Strong Convergence of Modified Averaging Iterative Algorithm for Asymptotically Nonexpansive Map

Abstract

Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T. Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T . Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T. No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T. Let H be a real Hilbert space and K a nonempty, closed and convex subset of H. Let T : K ! K be an asymptotically nonexpansive map with a nonempty xed points set. Let f ng1n=1 and ftng1n=1 be real sequences in (0,1). Let fxng be a sequence generated from an arbitrary x0 2 K by yn = PK [(1 tn)xn]; n 0 xn+1 = (1 n)yn + nT nyn; n 0: where PK : H ! K is the metric projection. Under some appropriate mild conditions on f ng1n=1 and ftng1n=1, we prove that fxng converges strongly to xed point of T . No compactness assumption is imposed on T and or K and no further requirement is imposed on the xed point set F ix(T ) of T.