The general linearization problem consists of finding the coefficients (G_{i,j,k}) in the expansion of the product of two polynomials (A_{i}(x)) and (B_{j}(x)) in terms of an arbitrary sequence of orthogonal polynomials ({P_{k}(x)}), that is, A_{i}(x) B_{j}(x)=sum _{k=0}^{i+j}G_{i,j,k} P_{k}(x).
where { a n ′ } Open image in new window, { b n ′ }, { c n ′ } ⊂ [ 0, 1 ] Open image in new window; a n ′ + b n ′ + c n ′ = 1 Open image in new window; and {u n } is an arbitrary sequence in K. Let K be a nonempty closed convex subset of an arbitrary Banach space X and T1,T2,…,T p (p≥2) be self-mappings of K. Let T1be a continuous ϕ -hemicontractive mapping and R(T2) is bounded.
Finally, using a generalized nonexpansive map on a metric space, we provide a necessary and sufficient condition for the convergence of an arbitrary sequence in to a fixed point of in terms of the approximating sequence.
For an arbitrary sequence in dataset, the frequencies of different motifs in the sequence were recorded.
Let be an arbitrary sequence in and (3.11).
Let be an arbitrary sequence in and define by (3.12).
Let { y n } be an arbitrary sequence in X.
Let ( u n ) be an arbitrary sequence in [ u −, u + ].
Now let { y n } be an arbitrary sequence in X.
Let ({u_m}) be an arbitrary sequence in (W^{1,1}(Omega )) which converges to (uin W^{1,1}(Omega )).
