.