Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let E be a real reflexive Banach space, which admits a akly sequentially continuous duality mapping of E into E*, and C be a nonempty closed convex subset of E. Let {T(t):t≥0} be a semigroup of nonexpansive self-mappings on C such that F:=∩t≥0Fix(T(t))≠O, where Fix(T(t))={x∈C: x=T(t)x}, and let f: C→C be a fixed contractive mapping. If {αn}, {βn}, {tn} satisfy some appropriate conditions, then a iterative process {xn} in C, defined by [image omitted]  converges strongly to q∈F, and q is the unique solution in F to the following variational inequality: 〈 (I-f)q, j(q-u)〉≤0 for all u∈F. Our results extend and improve corresponding ones of Suzuki [T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131 (2002), pp. 2133-2136.], Xu [H.K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (2005), pp. 371-379.] and Chen and He [R.D. Chen and H. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space, Appl. Math. Lett. 20 (2007), pp. 751-757.].
['Ljubomir Ćirić', 'Huimin He', 'Rudong Chen', 'Rade Lazović']
I am writing a research paper on a idea Strong convergence theorems for a semigroup of nonexpansive mappings in Banach spaces
[{"from": "human", "value": "I am writing a research paper on a idea Strong convergence theorems for a semigroup of nonexpansive mappings in Banach spaces"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let E be a real reflexive Banach space, which admits a akly sequentially continuous duality mapping of E into E*, and C be a nonempty closed convex subset of E. Let {T(t):t\u22650} be a semigroup of nonexpansive self-mappings on C such that F:=\u2229t\u22650Fix(T(t))\u2260O, where Fix(T(t))={x\u2208C: x=T(t)x}, and let f: C\u2192C be a fixed contractive mapping. If {\u03b1n}, {\u03b2n}, {tn} satisfy some appropriate conditions, then a iterative process {xn} in C, defined by [image omitted]\u00a0 converges strongly to q\u2208F, and q is the unique solution in F to the following variational inequality: \u2329 (I-f)q, j(q-u)\u232a\u22640 for all u\u2208F. Our results extend and improve corresponding ones of Suzuki [T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131 (2002), pp. 2133-2136.], Xu [H.K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (2005), pp. 371-379.] and Chen and He [R.D. Chen and H. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space, Appl. Math. Lett. 20 (2007), pp. 751-757.]."}]