New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems

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1 Applied Mathematical Sciences, Vol., 207, no. 49, HIKARI Ltd, New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems Wei-Shih Du and Yuan-Liang Liu Department of Mathematics National Kaohsiung Normal University Kaohsiung 82444, Taiwan Copyright c 207 Wei-Shih Du and Yuan-Liang Liu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we establish new convergence theorems and best proximity point theorems for approximate sequences. Mathematics Subject Classification: 4A7, 4A65 Keywords: Approximate sequence, MT (λ)-function, MT -function (or R-function), cyclic mapping, best proximity point, convergence theorem.. Introduction and preliminaries In the last decades, the theory of fixed points has become a crucial technique in the study of nonlinear functional analysis. However, as we know, the equation T x = x (i.e. d(t x, x) = 0) does not necessarily have a fixed points, where T is a self-mapping or non-self-mapping defined on the metric space (X, d). So, in such situations, we often turn to find the best approximation of the existence of solutions. In other words, it is quite natural to investigate an element x X such that d(t x, x) is minimum, that is, the point x and the point T x are close proximity to each other. In recent years, many researcher Corresponding author

2 2448 Wei-Shih Du and Yuan-Liang Liu have studied the use of new nonlinear conditions to promote the best proximity point theory; see, e.g., [-4, 7, 4-24] and references therein. Throughout this paper, we denote by N and R, the sets of positive integers and real numbers, respectively. Let A and B be nonempty subsets of a metric space (X, d) and T : A B A B be a self-mapping. Denote dist(a, B) = inf{d(x, y) : x A, y B}. A point x A B is called to be a best proximity point for T if d(x, T x) = dist(a, B). The concepts of cyclic mapping and best proximity point were introduced by Kirk, Srinavasan and Veeramani [9] in Definition. [9]. Let A and B be nonempty subsets of a metric space (X, d). A mapping T : A B A B is called cyclic if T (A) B and T (B) A. In 2006, Eldered and Veeramani [7] established some existence results about best proximity points of cyclic contraction mappings. Definition.2 [7]. Let A and B be nonempty closed subsets of a metric space (X, d). A cyclic mapping T : A B A B is called a cyclic contraction if for some α (0, ), the condition holds for all x A and y B. d(t x, T y) αd(x, y) + ( α)dist(a, B) Theorem. [7]. Let A and B be nonempty closed subsets of a metric space (X, d), T : A B A B be a cyclic contraction mapping. We defeine x A and x n+ = T x n, n N. Suppose {x 2n } has a convergent subsequence in A. Then there exists x A such that d(x, T x) = dist(a, B). Let f be a real-valued function defined on R. For c R, we recall that and lim sup f(x) = inf x c ε>0 lim sup f(x) = inf x c + ε>0 sup 0< x c <ε sup c<x<c+ε f(x) f(x). In 206, Du introduced the concept of MT (λ)-function [7, 9-4] as follows.

3 New nonlinear conditions for approximate sequences 2449 Definition.3. Let λ > 0. A function µ : [0, ) [0, λ) is said to be an MT (λ)-function [7, 9-4] if lim sup µ(s) < λ for all t [0, ). In particular, s t + if λ =, then µ : [0, ) [0, ) is called an MT -function (or R-function) [5-5]. In [7], Du established the following useful characterizations of MT (λ)- functions; see also [9-4]. Theorem.2 (see [7, Theorem 2.4]). Let λ > 0 and let µ : [0, ) [0, λ) be a function. Then the following statements are equivalent. () µ is an MT (λ)-function. (2) λ µ is an MT -function. (3) For each t [0, ), there exist ξ () t [0, λ) and ɛ () t µ(s) ξ () t for all s (t, t + ɛ () t ). (4) For each t [0, ), there exist ξ (2) t [0, λ) and ɛ (2) t µ(s) ξ (2) t for all s [t, t + ɛ (2) t ]. (5) For each t [0, ), there exist ξ (3) t [0, λ) and ɛ (3) t µ(s) ξ (3) t for all s (t, t + ɛ (3) t ]. (6) For each t [0, ), there exist ξ (4) t [0, λ) and ɛ (4) t µ(s) ξ (4) t for all s [t, t + ɛ (4) t ). > 0 such that > 0 such that > 0 such that > 0 such that (7) For any nonincreasing sequence {x n } n N in [0, ), we have 0 sup µ(x n ) < n N λ. (8) For any strictly decreasing sequence {x n } n N in [0, ), we have 0 µ(x n ) < λ. sup n N (9) For any eventually nonincreasing sequence {x n } n N (i.e. there exists l N such that x n+ x n for all n N with n l) in [0, ), we have 0 sup µ(x n ) < λ. n N (0) For any eventually strictly decreasing sequence {x n } n N (i.e. there exists l N such that x n+ < x n for all n N with n l) in [0, ), we have 0 sup µ(x n ) < λ. n N

4 2450 Wei-Shih Du and Yuan-Liang Liu Let us recall the concept of approximate sequence. Definition.4 [4]. Let A and B be nonempty subsets of a metric space (X, d) and τ : [0, ) [0, ) be a function. A sequence {x n } n N in A B is said to be approximate with respect to τ, if the following conditions are satisfied: (a) one of the following conditions holds: (i) {x 2n } n N A and {x 2n } n N B; (ii) {x 2n } n N A and {x 2n } n N B, (b) d (x n+, x n+2 ) τ(d(x n, x n+ ))d(x n, x n+ )+( τ(d(x n, x n+ )))dist(a, B) for all n N. The following convergence theorem for approximate sequences was essentially proved by using Theorem.2 in [4]. Theorem.3 [4]. Let A and B be nonempty subsets of a metric space (X, d) and ϕ : [0, ) [0, ) be an MT -function. If {x n } n N A B is an approximate sequence with respect to ϕ, then lim d(x n, x n+ ) = inf d(x n n, x n+ ) = n N dist(a, B). In this paper, we establish new convergence theorems and best proximity point theorems for approximate sequences. 2. Some new existence theorems In this section, we first give the following new nonlinear condition for approximate sequences. Theorem 2.. Let A and B be nonempty subsets of a metric space (X, d) and T : A B A B be a cyclic mapping. Suppose that (DL) there exists an MT -funtion ϕ : [0, ) [0, ) such that { d(t x, T y) ϕ (d (x, y)) max [d (x, T y) + 2d (T x, T y) + d (y, T x)], 4 } [d (x, T y) + 3d (x, T x) + 3d (y, T y) + d (y, T x)] 8 for all x A and y B. + [ ϕ(d (x, y))]dist (A, B)

5 New nonlinear conditions for approximate sequences 245 Then there exists an approximate sequence in A B with respect to ϕ. Proof. Because T is cyclic, we have T (A) B and T (B) A. Let x A be given. Define x n+ = T x n for all n N. Then {x 2n } n N A and {x 2n } n N B. In order to finish the proof it is sufficient to show that the sequence {x n } n N satisfies the following: d(x n+, x n+2 ) d(x n, x n+ ) for all n N, d(x n+, x n+2 ) ϕ(d(x n, x n+ ))d(x n, x n+ )+[ ϕ(d(x n, x n+ ))]dist(a, B) for all n N. Suppose that there exists j N such that ϕ (d (x j, x j+ )) = 0, then, by (DL), we obtain By (2.), we have d(x j+, x j+2 ) = d(t x j, T x j+ ) dist (A, B) d (x j, x j+ ). (2.) d(x j+, x j+2 ) ϕ (d (x j, x j+ )) d (x j, x j+ ) + [ ϕ(d (x j, x j+ ))]dist (A, B) and our claim for the case of ϕ (d (x j, x j+ )) = 0 is proved. For this reason we henceforth will assume that ϕ (d (x n, x n+ )) > 0 for all n N. By (DL), we have { d(x 2, x 3 ) ϕ (d (x, x 2 )) max 4 [d (x, x 3 ) + 2d (x 2, x 3 ) + d (x 2, x 2 )], Assume that 8 [d (x, x 3 ) + 3d (x, x 2 ) + 3d (x 2, x 3 ) + d (x 2, x 2 )] + [ ϕ(d (x, x 2 )]dist (A, B) { ϕ (d (x, x 2 )) max 4 d (x, x 2 ) d (x 2, x 3 ), 2 d (x, x 2 ) + } 2 d (x 2, x 3 ) Then (2.3) implies + [ ϕ(d(x, x 2 ))]dist(a, B). (2.2) 4 d (x, x 2 ) d (x 2, x 3 ) > 2 d (x, x 2 ) + 2 d (x 2, x 3 ). (2.3) By taking into account (2.2), (2.3) and (2.4), we get d (x 2, x 3 ) > d(x, x 2 ). (2.4) d(x 2, x 3 ) 4 ϕ (d (x, x 2 )) [d (x, x 2 ) + 3d (x 2, x 3 )] + [ ϕ(d(x, x 2 ))]dist(a, B). < ϕ(d(x, x 2 ))d(x 2, x 3 ) + [ ϕ(d(x, x 2 ))]d(x 2, x 3 ) = d(x 2, x 3 ), }

6 2452 Wei-Shih Du and Yuan-Liang Liu a contradiction. So we must have 4 d (x, x 2 ) d (x 2, x 3 ) 2 d (x, x 2 ) + 2 d (x 2, x 3 ) (2.5) and hence Taking into account (2.2), (2.5) and (2.6), we obtain d (x 2, x 3 ) d (x, x 2 ). (2.6) d(x 2, x 3 ) ϕ(d(x, x 2 ))d(x, x 2 ) + [ ϕ(d(x, x 2 ))]dist(a, B). Similarly, from (DL) again, we have { d(x 4, x 3 ) ϕ (d (x 3, x 2 )) max 4 [d (x 3, x 3 ) + 2d (x 4, x 3 ) + d (x 2, x 4 )], } 8 [d (x 3, x 3 ) + 3d (x 3, x 4 ) + 3d (x 2, x 3 ) + d (x 2, x 4 )] If then + ( ϕ (d (x 3, x 2 ))) dist (A, B) { ϕ (d (x 2, x 3 )) max 4 d (x 2, x 3 ) d (x 3, x 4 ), 2 d (x 2, x 3 ) + } 2 d (x 3, x 4 ) + [ ϕ(d(x 2, x 3 ))]dist(a, B). (2.7) 4 d (x 2, x 3 ) d (x 3, x 4 ) > 2 d (x 2, x 3 ) + 2 d (x 3, x 4 ), and we get from (2.7) that d(x 3, x 4 ) > d (x 2, x 3 ) d(x 3, x 4 ) 4 ϕ (d (x 2, x 3 )) [d (x 2, x 3 ) + 3d (x 3, x 4 )] + [ ϕ(d(x 2, x 3 ))]dist(a, B). < ϕ (d (x 2, x 3 )) d (x 3, x 4 ) +( ϕ(d(x 2, x 3 )))d(x 3, x 4 ) = d(x 3, x 4 ), which leads a contradiction. So it must be 4 d (x 2, x 3 ) d (x 3, x 4 ) 2 d (x 2, x 3 ) + 2 d (x 3, x 4 ), (2.8) and hence By (2.7), (2.8) and (2.9), we have d(x 3, x 4 ) d (x 2, x 3 ) (2.9) d(x 3, x 4 ) ϕ(d(x 2, x 3 ))d(x 2, x 3 ) + [ ϕ(d(x 2, x 3 ))]dist(a, B). Therefore, by induction, we prove that the sequence {x n } n N satisfies the following:

7 New nonlinear conditions for approximate sequences 2453 (a) d(x n+, x n+2 ) d(x n, x n+ ) for all n N, (b) d(x n+, x n+2 ) ϕ(d(x n, x n+ ))d(x n, x n+ )+[ ϕ(d(x n, x n+ ))]dist(a, B) for all n N. Therefore, we prove that {x n } n N is an approximate sequence in A B with respect to ϕ. Applying Theorems 2. and.3, we obtain the following new convergence theorem immediately. Theorem 2.2. Let A and B be nonempty subsets of a metric space (X, d) and T : A B A B be a cyclic mapping. Suppose that the condition (DL) as in Theorem 2. holds. Then there exists a sequence {x n } n N A B such that (a) d(x n+, x n+2 ) ϕ(d(x n, x n+ ))d(x n, x n+ )+[ ϕ(d(x n, x n+ ))]dist(a, B) for all n N; (b) lim n d(x n, x n+ ) = inf n N d(x n, x n+ ) = dist(a, B). Corollary 2.. Let A and B be nonempty subsets of a metric space (X, d) and T : A B A B be a cyclic mapping. Suppose that (DL) there exists an MT -funtion ϕ : [0, ) [0, ) such that d(t x, T y) ϕ (d (x, y)) [d (x, T y) + 2d (T x, T y) + d (y, T x)] 4 + [ ϕ(d (x, y))]dist (A, B) for all x A and y B. Then there exists a sequence {x n } n N A B such that (a) d(x n+, x n+2 ) ϕ(d(x n, x n+ ))d(x n, x n+ )+[ ϕ(d(x n, x n+ ))]dist(a, B) for all n N; (b) lim n d(x n, x n+ ) = inf n N d(x n, x n+ ) = dist(a, B). Corollary 2.2. Let A and B be nonempty subsets of a metric space (X, d) and T : A B A B be a cyclic mapping. Suppose that

8 2454 Wei-Shih Du and Yuan-Liang Liu (DL2) there exists an MT -funtion ϕ : [0, ) [0, ) such that d(t x, T y) ϕ (d (x, y)) [d (x, T y) + 3d (x, T x) + 3d (y, T y) + d (y, T x)] 8 + [ ϕ(d (x, y))]dist (A, B) for all x A and y B. Then there exists a sequence {x n } n N A B such that (a) d(x n+, x n+2 ) ϕ(d(x n, x n+ ))d(x n, x n+ )+[ ϕ(d(x n, x n+ ))]dist(a, B) for all n N; (b) lim n d(x n, x n+ ) = inf n N d(x n, x n+ ) = dist(a, B). Following a similar argument as the proof of [4,Theorem 2.5], we give an existence theorem for best proximity points. Theorem 2.3. In Theorem 2.2, if we further assume (H) d(t x, T y) d(x, y) for any x A and y B. Then the following statements hold. (a) If {x 2n } n N has a convergent subsequence in A, then there exists u A such that d(u, T u) = dist(a, B). (b) If {x 2n } n N has a convergent subsequence in B, then there exists v B such that d(v, T v) = dist(a, B). Remark 2.. (a) Theorems 2., 2.2 and 2.3 and Corollaries 2. and 2.2 are also true if the function ϕ is nonincreasing or nondecreasing. (b) In Theorem 2.3, if Theorem 2.2 is replaced with Corollary 2. or Corollary 2.2, then we can also obtain new best proximity point theorems. (c) As a direct application of Theorem 2.2, we can establish easily some new fixed point theorems.

9 New nonlinear conditions for approximate sequences 2455 Acknowledgments. This research was supported by grant no. MOST M of the Ministry of Science and Technology of the Republic of China. References [] M.A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), [2] S.S. Basha, Best proximity point theorems: An exploration of a common solution to approximation and optimization problems, Applied Mathematics and Computation, 28 (202), [3] S.S. Basha, N. Shahzed, R. Jeyaraj, Best proxmity point theorems: exposition of a significant non-linear porgramming porblem, Journal of Optimization Theory and Applications, 56 (203), [4] M. Derafshpour, Sh. Rezapour, N. Shahzad, On the existence of best proximity point of cyclic contractions, Advances in Dynamical Systems and Applications, 6 (20), [5] W.-S. Du, Some new results and generalizations in metric fixed point theory, Nonlinear Analysis: Theory, Methods and Applications, 73 (200), [6] W.-S. Du, On coincidence point and fixed point theorems for nonlinear multivalued maps, Topology and its Applications, 59 (202), [7] W.-S. Du, New existence results of best proximity points and fixed points for MT (λ)-functions with applications to differential equations, Linear and Nonlinear Analysis, 2 (206), no. 2, [8] W.-S. Du, Y.-L. Hung, A generalization of Mizoguchi-Takahashi s fixed point theorem and its applications to fixed point theory, International Journal of Mathematical Analysis, (207), no. 4, [9] W.-S. Du, New fixed point theorems for hybrid Kannan type mappings and MT (λ)-functions, Nonlinear Analysis and Differential Equations, 5 (207), no. 2,

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11 New nonlinear conditions for approximate sequences 2457 metric spaces, Bangmod International Journal of Mathematical & Computational Science, (205), no., 2-2. [2] C. Mongkolkeha, P. Kuman, Best porximity point theorems for generalized cyclic contractions in ordered metric spaces, Journal of Optimization Theory and Applications, 55 (202), [22] M. Pǎcurar, I.A. Rus, Fixed point theory for cyclic ϕ-contractions, Nonlinear Analysis: Theory, Methods and Applications, 72 (200), [23] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Am. Math. Soc., 36 (2008), no. 5, [24] T. Suzuki, M. Kikkawa, C. Vetro, The existence of the best proximity points in metric spaces with the property UC, Nonlinear Analysis: Theory, Methods and Applications, 7 (2009), Received: September 9, 207; Published: September 26, 207