Sentence examples for for any given i from inspiring English sources

(40) for any given (i in bar{p}).

right))-set, for any given (i in bar{p}) is a pair of quasi-best proximity points if (F_{x,Tx} left( {hat{D}; + varepsilon } right) > 1 - lambda) for any given (varepsilon, in {mathbf{R}}_) and (F_{x,Tx} left( D right) = F_{x,Tx} left( {D^ right) = 0).

Let (u in clA_{i}), (v in clA_{j}) be best proximity points for any given (i,j in bar{k}) such that (u = T^{np} u = T^{p} u) and (v = T^{np} v = T^{p} v), (forall n in {mathbf{Z}}_) so that there is (z in bigcupnolimits_{{i in bar{p}}} {clA_{i} }) such that (hbox{min},left( {alpha left( {u,z,t} right),;alpha left( {v,z,t} right)} right) ge 1), (forall tleft( { > D} right) in {mathbf{R}}_).

Then there is a (non-necessarily unique) sequence ({ x_{n} }) of (alpha ( x_{n} ))-fuzzy best proximity points of X through T with first element (x in X_{i}), being subject to free choice if (operatorname{card} X_{i} ge 2), for any given (i in bar{p}) such that the sequence of distances ({ d ( x_{n}, x_{n + 1} ) }) is non-increasing A possible choice is a constant sequence.

Only one of k i and e i can evaluate to 1 for any given i.

If for any given 1 ≤ l ≤ k, (i) lim n → ∞ | | x n - T l x n | | = 0 ;   (ii) there exists a constant α > 0 such that || x n - T l x n || ≥ αd(x n, F) for all n ≥ 1.  . lim n → ∞ | | x n - T l x n | | = 0 ; there exists a constant α > 0 such that || x n - T l x n || ≥ αd(x n, F) for all n ≥ 1.

Thus, (F_{u,v}} left( {varphi left( t right)} right) = F_{{T^{p} u,,T^{p} v}} left( {varphi left( t right)} right) = 1), (forall t in {mathbf{R}}_) and then (u = v) is a unique fixed point of (T^{p} :left. {bigcupnolimits_{{j in bar{p}}} {A_{j} } } right|A_{i} to A_{i}) for any arbitrary given (i in bar{p}), (u in clA_{i}) and (Tu in clA_{i + 1}) are unique adjacent best proximity points.

Theorem 1 For any given f i 1 ( x ), f i 2 ( x ) ∈ L ∞ ( i = 1, …, d, where d = 1 or 3 ), there exists a function u ∈ g − B V [ 0, 1 ] minimizing the energy functional E 1 in (3.3).

For any given f i 1 ( x ), f i 2 ( x ) ( i = 1, …, d ), if u ( x ) is any minimum of the energy functional E 1 defined in (3.3), then for almost every μ ∈ [ 0, 1 ], the characteristic function 1 Ω c = 1 { x : u ( x ) > μ } is a global minimum of the functional E 1 where c is the boundary of the set Ω c.

Furthermore, for any given f i 1, f i 2, model (3.1) can be converted into a simpler form.

Therefore, for any given node i, − 1 ≤ s i ≤ 1.