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In this paper, CQ Algorithms for iterative approximation of a common fixed point of a finite family of nonlinear maps were introduced and sufficient conditions for the strong convergence of this process to a common fixed point of the family of Total asymptotically Nonexpansive maps (TAN) were proved.

Preliminaries

Let H be a normed space, K be a nonempty closed convex subset of H and T:KK be a map. The mapping T is called asymptotically nonexpansive mapping if and only if there exists a sequence {μn}n1[0,+), with limnμn=0 such that for all x.yK,

T n x T n y ( 1 + μ n ) x y        n N

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalisation of nonexpansive mappings. As further generalisation of class of nonexpansive mappings, Alber et al. [2] introduced the class of total asymptotically nonexpansive mappings, where a mapping T:KK is called total asymptotically nonexpansive (TAN) if and only if there exist two sequences {μn}n1,{ηn}n1[0,+), with limnμn=0=limnηn and nondecreasing continuous function ϕ:[0,+)[0,+) with ϕ(0)=0 such that for all x,yK,

T n x T n y x y + μ n ϕ ( x y ) + η n      n 1

In Ofoedu and Nnubia [3], an example to show that the class of asymptotically nonexpansive mappings is properly contained in the class of total asymptotically nonexpansive mappings was given. The class of asymptotically nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense. These classes of mappings had been studied extensively by several authors (see e.g., [4]–[9]).

A map T is said to satisfies condition B if there exists f:[0,)[0,) strictly increasing, continuous, f(0)=0,f(r)>0  r>0 such that for all xD(T),xTxf(d(x,F)) where F=F(T)={xD(T):x=Tx}  and  d(x,F)=inf{xy:yF}.

Lemma 1.1 Takahashi [10]

Let {μn},{βn},{γn} be sequences of nonnegative numbers satisfying the conditons: n0βn=,  βn0 as n and γn=0(βn). Suppose that where ψ:[0,1)[0,1) is a strictly increasing function with ψ(0)=0. Then μn0 as n.

μ n + 1 2 μ n 2 β n ψ ( μ n + 1 ) + γ n ;     n = 1 , 2 , . . .

Lemma 1.2 Moore and Nnoli [11]

Let K be closed convex nonempty subset of a real Hilbert Space H. Let {xn} be a sequence in H, yH and z=PKy be such that ωw(xn)K and xnyyzn1, then {xn} converges strongly to z.

ω w ( x n ) = { z H : { x n j ( z ) } { x n } x n j ( z ) w z   as   j }

Lemma 1.3 Ofoedu and Nnubia [9, p. 703]

Let E be a reflexive Banach space with weakly continuous normalized duality mapping. Let K be a closed convex subset of E and T:KK a uniformly continuous total asymptotically nonexpansive mapping with bounded orbits. Then IT is demiclosed at zero.

Proposition 1.1 Ofoedu and Nnubia [9, p. 704]

Let H be a real Hilbert space, let K be a nonempty closed convex subset of H and let Ti:KK,where iI={1,2,...,m}, be m uniformly continuous total asymptotically nonexpansive mappings from K into itself with sequences {μn,i}n1,{ηn,i}n1[0,+) such that limnμn,i=0=limnηn,i and with function ϕi:[0,+)[0,+) satisfying ϕi(t)M0t    t>M1 for some constants M0,M1>0. Let μn=maxiI{μn,i} and ηn=maxiI{ηn,i} and, ϕ(t)=maxiI{ϕi(t)} t[0,). Suppose that F(T)=i=1mF(Ti), then F(T) is closed and convex.

Proposition 1.2 Nnubia and Bishop [6, p. 74]

Let K be a nonempty subset of a real normed space E and Ti:KK,where iI={1,2,...,m}, be m total asymptotically nonexpansive mappings, then there exist sequences {μn}n1,{ηn}n1[0,+), with limnμn=0=limnηn and nondecreasing continuous function ϕ:[0,+)[0,+) with ϕ(0)=0 such that for all x,yK,

T i n x T i n y x y + μ n ϕ ( x y ) + η n ;      n 1 ,   i I .

Main Result

Proposition 2.1 Result

Suppose that there exist c>0,k>0 constants such that ϕ(t)ct tk, then T is total asymptotically nonexpansive if νn=μnc and γn=μnc0+ηn such that

T n x T n y ( 1 + ν n ) x y + γ n

Proof

Suppose T is total asymptotically nonexpansive, that is, let T be such that

T n x T n y x y + μ n ϕ ( x y ) + η n      n 1

Since ϕ is continuous, it follows that ϕ attains its maximum (say c0) on the interval [0,k]; moreover, ϕ(t)ct whenever t>k. Thus,

ϕ ( t ) c 0 + c t      t [ 0 , + ) .

So, we have, where νn=μnc and γn=μnc0+ηn Thus completing the proof.

T n x T n y x y + μ n ( c 0 + c x y ) + η n      n 1 = ( 1 + μ n c ) x y + μ n c 0 + η n = ( 1 + ν n ) x y + γ n

Theorem 2.1 Let H, K, Ti, and F be as in Propostion 1.1, then {xn}n1 generated iteratively by: converges to PFx0 where σn=αn(km(n),i(n)21)(diam.K)2+αn(2km(n),i(n)diam.K+vm(n),i(n))vm(n),(n) and {αn}[a,b](0,1).

y n = ( 1 α n ) x n + α n T i ( n ) m ( n ) x n ; i ( n ) n mod m      n Z ; m ( n ) = 1 + [ n m ] K n = { z K : y n z 2 x n z 2 α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 + σ n } Q n = { z K : x n z , x o x n 0 } x n + 1 = P K n Q n x 0 .

Proof

Let xF.

y n x 2 = ( 1 α n ) ( x n x ) + α n ( T i ( n ) m ( n ) x n x ) 2 ( 1 α n ) x n x 2 + α n T i ( n ) m ( n ) x n x 2 α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 ( 1 α n ) x n x 2 + α n ( 1 + k i ( n ) m ( n ) ) 2 x n x 2 + ( 2 ( 1 + k i ( n ) m ( n ) ) x n x + v i ( n ) m ( n ) ) v i ( n ) m ( n ) α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 = ( 1 + α n [ 1 + k i ( n ) m ( n ) 1 ] ) x n x 2 + ( 2 ( 1 + k i ( n ) m ( n ) ) x n x + v i ( n ) m ( n ) ) v i ( n ) m ( n ) α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 x n x 2 α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 + α n [ ( 1 + k i ( n ) m ( n ) ) 2 1 ] ( d i a m . K ) 2 + α n [ 2 ( 1 + k i ( n ) m ( n ) ) ( d i a m . K ) + v i ( n ) m ( n ) ] v i ( n ) m ( n ) = x n x 2 α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 + σ n .

So that xKnn. Hence, FKnn. For n=0,Qo=K. FQo.

Let FQv, we show that FQv+1. Now, xv+1 is the projection of xo onto KvQv. then (i) xv+1z,xoxv+10.zKvQv.

Since, FKvQv, then xv+1x,xoxv+10xF. So, FQv+1 and hence FQnn0.

Now,

x o P Q n x o x o y y Q n

Hence, n and since FQn, then xnxoxoxxF.

x n x o = x o P Q n x o x o y y Q n .

In particular, xnxoxox;x=PFxo.

Since, xn+1Qn,xn+1xn,xnxo0.

So,

x n + 1 x n 2 = x n + 1 x o ( x n x o ) 2 = x n + 1 x o 2 x n x o 2 2 x n + 1 x n , x n x o x n + 1 x o 2 x n x o 2

Then, xnxoxn+1xo.

Since {xnxo} is bounded, then limnxnxo exists.

Thus,

lim n x n + 1 x n = 0.

Observe that by (7)

lim n x n + i x n = 0 = lim n x n i x n   i { 1 , . . . , m } .

Now,

α n 2 x n T i ( n ) m ( n ) x n 2 = y n x n 2 ( y n x n + 1 + x n + 1 x n ) 2 = y n x n + 1 2 + 2 y n x n + 1 . x n + 1 x n + x n + 1 x n 2

xn+1Kn, so that

y n x n + 1 2 x n x n + 1 2 α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 + σ n .

Hence, and hence, αn2xnTi(n)m(n)xn22xn+1xn2+2ynxn+1.xn+1xn+σn

α n 2 x n T i ( n ) m ( n ) x n 2 x n x n + 1 2 α n ( 1 α n ) x n T i ( n ) m ( n ) x n 2 + σ n + 2 y n x n + 1 . x n + 1 x n + x n + 1 x n 2

Thus,

lim n x n T i ( n ) m ( n ) x n = 0

Now,   n>m, we have n=(nm)(modm) and since n=(m(n)1)m+i(n), we obtain nm=(m(n)1)N+i(n)m=(m(nm)1)+i(nm), so that nm=[(m(n)1)1]m+i(n)=(m(nm)1)m+i(nm). Hence, m(n)1=m(nm) and i(n)=i(nm). Similarly, m(n+1m)=m(n+1)1 and i(n + 1 − N) = i(n + 1). Using this we obtain but, so that by hypothesis, (9), (8) and the uniform countinuity of Ti i I we have:

x n T i ( n + 1 ) x n + 1 x n T i ( n + 1 ) m ( n + 1 ) x n + 1 + T i ( n + 1 ) m ( n + 1 ) x n + 1 T i ( n + 1 ) x n + 1 x n x n + 1 + x n + 1 T i ( n + 1 ) m ( n + 1 ) x n + 1 + T i ( n + 1 ) m ( n + 1 ) x n + 1 T i ( n + 1 ) x n + 1
T i ( n + 1 ) m ( n + 1 ) 1 x n + 1 x n + 1 T i ( n + 1 ) m ( n + 1 ) 1 x n + 1 T i ( n + 1 ) m ( n + 1 ) 1 m ( n + 1 ) 1 n + 1 m + T i ( n + 1 ) m ( n + 1 ) 1 x n + 1 m x n + 1 m + x n + 1 m x n + 1 ( 2 + k m ( n + 1 ) 1 ) x n + 1 x n + 1 m + T i ( n + 1 m ) m ( n + 1 m x n + 1 m x n + 1 m + v m ( n + 1 ) 1
lim n x n T i ( n + 1 ) x n + 1 = 0

Furthermore,

x n + 1 T i ( n + 1 ) x n + 1 x n + 1 x n + x n T i ( n + 1 ) x n + 1 .

So that using (7) and (11) we have,

lim n x n + 1 T i ( n + 1 ) x n + 1 = 0.

Now, let kI be arbitrarily choosen, then,

x n T i ( n + k ) x n x n x n + k + x n + k T i ( n + k ) x n + k + T i ( n + k ) x n + k T i ( n + k ) x n

By uniform continuity of TiiI and from (8) and (13) we have that

lim n x n T i ( n + k ) x n = 0 k I = { 1 , . . . , m }  

Observe that kI={1,..,m}    ηk I  such that  i(n)+ηkkmodm, put in another way, kI    ik I  such that  i(n+k)imodN Hence,

lim n x n T i x n = 0 ; i { 1 , . . . , m } .

Since (1Ti) is demiclosed at 0 Hi.{xn} is bounded and H is reflexive,

so, zK and {xnj}{xn} such that, xnjwz as j. Since, xnjTixnj0 as ji then zF(Ti)i and so zF=i=1mF(Ti)

Let qωw(xn) arbitrary. Then {xnr}{xn}xnrwq and xnrTixnr0 as ri. So that since 1Ti is demiclosed at 0i,qF. Hence, ωw(xn)F. Moreover, xnxoxoxn0 where x=PFxo. Then by the lemma 1.2 {xn} converges strongly to x=PFxo (that is the common fixed point nearest to x0).

Theorem 2.2 Let H, K, Ti, and F be as in Theorem 2.1, then {xn}n1 generated iteratively by converges to PFx0 where σn,i=doj=1i[(kn,j21)+vn,j], iI,0<aαnb<1.

y n , 0 = x n ;   y n , i = ( 1 α n ) x n + α n T i n y n , i 1 ;   i = 1 , . . . , m K n , i = { z K : y n , i z 2 x n z 2 α n ( 1 α n ) a i j = 0 i 1 x n T i j n y n , i j 1 2 + σ n , i } K n = i = 1 m K n , i
Q n = { z K : x n z , x o x n 0 } x n + 1 = P K n Q n x 0 .

Proof

Let xF.

y n , i x 2 ( 1 α n ) x n x 2 + α n k n , i 2 y n , i 1 x 2 + α n ( 2 k n , i y n , i 1 x + v n , i ) η n , i α n ( 1 α n ) x n T i n y n , i 1 2

So,

y n , 1 x 2 ( 1 + α n ( k n , 1 2 1 ) ) x n x 2 + α n ( 2 k n , 1 x n x + v n , 1 ) v n , 1 α n ( 1 α n ) x n T 1 n x n

So, where do=bqimax{d1,(diamK)2} so, x Kn,i i and hence xi=1mKn,i  n. So, FKn,i n0,  i and hence, FKn. Following the argument as in Theorem 2.1, we have that FQnn0 and

y n , i x 2 ( 1 + j = 0 i 1 α n j + 1 Π t = 0 j 1 k n , i t 2 ( k n , i j 2 1 ) ) x n x 2 + j = 0 i 1 α n j + 1 ( 2 k n , i j y n , i j 1 x 2 + v n , i j ) v n , i j Π t = 0 j 1 k n , i t 2 j = 0 j 1 α n j + 1 ( 1 α n ) x n T i j n y n , i j 1 2 Π t = 0 j 1 k n , i t 2 ( 1 + b q i j = 1 i ( k n , j 2 1 ) ) x n x 2 + b q i d 1 j = 1 i v n , j a i ( 1 b ) j = 1 i x n T j n y n , j 1 2 x n x 2 a i ( 1 b ) j = 1 i x n T j n y n , j 1 2 + σ n , i
lim n x n + 1 x n = 0.

Now, but xn+1Kn, so, iI so that iI and so iI

α n 2 x n T i n y n , i 1 2 y n , i x n + 1 2 + 2 y n , i x n + 1 . x n + 1 x n + x n + 1 x n 2
y n , i x n + 1 2 x n x n + 1 2 α n ( 1 α n ) a i j = 1 i x n T j n y n , j 1 2 + σ n , i .
α n 2 x n T i n y n , i 1 2 x n + 1 x n 2 + σ n , i α n ( 1 α n ) a i j = 1 i x n T j n y n , j 1 2 + 2 y n , i x n + 1 . x n + 1 x n + x n + 1 x n 2
a 2 x n T i n y n , i 1 2 α n 2 x n T i n y n , i 1 2 2 x n + 1 x n 2 + σ n , i + 2 y n , i x n + 1 . x n + 1 x n

Hence, limnxnTinyn,i1=0 iI and hence, limnyn,ixn=0 iI

x n T i n x n x n T i n y n , i 1 + k n , i y n , i 1 x n + v n , i

Thus, limnxnTinxn=0 iI.

x n T i x n x n T i n x n + T i n x n T i x n
x n T i n 1 x n ( 1 + k n 1 , i ) x n x n 1 + x n 1 T i n 1 x n 1

Thus, limnxnTin1xn=0 iI, so that by uniform continuity of Ti iI, limnxnTixn=0. Now, since (1Ti) is demiclosed at 0, the same argument in Theorem 2.1 completes the Proof.

Conclusion

CQ Algorithms for iterative approximation of a common fixed point of a finite family of nonlinear maps were introduced and sufficient conditions for the strong convergence of this process to a common fixed point of the family of Total asymptotically Nonexpansive maps were proved. Our iterative processes generalise some of the existing ones, our theorems improve, generalise and extend several known results and our method of proof is of independent interest.

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