Fixed Point Theorems for Nonlinear Contractions in Menger Spaces

and Applied Analysis 3 Definition 2.7. Let X,F, t be a Menger space, and let T : X → X be a selfmapping. For each p ∈ X, x > 0 and n ∈ N, let M ( p, x, n ) min { FTkp,Tlp x : k, l ≤ n and k, l ∈ Z } , M1 ( p, x, n ) min { FTkp,Tlp x : k, l ≤ n and k, l ∈ N } , M2 ( p, x, n ) min { Fp,Tlp x : l ≤ n and l ∈ N } , O ( p, n ) { Tp : k ≤ n and k ∈ Z } , O ( p,∞ { Tp : k ∈ Z } , 2.3 where it is understood that T0p p. A Menger space X,F, t is said to be T orbitally complete if and only if every Cauchy sequence which is contained in O p,∞ for some p ∈ X converges in X. From Definition 2.1∼ Definition 2.5, we can prove easily the following lemmas. Lemma 2.8 see 20 . Let X, d be a metric space, and let T : X → X be a selfmapping on X. Define F : X ×X → L by [ F ( p, q )] x ≡ Fp,q x H ( x − dp, q 2.4 for all p, q ∈ X and x ∈ R, where {Fp,q : p, q ∈ X} ⊆ L. Suppose that t-norm t : 0, 1 × 0, 1 → 0, 1 is defined by t a, b min{a, b} for all a, b ∈ 0, 1 . Then, 1 X,F, t is a Menger space; 2 If X, d is T orbitally complete, then X,F, t is T orbitally complete. Menger space generated by a metric is called the induced Menger space. Lemma 2.9. In a Menger space X,F, t , if t x, x ≥ x for all x ∈ 0, 1 , then t a, b min{a, b} for all a, b ∈ 0, 1 . 3. Ćirić-Type Fixed Point Theorems In 2010, Ćirić proved the following theorem. Theorem A see Ćirić 9 , 2010 . Let X,F, t be a complete Menger space under a t-norm t of H type. Let T : X → X be a generalized φ-probabilistic contraction, that is, FTp,Tq ( φ x ) ≥ Fp,q x ∗1 for all p, q ∈ X and x > 0, where φ : 0,∞ → 0,∞ satisfies the following conditions: φ 0 0, φ x < x, and limr→x infφ r < x for each x > 0. Then, T has a unique fixed point u ∈ X and {Tn p } converges to u for each p ∈ X. 4 Abstract and Applied Analysis Definition 3.1. Let X,F, t be aMenger space with t x, x ≥ x for all x ∈ 0, 1 , and let T : X → X be a mapping of X. We will say that T is Ćirić-type-generalized contraction if FTp,Tq ( φ x ) ≥ min{Fp,q x , Fp,Tp x , Fq,Tq x , Fp,Tq x , Fq,Tp x } ∗2 for all p, q ∈ X and x > 0, where φ : 0,∞ → 0,∞ is a mapping and for all p, q ∈ X and x ∈ R, Fp,q x is the same as in Definition 2.2. It is clear that ∗1 implies ∗2 . The following example shows that a Ćirić-type-generalized contraction need not be a generalized φ-probabilistic contraction. Example 3.2. Let X 0,∞ , T : X → X be defined by Tx x 1, and let φ : 0,∞ → 0,∞ be defined by φ x ⎧ ⎪⎨ ⎪⎩ x 1 x , 0 ≤ x ≤ 1, x − 1, 1 < x. 3.1 For each p, q ∈ X, let Fp,q : R → R be defined by Fp,q x H x−d p, q for all x ∈ R, where H is the same as in Definition 2.1, and d is a usual metric on R × R. Then, since max{|p − q − 1|, |q − p − 1|} |p − q| 1 for all p, q ∈ X, we have FTp,Tq φ x ≥ min{Fp,Tq x , Fq,Tp x } for all p, q ∈ X and x > 0. Thus, FTp,Tq ( φ x ) ≥ min{Fp,q x , Fp,Tp x , Fq,Tq x , Fp,Tq x , Fq,Tp x } 3.2 for all p, q ∈ X and x > 0, which satisfies ∗2 . If x 2, p 0 and q 3/2, then FT0,T3/2 φ 2 0 and F0,3/2 2 1. Thus, FT0,T3/2 φ 2 < F0,3/2 2 , which does not satisfy ∗1 . In the next example, we shall show that there exists T that does not satisfy ∗2 with φ t kt, 0 < k < 1. Example 3.3. Let X 0,∞ , T : X → X be defined by Tx 2x and let φ : 0,∞ → 0,∞ be defined by φ x kx, 0 < k < 1. For each p, q ∈ X, let Fp,q : R → R be defined by Fp,q x H x − d p, q for all x ∈ R, where H is the same as in Definition 2.1, and d is a usual metric on R × R. If p 0, q 1 and x 2/k > 0, then for simple calculations, FT0,T1 φ 2/k 0 and min { F0,1 ( 2 k ) , F0,T0 ( 2 k ) , F1,T1 ( 2 k ) , F0,T1 ( 2 k ) , F1,T0 ( 2 k )} 1. 3.3 Therefore, for p 0, q 1, and x 2/k > 0, the mapping T does not satisfy ∗2 . Thus, we showed that there exists T that does not satisfy ∗2 withφ t kt, 0 < k < 1. Abstract and Applied Analysis 5 Definition 3.4. Let X,F, t be a Menger space with t x, x ≥ x for all x ∈ 0, 1 and let T : X → X be a self mapping of X. We will say that T is a mapping of type U if there exists p ∈ X such thatand Applied Analysis 5 Definition 3.4. Let X,F, t be a Menger space with t x, x ≥ x for all x ∈ 0, 1 and let T : X → X be a self mapping of X. We will say that T is a mapping of type U if there exists p ∈ X such that Fp,Tp (( I − φ x ) ≤ inf { FTkp,Tlp x : k, l ∈ Z } ∀x > 0, ∗3 where φ : 0,∞ → 0,∞ is a mapping, and I : 0,∞ → 0,∞ is identity mapping. The following example shows that T has no fixed point, even though T satisfies ∗2 and ∗3 . Example 3.5. Let X 0,∞ , T : X → X be defined by Tx x 4, and let φ : 0,∞ → 0,∞ be defined by φ x ⎧ ⎨ ⎩ x 2 , 0 ≤ x ≤ 4, x − 2, 4 < x. 3.4 For each p, q ∈ X, let Fp, q : R → R be defined by Fp,q x H x−d p, q for all x ∈ R, where H is the same as in Definition 2.1, and d is a usual metric on R × R. Then, since max{|p − q − 4|, |q − p − 4|} |p − q| 4 for all p, q ∈ X, we have FTp,Tq φ x ≥ min{Fp,Tq x , Fq,Tp x } for all p, q ∈ X and x > 0. Thus, FTp,Tq ( φ x ) ≥ min{Fp,q x , Fp,Tp x , Fq,Tq x , Fp,Tq x , Fq,Tp x } 3.5 for all p, q ∈ X and x > 0, which implies ∗2 . It is easy to see that there exists p 1 ∈ X such that Fp,Tp (( I − φ x ) ≤ inf { FTkp,Tlp x : k, l ∈ Z } ∀x > 0, 3.6 which implies ∗3 . But T has no fixed point. Remark 3.6. It follows from Example 3.5 that T must satisfy ∗2 , ∗3 , and other conditions in order to have fixed point of T . The following is Ćirić-type fixed point theorem which is generalization of Ćirić’s fixed point theorems 6, 9 . Theorem 3.7. Let X,F, t be a Menger space with continuous t norm and t x, x ≥ x for all x ∈ 0, 1 , let T be a self-mapping on X satisfying ∗2 and ∗3 . Let X,F, t be T orbitally complete. Suppose that φ : R → R is a mapping such that i φ x < x for all x > 0 and limx→∞ I − φ x ∞, where I : R → R is identity mapping, ii φ and I − φ are strictly increasing and onto mappings, iii limn→∞φ−n x ∞ for each x > 0, where φ−n is n-time repeated composition of φ−1 with itself. 6 Abstract and Applied Analysis Then, a M p, x, n min{M1 p, x, n ,M2 p, x, n } for all p ∈ X, x > 0, and n ∈ N, b M p, x, n M2 p, x, n for all p ∈ X, x > 0 and n ∈ N, c {Tnp} is Cauchy sequence for each p ∈ U, where U { p ∈ X∣∣Fp,Tp (( I − φ x ) ≤ inf [ FTkp,Tlp x : k, l ∈ Z ] ∀x > 0 } , 3.7 d T has a unique fixed point in X. Proof. Let p ∈ X, x > 0 and n ∈ N be arbitrary. By Definition 2.2 and Definition 2.7, clearly, we haveM p, x, n min{M1 p, x, n ,M2 p, x, n }which implies a . From i , ii , and ∗2 , we have M1 ( p, φ x , n ) min { FTkp,Tlp ( φ x ) | k, l ≤ n and k, l ∈ N } min { FTTk−1p,TTl−1p ( φ x ) | k, l ≤ n and k, l ∈ N } ≥ min { min [ FTk−1p,Tl−1p x , FTk−1p,Tkp x , FTl−1p,Tlp x , FTk−1p,Tlp x , FTl−1p,Tkp x ] : k, l ≤ n and k, l ∈ N }