t-Best approximation in fuzzy normed linear spaces

Journal of Soft Computing and Applications 2014 (2014) 1-6

Available online at www.ispacs.com/jsca Volume 2014, Year 2014 Article ID jsca-00043, 6 Pages doi:10.5899/2014/jsca-00043 Research Article

t-Best approximation in fuzzy normed linear spaces M. Ertefaat1 , H. Mazaheri1∗ , S. M. S. Moddares1 (1) Faculty of Mathematics, Yazd University, Yazd, Iran. c M. Ertefaat, H. Mazaheri and S. M. S. Moddares. This is an open access article distributed under the Creative Commons Copyright 2014 ⃝ Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper we consider t-best approximation in fuzzy normed linear spaces. We consider fuzzy cyclic contraction map T , and show that there exists a point x such that M(x, T x,t) = M(A, B,t). Keywords: Fuzzy cyclic contraction map, Fuzzy approximatively compact.

1 Introduction In this section we recall some definitions and lemmas that we use to proof of some theorems in other sections. Definition 1.1. A binary operation ∗ : [0; 1] × [0; 1] → [0; 1] is a continuous t-norm if ∗ satisfying conditions: (1) ∗ is commutative and associative; (2) ∗ is continuous; (3) a ∗ 1 = a for all a ∈ [0; 1]; (4) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, and a; b; c; d ∈ [0; 1]. Definition 1.2. The 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X × X × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0 (M1) M(x, y,t) > 0, (M2) M(x, y,t) = 1 if and only if x = y, (M3) M(x, y,t) = M(y, x,t), (M4) M(x, y, s + t) ≥ M(x, z, s) ∗ M(z, y,t), (M5) the function M(x, y, .) : (0, ∞) → [0, 1] is continuous. Definition 1.3. Let (X, M, ∗) be a fuzzy metric space and A, B ̸= 0/ are two subsets of X. M(A, B,t) is defined as follows: M(A, B,t) = sup{M(x, y,t) : (x, y) ∈ A × B}. Definition 1.4. Let X be a linear space on R. A function N : X × R → [0, 1] is called a fuzzy norm if and only if for every x, y ∈ X and every c ∈ R the following properties are satisfy (N1) N(x,t) = 0 for every t ∈ R− ∪ {0}, (N2) N(x,t) = 1 if and only if x = 0 for every t ∈ R+ , t ) for every c ̸= 0 and t ∈ R+ , (N3) N(cx,t) = N(x, |c| (N4) N(x + y, s + t) ≥ min{N(x, s), N(y,t)} for every s,t ∈ R and x, y ∈ X, (N5) the function N(x, .) is nondecreasing on R, and limt→∞ N(x,t) = 1. ∗ Corresponding

author. Email address: [email protected]

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00043/

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A pair (X, N) is called a fuzzy normed space. We say that (X, N) satisfies conditions N6 and N7 if: (N6) for each t > 0, N(x,t) > 0 implies x = 0. (N7) for x ̸= 0, N(x, .) is a continuous function of R and strictly increasing on the subset {t : 0 < N(x,t) < 1} of R. Definition 1.5. Let (X, N) be a fuzzy normed linear space. A sequence {xn } in X is said to be a Cauchy sequence if for each 0 < ε < 1 and t ∈ (0, ∞) there exits N0 such that N(xm − xn ,t) > 1 − ε for each m > n ≥ N0 . Definition 1.6. Let (X, N) be a fuzzy normed linear space and {xn } a sequence in X. Then {xn } is said convergent to x ∈ X if for each 0 < ε < 1 and t ∈ (0, ∞) there exits N0 such that N(xn − x,t) > 1 − ε for each n ≥ N0 . Definition 1.7. Let (X, N) be a fuzzy normed linear space. A subset A of X is said to be fuzzy bounded (f-bounded) if there exits 0 < α < 1 and t > 0 such that N(x,t) > 1 − α for each x ∈ A. Definition 1.8. Let (X, N) be a fuzzy normed linear space also A and B are two nonempty subsets of X. Then N(A − B,t), for t > 0 is defined as follows N(A − B,t) = sup{N(x − y,t) : (x, y) ∈ A × B}. ∪

∪

Let A, B be nonempty subset of a fuzzy normed linear space (X, N, ∗) and a map T : A B → A B is said to be a relatively nonexpansive map fuzzy if it satisfies (1) T (A) ⊆ B and T (B) ⊆ A and (2) N(T x − Ty,t) ≤ N(x − y,t), for all (x, y) ∈ A × B. Note that a relatively nonexpansive map fuzzy need not be continuous in general. But if A ∩ B is nonempty, then the map T restricted to A ∩ B is a nonexpansive self map. If the fixed point equation T x = x does not possess a solution it is natural to explore to find an x0 ∈ A satisfying N(x0 − T x0 ,t) = M(A, B,t). Definition 1.9. A convex pair (K1 , K2 ) in a fuzzy normed linear space is said to have t-proximal normal structure if for any closed, bounded, convex t-proximal pair (H1 , H2 ) ⊆ (K1 , K2 ) for which M(H1 , H2 ) = M(K1 , K2 ) and δ (H1 , H2 ) > M(H1 , h2 .t), there exists (x1 , x2 ) ∈ H1 × H2 such that

δ (x1 , H2 ) < δ (H1 , H2 ), δ (x2 , H1 ) < δ (H1 , H2 ) where δ (H1 , H2 ) = sup{N(h1 , h2 ,t) : (h1 , h2 ) ∈ H1 × H2 }. Theorem 1.1. let (A, B) be a nonempty, fuzzy subset∪in a fuzzy normed linear space (X, N, ∗) and suppose (A, B) ∪ has t-proximal normal structure. Let T : A B → A B be a relatively nonexpansive fuzzy map, then there exists (x, y) ∈ A × B such that N(x − T x,t) = N(Ty − y,t) = M(A, B,t). The proof of the above theorem invokes zorn’s lemma and the proximal normal structureidea. Also it has been proved that every closed bounded convex pair fuzzy (A, B) of uniformly convex fuzzy Banach space has proximal normal structure and every compact convex pair fuzzy has proximal normal structure. In this paper, by using a convergence theorem we attempt to prove the existence of a t-best proximity point without invoking Zorn’s lemma. 2 preliminaries and notations In this section we give some basic definitions and concepts which are useful and related to the context of our resylts. we shall say that a pair (A, B) of sets in a fuzzy linear space satisfies a property. We say that (A, B) is to be convex if both A and B are convex, (C, D) ⊆ (A, B) ⇔ C ⊆ A, D ⊆ B, M(A, B.t) = inf{M(a, b,t) : (a, b) ∈ A × B},

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A0 = {x ∈ X : N(x − y,t) = M(A, B,t) f or some y ∈ B} and B0 = {y ∈ B : N(x − y,t) = M(A, B,t) f or some x ∈ A} Let X be a fuzzy normed linear space and C be a nonempty subset of X. Then the metric projection operator fuzzy PCt : X → 2C is defined as PCt (x) = {y ∈ C : N(x − y,t) = M(x,C,t) f or each x ∈ X}. It is well known that the metric projection operator fuzzy PC on a fuzzy linear space X is a single valued map from X to C, where C is a nonempty fuzzy compact convex subset of X. Lemma 2.1. Let X be a fuzzy linear space and A be a nonempty subset of X. Then A0 and B0 are nonempty and satisfy PBt (A0 ) ⊆ B0 , PAt (B0 ) ⊆ A0 . Lemma 2.2. If A and B are nonempty subsets of a fuzzy normed linear space X such that M(A, B,t) > 0, then A0 ⊆ ∂ (A) and B0 ⊆ B. Suppose (A, B) is a nonempty fuzzy convex pair of subset in a fuzzy normed linear space X. Consider the P : A ∪ B → A ∪ B defined as P(x) = PBt (x) i f x ∈ A, PAt (x) i f x ∈ B. (∗) Lemma 2.3. If X a fuzzy strictly convex Banach space, then the map P is a single value map and satisfies P(A) ⊆ B, P(B) ⊆ A. Proposition 2.1. Let A, B be nonempty fuzzy weakly compact convex subset of a fuzzy strictly convex Banach space X. Let T : A ∪ B → A ∪ B be a relatively nonexpansive map fuzzy and P : A ∪ B → A ∪ B be a map defined as in (*). Then T P(x) = P(T x), for all x ∈ A0 ∪ B0 . Proof. Suppose x ∈ A0 , then there exist y ∈ B such that N(x − y,t) = M(A, B,t). By the uniqueness of the metric projection on a fuzzy strictly convex Banach space, we have PB (x) = y, PA (y) = x. Since T is relatively nonexpansive fuzzy, we have N(T x − Ty,t) ≤ N(x − y,t) = M(A, B,t). ie PA (T x) = ty. This implies that PA (T x) = T PB (x). This observation will play an important role in this article. Eldred and veeramani introduced a notion of fuzzy cyclic contraction and studied the existence of t-best proximity point for such maps. We make use of the main results proved in [3] to obtain t-best proximity pair theorems for relatively nonexpansive mapping fuzzy. Definition 2.1. Let A and B be nonempty subsets of a fuzzy metric space X. A map T : A ∪ B → A ∪ Bis said to be a fuzzy cyclic contraction map if it satisfies: 1)T (A) ⊆ B, T (B) ⊆ A. 2)there exists k ∈ (0, 1) such that M(T x, Ty,t) ≤ kM(x, y,t) + (1 − k)M(A, B,t) for each x ∈ A, y ∈ B. we can easily see that every fuzzy cyclic contraction map satisfies M(T x, Ty,t) ≤ M(x, y,t), for all x ∈ A, y ∈ B. In [3], a simple existence result for a t-best proximity point of a fuzzy cyclic contraction map has been given. It states as follows: Theorem 2.1. Let A and B be nonempty closed subset of a complete fuzzy metric space X. Let T : A ∪ B → A ∪ B be a fuzzy cyclic contraction map, Let x0 ∈ A and define xn+1 = T xn . Suppose {x2n } has a convergent subsequence in A. Then there exists a x ∈ A such that M(x, T x,t) = M(A, B,t). In uniformly convex Banach space setting, the the following result proved in [3] ensures the existence, uniqueness and convergence of a t-best proximity point for a fuzzy cyclic contraction map. we use this result to prove our main results. Theorem 2.2. Let A and B nonempty closed and convex subset of a fuzzy uniformly convex Banach space. Suppose T : A ∪ B → A ∪ B is a fuzzy cyclic contraction map, then there exist a unique t-best proximity point x ∈ A (that is with N(x − T x,t) = M(A, B,t). Further, if x0 ∈ A and xn + 1 = T xn , then {x2n } converges to the t-best proximity point.



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We need the notion of “approximatively compact set” to prove a convergence result in the next section. Definition 2.2. Let X be a metric space. A subset C of X is said to be fuzzy approximatively compact if for any y ∈ x, and for any sequence {xn } in C of X is said to be fuzzy approximatively compact if for any y ∈ x, and for any sequence {xn } in C such that M(xn , y,t) → M(y, c,t) as n → ∞, then {xn } has a subsequence which fuzzy converges to a point in C. In a metric space, every fuzzy approximatively compact set is closed and every compact set is fuzzy approximatively compact. Also a closed convex subset of a fuzzy uniformly convex Banach space is fuzzy approximatively compact. 3 main results Theorem 3.1. Let X be a strictly convex fuzzy of Banach space and A a nonempty approximatively compact convex subset of X and B be a nonempty closed subset of X. Let {xn } be a sequence in A and y ∈ B. Suppose N(xn − y,t) → M(A, B,t), then xn → PA (y). Proof. Suppose that {xn } does not fuzzy convergence to PA (y), then there exists a ε > 0 and a subsequence xnk of xn such that N(xnk ,t) ≥ ε (3.1) Since {xnk } is a sequence in A such that N(xnk ,t) → M(A, B,t), and A is approximatively compact, {xnk } has a convergent subsequence {xnk′ } such that xnk′ → x for some x ∈ A. Then ′ N(xnk ,t) → N(x − y,t) ′ − y,t) → M(A, B,t) implies N(x − y) = M(A, b,t). also, N(xnk By the uniqueness of PA we have x = PA (y). But from (3.1) we have ′ ε ≤ N(xnk − PA (y),t) ⇒ 0 < ε ≤ N(x − PA (y),t) ⇒ x ̸= PA (y)

which is a contradiction. Hence xn −→ PA (y). Theorem 3.2. Let X be a normed linear space, let A be a nonempty closed convex subset of X. If A0 is compact, then B0 is also compact. Proof. Let B0 is empty, then nothing to prove. Assume B0 is nonempty. Let {yn } be a sequence in B0 . Then for each n ∈ N, there exists xn ∈ A0 such that N(xn − yn ,t) = M(A, B,t). Since A0 is compact, there exists a convergent subsequence {xnk } which converges to some x ∈ A0 . Consider the inequality, N(ynk − x,t) ≤ N(ynk − xn k,t) + N(xnk − x,t) → M(A, B,t). Since B is approximatively compact, {ynk } has a convergent subsequence {ynk′ } converges to some y ∈ B. Since B0 is closed, it implies that y ∈ B0 . Hence B0 is compact. Theorem 3.3. Let X be a uniformly convex Banach space. Let A be a nonempty closed bounded convex subset of X and B be a nonempty closed convex subset of X. Let T : A ∪ B → A ∪ B be a relatively nonexpansive map. Then there exist a sequence {xn } in A0 and x∗ ∈ A0 such that 1)xn → x∗ 2)N(x∗ − T x∗ ,t) ≤ M(A, B,t) + lim infn N(T xn − T x∗ ,t). Proof. By lemma 2.1, A0 is nonempty, hence there exist x0 ∈ A0 and y0 ∈ B0 such that N(x0 − y0 ,t) = M(A, B,t). For each n ∈ N, define a map Tn : A ∪ B → A ∪ B by Tn (x) = 1/ny0 + (1 − 1/n)T x, i f x ∈ A and 1/nx0 + (1 − 1/n)T x,

ifx ∈ B



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Since A and B are convex and T is a relatively nonexpansive map, for each n ∈ N, Tn (A) ⊆ B, Tn (B) ⊆ A. Also for each x ∈ A, y ∈ B, N(Tn (x) − Tn (y),t) ≤ (1/n)N(x0 − y0 ,t) + (1 − 1/n)N(T x − Ty,t) ≤ (1 − 1/n)N(x − y,t) + 1/nM(A, B,t). This implies that for each n ∈ N, Tn is a fuzzy cyclic contraction on A ∪ B. Hence for each n ∈ N, there exists xn ∈ A such that N(xn − Tn xn ,t) = M(A, B,t). Hence xn ∈ A0 , for each n ∈ N. Since A0 is bounded, B0 is also bounded, and T (A0 ) ⊆ B0 , Tn (A0 ) ⊆ B0 . Also observe that for any x ∈ A0 , N(Tn x − T x,t) ≤ 1/nN(y0 − T x,t) ≤ 1/nδ (B0 ) → 0 as n → ∞. since A0 is a closed bounded convex set, xn has a weakly convergent subsequence. Without loss of generality, let us assume that xn itself weakly converges to x∗ , for some x∗ ∈ A0 . Then xn − T x∗ → x∗ − T x∗ . Since (X, N, .) is weakly lower semi continuous, and by 3.4, 3.5 we have N(x∗ − T x∗ ,t) ≤ lim inf N(xn − T x∗ ,t) n

≤ lim inf N(xn − Tn xn ,t) + N(Tn xn 0T xn ,t) + N(T xn − T x∗ ,t) n

≤ lim inf M(A, B,t) + 1/nδ (B0 ) + N(T xn − T x∗ ,t) n

≤ M(A, B,t) + lim inf N(T xn − T x∗ ,t). n

we use the above theorem to prove: Theorem 3.4. Let X be a uniformly convex Banach space. Let A be a nonempty closed bounded convex subset of X such that A0 is compact, and B be a nonempty closed convex subset of X. Let T : A ∪ B → a ∪ B be a relatively nonexpansive map. Then there exist x∗ ∈ A such that N(x∗ − T x∗ ,t) = M(A, B,t). Proof. By Theorem 3.4, there exist a sequence xn inA0 and x∗ ∈ A0 such that xn → x∗ and satisfies the inequality N(x∗ − T x∗ ,t) ≤ M(A, B,t) + lim inf N(T xn − T x∗ ,t) n

Since A0 is compact,xn converges to x∗ strongly. The proof will be complete if we show that N(T xn − T x∗ ,t) → 0. Claim:N(T xn − T x∗ ,t) → 0asn → ∞. It is enough to show that N(T xn − pA (T x∗ ),t) → M(A, B,t) as n → ∞. Then by Theorem 3.1, we have T xn → pB (PA (T x∗ )) = T x∗ . Consider N(xn − PB x∗ ,t) ≤ N(xn − x∗ ,t) + N(x∗ − PB x∗ ,t) → M(A, B,t). Since T is relatively nonexpansive we have, T (xn − PA T x∗ ,t) = N(T xn − T (PB x∗ ),t) ≤ N(xn − PB x∗ ,t) → M(A, B,t). this ends the claim and hence the theorem. Theorem 3.5. Let x be a strictly convex Banach space, let A be a nonempty closed convex subset of X such that A0 is is a nonempty compact set and B be a nonempty closed convex subset of X. Let T : A ∪ B → A ∪ B be a relatively nonexpansive map. Then there exists x∗ ∈ A such that N(x∗ − T x∗ ,t) = M(A, B,t). Proof. Since A0 is nonempty and compact, we can construct a sequence of cyclic contraction maps Tn : A ∪ B → A ∪ B as in Theorem 3.4. We use Theorem 3.5 for an existence of t-best proximity point xn ∈ A0 such that N(xn − Tn xn ,t) = M(A, B,t). since A0 is compact, xn has a convergent subsequence xn k such that xn k → x∗ for some x∗ ∈ A0 . As in the proof of Theorem 3.5, We can show T xn k → T x∗ . The proof ends by considering the following inequality, N(x∗ − T x∗ ,t) ≤ N(x∗ − xn k,t,t) + N(xn k − Tn kxn k,t) + N(Tn kxn k − T xn k,t) + N(T xn k − T x∗ ,t). and by observing N(Tn kxn k − T xn k,t) ≤ 1/nk δ (B0 ) → 0.



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we give below some situations where A0 is a compact subset of A. Theorem 3.6. Let A be a unit ball in a strictly convex Banach space X and B be a closed convex subset of X with M(A, B,t) > 0. Then A0 contains almost one point. Proof. Clearly A0 is a bounded convex subset of A, moreover by Lemma 2.2, A0 is contained in the boundary of A. ie A0 ⊆ δ A. Suppose x1 , x2 ∈ A0 with x1 ̸= x2 , then by strict convexity N((x1 + x2 )/2,t) < 1 which implies that (x1 + x2 )/2δ A, a contradiction to the convexity of A0 . Hence A0 contains almost one point. References [1] S. Cobzas, Geometric properties of Banach spaces and the existence of nearest and farthest points, Abstr. Appl Anal, 3 (2005) 259-285. http://dx.doi.org/10.1155/AAA.2005.259 [2] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy math, 11 (3) (2003) 687-705. [3] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy sete and systems, 48 (1992) 239-248. http://dx.doi.org/10.1016/0165-0114(92)90338-5 [4] A. George, P. Veeramani, On some resulte of fuzzy metric space, Fuzzy sete and systems, 64 (1994) 395-399. http://dx.doi.org/10.1016/0165-0114(94)90162-7 [5] A. George, P. Veeramani, On some resulte of analysis for fuzzy metric spaces, Fuzzy sete and systems, 90 (1997) 365-368. http://dx.doi.org/10.1016/S0165-0114(96)00207-2 [6] H. Mazaheri, F. M. Maalek Ghaini, Quasi-orthogonality of the best approximant sets, Nonlinear Anal, 65 (3) (2006) 534-536. http://dx.doi.org/10.1016/j.na.2005.09.026 [7] S. M. Vaezpour, F. Karimi, t-nearest point in fuzzy normed spaces, Iranian Journal of fuzzy systems, 5 (2) (2001) 75-80. [8] B. M. Pu, Y. M. Liu, Fuzzy topology I: Neighborhood stucture of a fuzzy point and Moore- Smith convergence, J. Math. Anal. Appl, 76 (1980) 517-599. [9] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
