Sentence examples for mappings for i from inspiring English sources

Let (A: C rightarrow E) be an m-accretive mapping, and (S_{i}: C rightarrow C) be non-expansive mappings, for (i = 1,2,ldots ) .

Let (A_{i}: C rightarrow E) be m-accretive mappings, for (i = 1,2,ldots ) .

Let (A_{i}: C rightarrow E) be m-accretive mappings, (B_{i}: C rightarrow E) be (mu_{i} -inversely strongly accretive mu_{i} -inverselyinmathbb{N^).

Also if x ∗ ∈ F ( S ), then x ∗ ∈ F ( S n ) for all integer n ≥ 1. Assume that S i : H → H is asymptotically λ i -strictly pseudocontractive mappings for i = 1, 2 with ⋂ i = 1 2 F ( S i ) ≠ ∅.

Let A i, B j : C → E be m-accretive mappings, for i = 1, 2, … N, and j = 1, 2, …, M. Suppose that the duality mapping J : E → E ∗ is weakly sequentially continuous and D ≠ ∅.

where T n = T n ( mod N ), z n ∈ T n u n, α n + β n + γ n = 1 for all n ≥ 1 and f i, T i are finite families of nonspreading mappings and multivalued nonexpansive mappings for i = 1, 2, …, N, respectively.

From the definition of G i, we have G i = P C ( I − η B i ) are nonexpansive mappings for all i = 1, 2, …, N. Since x ∗ ∈ F, by Lemma 2.2, we have x ∗ = G i x ∗ = P C ( I − η B i ) x ∗, ∀ i = 1, 2, …, N. (3.7).

Proof The first part of this proof - J r n, i T i (resp. J r n, i S i ) are relatively nonexpansive mappings for each i ≥ 1 - is essentially due to Matsushita and Takahashi (cf. [[27], p.265]) which we include for completeness.

Corollary 3.4 Let H be a real Hilbert space and K be a nonempty closed convex subset of H. Let S i : K → K be κ i -inverse-strongly monotone mappings for each i = 1, 2 and f : K → K be an α-contraction.

If C is any bounded subset of K, then lim n → ∞ sup x ∈ C ∥ W x − W n x ∥ = 0. Theorem 3.1 Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Let B i : K → H be μ i -inverse-strongly monotone mappings for each i = 1, 2, and f : K → K be an α-contraction.

where y ∗ = P C [ x ∗ − μ 2 B 2 x ∗ ], μ i ∈ ( 0, 2 θ i ) and B i : C → C is for the θ i -inverse strongly monotone mappings for each i = 1, 2. Assumption 2.1 [14].