RESEA R C H Open Access Convergence theorem for finite family of lipschitzian demi-contractive semigroups Bashir Ali * and Godwin Chidi Ugwunnadi * Correspondence: bashiralik@yahoo.com Department of Mathematical Sciences, Bayero University, Kano, Nigeria Abstract Let E be a real Banach space and K be a nonempty, closed, and convex subset of E. Let {J i } N i = 1 be a finite family of Lipschitzian demi-contractive semigroups of K,with sequences of bounded measurable functions L i : [0, ∞) ® (0, ∞) and bounded functions l i : [0, ∞) ® (0, ∞), respectively, where J i := {T i ( t ) : t ≥ 0 } , i = 1,2, , N. Strong convergence theorem for common fixed point for finite family {J i } N i = 1 is proved in a real Banch space. As an application, a new convergence theorem for finite family of Lipschitzian demi-contractive maps is also proved. Mathematics subject classification (2000) 47H09, 47J25 Keywords: Demi-contractive maps, Demi-contractive semigroup, Demicompact maps, Fixed point 1. Introduction Let E be a real Banach space and E* be the dual space of E. The normalized duality mapping J : E → 2 E ∗ is defined by, x Î E, Jx = { x ∗ ∈ E ∗ : x, x ∗ = || x || || x ∗ || , || x ∗ || = || x ||}, where 〈., .〉 denotes the normalized duality pairing. For any x Î E, an element of Jx is denoted by j(x). Let K be a nonempty, closed and convex subset of E.LetT : K ® K be a map, a point x Î K is called a fixed point of T if Tx = x, and the set of all fixed points of T is denoted by F(T). The mapping T is called L-Lipschitzian or simply Lipschitz if ∃L>0, such that ||Tx -Ty|| ≤ L||x - y|| ∀x, y Î K and if L = 1, then the map T is called nonexpansive. A one parameter family J = {T ( t ) : t ≥ 0 } of self mapping of K is called a nonexpan- sive semigroup if the following conditions are satisfied, (i) T(0)x = x ∀ x Î K; (ii) T(t + s)=T(t) ∘ T(s) ∀ t, s ≥ 0; (iii) for each x Î K, the mapping t ® T(t)x is continuos; (iv) for x, y Î K and t ≥ 0, ||T(t)x -T(t)y|| ≤ ||x - y||. If the family J = {T ( t ) : t ≥ 0 } satisfies conditions (i)-(iii), then it is called (a) pseudocontractive semigroup if for any x, y Î K,thereexistsj(x - y) Î J(x - y) such that T ( t ) x − T ( t ) y, j ( x − y ) ≤ ||x − y|| 2 ; Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 © 2011 Ali and Ugwunnadi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Common s Attribu tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (b) strictly pseudocontractive semigroup if there exists a bounded function l :[0,∞) ® (0, ∞) and j(x - y) Î J(x - y) such that T ( t ) x − T ( t ) y, j ( x − y ) ≤ ||x − y|| 2 − λ ( t ) || ( I − T ( t )) x − ( I − T ( t )) y|| 2 for all x, y Î K; (c) demi-contractive semigroup if F(T(t)) ≠ ∅∀t ≥ 0, there exists a bounded function l : [0, ∞) ® (0, ∞), and j(x - y) Î J(x - y) such that T ( t ) x − q, j ( x − q ) ≤ ||x − q|| 2 − λ ( t ) ||x − T ( t ) x|| 2 for any x Î K and q Î F(T(t)); (d) Lipschitzian semigroup if there is a bounded measurable function L : [0, ∞) ® (0, ∞) such that for x, y Î K and t ≥ 0, ||T ( t ) x − T ( t ) y|| ≤ L ( t ) ||x − y|| . It is known that every strictly pseudocontractive semigroup is Lipschitzian, and every strictly pseudocontractive semigroup with fixed point is demi-contractive semi-group. Let E be a real Banach space and let K be a nonempty closed convex subset of E.A mapping T : K ® K is demicomp act if for every bounded sequence {x n }inK such that {xn - Tx n } converges, and there exists a subsequence, say {x n j } of {x n }thatconverges strongly to some x*inK. T is said to be demi-contractive if F(T) ≠ ∅, and there exists l >0 such that 〈Tx- q, j(x - q)〉 ≤ ||x - q|| 2 - l||x - Tx|| 2 ∀ x Î K, q Î F(T)andj(x - q) Î J (x - q). Let T 1 , T 2 , , T N be a family of self-mappings of K such that F := ∩ N i =1 F(T i ) = ∅ . Then, the family is said to satisfy condition C if there exists a nondecreasing function f :[0,∞) ® [0, ∞)withf (0) = 0 and f (r ) >0 ∀ r Î (0, ∞) such that f (d(x, F)) ≤ ||x - T s x|| for some s in {1, 2, , N} and for all x Î K, where d(x, F)=inf {||x - q|| : q Î F}. Existence theorems for family of nonexpansive mappings a re proved in [1-5] and actually many others. Recently, Suzuki [6] proved the equivalence between the fixed point property for nonexpansive mappings and that of the nonexpansive semi-groups. Both implicit and explicit, Mann, Ishikawa, and Halpern-typ e schemes we re studied for approximation of common fixed points of family o f nonexpansive semigroups and their generalizations in various spaces; see, for example [6-13], to list but a few. In 1998, Shoiji and Takahashi [7] introduced and studied a Halpern- type sche me for common fixed point of a family of asymptotically nonexpansive semigroup in the fra- mework of a real Hilbert space. Suzuki [8] proved that the implicit scheme defined by x, x 1 Î K, x n = α n T ( t n ) x n + ( 1 − α n )x converges strongly to a common fixed point of the family of nonexpansive semigroup in a real Hilbert space. Xu [9] extended the result of Suzuki to a more general real uni- formly convex Banach space having a weakly sequentially continuous duality mapping. In 2005, Aleyner and Reich [10] proved the strong convergence of an explicit Halpern- type scheme defined by x, x 1 Î K, x n+1 = α n T ( t n ) x n + ( 1 − α n )x Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 2 of 10 to a common fixed point of the family {T(t):t ≥ 0} of nonexpansive semigroup in a reflexive Banach space with uniformly Gatéu ax differentiable norm. Recently, Zhang et al. [11] introduced and studied a composite iterative scheme defined by x, x 1 Î K, x n+1 = α n y n + ( 1 − α n ) x; y n = β n T ( t n ) x n + ( 1 − β n ) x n . Those authors proved strong convergence of the sequence {x n } to a common fixed point of the family {T(t):t ≥ 0} of nonexpansive semigroup. Very recently, Chang et al. [12] proved a strong convergence theorem which extended and improved the results in [10,9] and some others. They proved the follow- ing theorem. Theorem 1.1. Chang et al. [12]Let K be a nonempty, closed, and convex subset of a real Banach space E: Let J := {T ( t ) : t ≥ 0 } be a Lipschitzian demi-contractive semi- group of K with bounded measurable function L :[0,∞) ® (0, ∞) and bounded func- tion l : [0, ∞) ® (0, ∞) such that L := sup t≥0 {L(t ) } < ∞, λ := inf t ≥ 0 {λ(t)} > 0andF := ∩ t≥0 F(T(t )) = ∅ . Let {t n } be an increasing sequence in [0, ∞) and {a n } be a sequence in (0,1) satisfying the following conditions, (i) ∞ n =1 (1 − α n )= ∞ ;(ii) ∞ n =1 (1 − α n ) 2 < ∞ . Assume that there exists a compact subset C of E such that ∪ t≥0 T(t)(K) ⊂ C and for any bounded set D ⊂ K lim n→∞ sup x∈D , s∈R + ||T(s + t n )x − T(t n )x|| =0 . Let {x n } be generated by x 1 Î K, x n+1 = α n x n + ( 1 − α n ) T ( t n ) x n . (1:1) Then, the sequence {x n } converges strongly to some element in F. Thepurposeinthisarticleistoproveastrong convergence theorem for common fixed point for finite families { J i } N i = 1 of demi-contractive semigroups in a real Banach space. As application, we al so prove convergence theorem for finite family of demi- contractive mappings. Our theorems generalize and improve several recent results. For instance, Theorem 1.1, which generalized, extended and improved several recent results, is a special case of our Theorem. 2. Preliminaries We shall make use of the following lemmas. Lemma 2.1. Let E be a real normed linear space. Then, the following inequality holds: ||x + y|| 2 ≤||x|| 2 +2y, j ( x + y ) , ∀ x, y ∈ Eandj ( x + y ) ∈ J ( x + y ). Lemma 2.2. (Xu [14]) Let {a n }and{b n } be sequences of nonnegative real numbers satisfying the inequality a n+1 ≤ ( 1+b n ) a n , n ≥ 1 . If ∞ n =1 b n < ∞ , then lim n →∞ a n exists. If in addition {a n } has a subsequence which con- verges strongly to zero, then lim n →∞ a n =0 . Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 3 of 10 Lemma 2.3. (Suzuki [15]) Let {x n } and {y n } be bounded sequences in a Banach space E and let {b n } be a sequence in [0, 1] with 0 <lim inf b n ≤ lim supb n <1. Suppose x n+1 = b n y n +(1 -b n )x n for all integers n ≥ 1 and lim sup(||y n+1 - y n || - ||x n+1 - x n ||) ≤ 0. Then, lim ||y n - x n || = 0. 3. Main Results Let E be a rea l Banach space, and K be a nonempty, closed convex subset of E. For some fixed i Î N,let J i := {T i ( t ) : t ≥ 0 } be a Lipschitzian demi-contractive semi- group with bounded measurable function L i :[0,∞) ® (0, ∞) and bounded function l i :[0,∞) ® (0, ∞) such that L i := sup t≥0 {L i (t )} < ∞, λ i := inf t ≥ 0 {λ i (t )} > 0andF i := ∩ t≥0 F(T i (t )) = ∅ . Then, for x, y Î K, q Î F i and t ≥ 0, T i ( t ) x − q, j ( x − q ) ≤||x − q|| 2 − λ i ||x − T i ( t ) x|| 2 and | |T i ( t ) x − T i ( t ) y|| ≤ L i ||x − y|| . Consider a family {J i } N i = 1 of Lipschitzian demi-contractive semigroups of K and let L := max 1 ≤ i ≤ N {L i } , L := max 1 ≤ i ≤ N {L i } and λ := min 1 ≤ i ≤ N {λ i } Clearly L<∞ and l >0andforx, y Î K, q ∈ F , t ≥ 0 and any i Î {1, 2, , N}, T i ( t ) x − q, j ( x − q ) ≤||x − q|| 2 − λ||x − T i ( t ) x|| 2 and ||T i ( t ) x − T i ( t ) y|| ≤ L||x − y|| . For a fixed δ Î (0, 1) and t ≥ 0 define a family S i (t):K ® Ki= 1, 2, , N by S i ( t ) x := ( 1 − δ 2 ) x + δ 2 T i ( t ) x, ∀x ∈ K . (3:1) Then, for x, y Î K and q ∈ F , S i (t )x − q, j(x − q) =(1− δ 2 )x − q, j(x − q) + δ 2 T i (t )x − q , j(x − q) ≤ (1 − δ 2 )||x − q|| 2 + δ 2 [||x − q|| 2 − λ||x − T i (t )x || 2 ] = ||x − q|| 2 − λδ 2 ||x − T i ( t ) x|| 2 . Let ¯ λ = λδ 2 > 0 , then S i ( t ) x − q, j ( x − q ) ≤ ||x − q|| 2 − ¯ λ||x − T i ( t ) x|| 2 . (3:2) Also, ||S i (t )x − S i (t )y|| = ||(1 − δ 2 )(x − y)+δ 2 (T i (t )x − T i (t )y)| | ≤ (1 − δ 2 )||x − y|| + δ 2 L||x − y|| =[1− δ 2 + δ 2 L]||x − y|| ≤ ( 1+δ 2 L ) ||x − y||. Let ¯ L =1+ δ 2 L . Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 4 of 10 Then, | |S i ( t ) x − S i ( t ) y|| ≤ ¯ L||x − y|| . (3:3) Hence, for each i Î {1, 2, N}, S i is Lipschitz with Lipschitz constant ¯ L > 0 . Lemma 3.1. Let E be a real Banach space and K be a nonempty closed convex subset of E. Let {J i } N i = 1 be a finite family of Lipschitzian demi-contractive semigroups of K with sequences of bounded measurable functions L i :[0,∞) ® (0, ∞) and bounded functions l i : [0, ∞) ® (0, ∞) i = 1, 2, , N such that for each i = 1, 2, , N, L i := sup t≥0 {L i (t )} < ∞, λ i := inf t ≥ 0 {λ i (t )} > 0andF i := ∩ t≥0 F(T i (t )) = ∅ . Let F := ∩ 1 ≤ i ≤ N { ∩ t≥0 F(T i (t ))} = ∅ ,{t n }be an increasing sequence in [0, ∞) and {a n } be a sequence in (0,1) satisfying the following conditions: (i) ∞ n =1 (1 − α n )= ∞ , (ii) ∞ n =1 (1 − α n ) 2 < ∞ . Assume ∀ i Î {1,2, , N} for any bounded set D ⊂ K the relation lim n→∞ sup x∈D , s∈R + ||T i (s + t n )x − T i (t n )x|| = 0 (3:4) holds. Let {x n } be a sequence generated by x 1 Î K, x n+1 = α n+1 x n + ( 1 − α n+1 ) S n+1 ( t n+1 ) x n , n ≥ 1 (3:5) where T n (t n )=T n modN (t n ). Then, (a) lim n →∞ ||x n − q| | exists for all q ∈ F . (b) lim inf n →∞ ||x n − T i (t n )x n || = 0 for all i Î {1,2,3, , N}. Proof. For any fixed q ∈ F using (3.5), we have x n+1 − q = ( x n − q ) + ( 1 − α n+1 )( S n+1 ( t n+1 ) x n − x n ). Thus, | |x n+1 − q|| 2 = ||(x n − q)+(1− α n+1 )(S n+1 (t n+1 )x n − x n )|| 2 ≤||x n − q|| 2 +2(1− α n+1 )S n+1 (t n+1 )x n − x n , j(x n+1 − q) = ||x n − q|| 2 +2(1− α n+1 ) S n+1 (t n+1 )x n − S n+1 (t n+1 )x n+1 , j(x n+1 − q) +S n+1 (t n+1 )x n+1 − q, j(x n+1 − q)−x n+1 − q, j(x n+1 − q) +x n+1 − x n , j(x n+1 − q) ≤||x n − q|| 2 +2(1− α n+1 )( ¯ L +1)||x n − x n+1 ||x n+1 − q|| −2(1 − α n+1 ) ¯ λ||x n+1 − T n+1 (t n+1 )x n+1 || 2 ≤||x n − q|| 2 +2(1− α n+1 ) 2 (1 + ¯ L) 2 ||S n+1 (t n+1 )x n − x n || ||x n − q|| −2(1 − α n+1 ) ¯ λ||x n+1 − T n+1 (t n+1 )x n+1 || 2 ≤||x n − q|| 2 +2(1− α n+1 ) 2 (1 + ¯ L) 3 ||x n − q|| 2 −2(1 − α n+1 ) ¯ λ||x n+1 − T n+1 (t n+1 )x n+1 || 2 =(1+σ n+1 )||x n − q|| 2 − 2(1 − α n+1 ) ¯ λ||x n+1 − T n+1 (t n+1 )x n+1 || 2 ≤ ( 1+σ n+1 ) ||x n − q|| 2 , (3:6) Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 5 of 10 where σ n+1 =2 ( 1+ ¯ L ) 3 ( 1 − α n+1 ) 2 . Since ∞ n =1 (1 − σ n+1 ) 2 < ∞ , by lemma 2.2, it follows that lim n →∞ ||x n − q| | exists. Hence, {x n } is bounded, which implies that {T n (t n )x n } and {S n (t n )x n } are also bounded. From (3.6) ||x n+1 − q|| 2 ≤||x n − q|| 2 +2(1− α n+1 ) 2 (1 + ¯ L) 3 ||x n − q|| 2 −2(1 − α n+1 ) ¯ λ||x n+1 − T n+1 (t n+1 )x n+1 || 2 ≤||x n − q|| 2 − 2 ( 1 − α n+1 ) ¯ λ||x n+1 − T n+1 ( t n+1 ) x n+1 || 2 +2 ( 1 − α n+1 ) 2 M , where, M := (1 + ¯ L) 3 sup n ∈ N (||x n − q|| 2 ) . Hence, for some m Î N, 2 ¯ λ m n=1 (1 − α n+1 )||x n+1 − T n+1 (t n+1 )x n+1 || 2 ≤ m n=1 (||x n − q|| 2 −||x n+1 − q|| 2 ) +2M m n=1 (1 − α n+1 ) 2 ≤||x 1 − q|| 2 +2M m n =1 (1 − α n+1 ) 2 < ∞. Since m Î N is arbitrary, we have 2 ¯ λ ∞ n =1 (1 − α n+1 )||x n+1 − T n+1 (t n+1 )x n+1 || 2 < ∞ which implies lim inf n →∞ ||x n+1 − T n+1 (t n+1 )x n+1 || =0 . (3:7) Next, we show that, lim n → ∞ ||x n+1 − x n || =0 . Let {b n }and{y n } be two sequences define by b n := δ(1 - δ)a n+1 + δ 2 and y n := x n+1 −x n +β n x n β n . Then, using the definition of {b n }and{S n }weobtainthat y n := δα n+1 x n +δ 2 (1−α n+1 )T n+1 (t n+1 )x n β n . Then, y n+1 − y n = δα n+2 β n+1 [x n+1 − x n ]+δ α n+2 β n+1 − α n+1 β n x n + δ 2 (1 − α n+2 ) β n+1 [T n+2 (t n+2 )x n+1 − T n+2 (t n+2 )x n ] + δ 2 1 − α n+2 β n+1 − 1 − α n+1 β n T n+2 (t n+2 )x n + δ 2 (1 − α n+1 ) β n [T n+2 (t n+2 )x n − T n+1 (t n+1 )x n ]. Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 6 of 10 Therefore, | |y n+1 − y n || − ||x n+1 − x n || ≤ δα n+2 β n+1 + δ 2 L(1 − α n+2 ) β n+1 − 1 ||x n+1 − x n || + δ α n+2 β n+1 − α n+1 β n ||x n || + δ 2 1 − α n+2 β n+1 − 1 − α n+1 β n ||T n+2 (t n+2 )x n || + δ 2 (1 − α n+1 ) β n ||T n+2 (t n+2 )x n − T n+1 (t n+1 )x n || . Hence, lim sup n →∞ (||y n+1 − y n || − ||x n+1 − x n ||) ≤ 0, and by lemma 2.3, lim n → ∞ ||y n − x n || =0 . Thus, | |x n+1 − x n || = β n || y n − x n || → 0 as n →∞ . This implies that, || x n+i − x n || → 0 as n →∞, ∀ i ∈ { 1, 2, 3, , N }. But, for i Î {1,2,3, , N}, | |x n − S n+i (t n+i )x n || ≤ δ 2 ||x n − x n+i || + ||x n+i − T n+i (t n+i )x n+i || + ||T n+i (t n+i )x n+i − T n+i (t n+i )x n || ≤ δ 2 [ ( 1+L ) ||x n+i − x n || + ||x n+i − T n+i ( t n+i ) x n+i ||] . Therefore, lim inf n →∞ ||x n − S n+i (t n+i )x n || =0 . Hence, lim inf n→∞ ||T n+i (t n+i )x n − x n || = lim inf n→∞ [ 1 δ 2 ||S n+i (t n+i )x n − x n ||]=0 . From the relation, | |T n+i (t n )x n − x n || ≤ ||T n+i (t n )x n − T n+i ((t n+i − t n )+t n )x n || +||T n+i (t n+i )x n − x n || ≤ sup z∈ { x n } ,s∈R + ||T n+i (t n )z − T n+i (s + t n )z|| + ||T n+i (t n+i )x n − x n || , and condition (3.4) we get lim inf n →∞ ||T n+i (t n )x n − x n || =0 . (3:8) It follows from (3.8) that lim inf n → ∞ ||T l (t n )x n − x n || =0∀ l ∈{1, 2, 3, , N } .Thiscom- pletes the proof. □ Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 7 of 10 Theorem 3.2. Let E, K, F ,{a n }, {t n }, {J i } N i = 1 and {x n } be as in lemma 3.1. Assume that, for at least one i Î {1, 2, , N}, there exi sts a compact subset C of E such that ∪ t≥0 T i (t) (K) ⊂ C. Then, the sequence {x n } converges to some element F . Proof. By Lemma 3.1, we have lim inf n →∞ ||T l (t n )x n − x n || =0∀ l ∈{1, 2, 3, , N } . If ∪ t≥0 T s (t)(K) ⊂ C for some compact subet C of E and some s Î {1,2, ,N}, then there exists a subsequence {x n k } ,of{x n } and q* Î K, such that x n k → q ∗ and ||T s (t n k )x n k − x n k || → 0asn →∞ . (3:9) Observe that for t>0, | |T s (t )x n k − x n k || ≤ ||T s (t )x n k − T s (t )T s (t n k )x n k || + ||T s (t )T s (t n k )x n k − T s (t n k )x n k || + ||T s (t n k )x n k − x n k || ≤||T s (t + t n k )x n k − T s (t n k )x n k || +(1+L) ||T s (t n k )x n k − x n k || . From the above we have lim k →∞ ||T s (t )x n k − x n k || = 0 . Using (3.9) and the fact that Ts is Lipschitzian, we get q* Î ∩ t≥0 F(T s (t)). Now, for any l Î {1,2, ,N }, since lim inf k → ∞ ||T l (t n k )x n k − x n k || = 0 , there exists a subse- quence {x n k j } of {x n k } such that lim j →∞ ||T l (t n k j )x n k j − x n k j || = lim inf k→∞ ||T l (t n k )x n k − x n k || = 0 .Then,similarlyfort ≥ 0, we obtain | |T l (t )x n k j − x n k j || ≤ ||T l (t )x n k j − T l (t )T l (t n k j )x n k j || +||T l (t )T l (t n k j )x n k j − T l (t n k j )x n k j || + ||T l (t n k j )x n k j − x n k j || ≤||T l (t + t n k j )x n k j − T l (t n k j )x n k j || +(1+L) ||T l (t n k j )x n k j − x n k j || . This implies that lim j →∞ ||T l (t )x n k j − x n k j || = 0 and hence q* Î ∩ t≥0 F(T l (t)). Since l Î {1, 2, N} is arbitrarily chosen, we have q ∗ ∈ F .Asthelimit lim n →∞ ||x n − q ∗ | | exists, the conclusion of the theorem follows immediately and this completes the proof. □ Remark 3.3. Observe that considering a single one-parameter family of demi-contrac- tive semigroup in Theorem 3.2, we obtain the conclusion of Theorem 1.1. Let T 1 , T 2 , , T N be a finite family of Lipschitzian demi-contractive self-mapping of K with positive constants l 1 , l 2 , , l N and Lipschitz constants L 1 ,L 2 , , L N , respectively. Let F := ∩ 1 ≤ i ≤ N F(T i ) = ∅ . For a fixed δ Î (0, 1), define S n : K ® K by S n x := ( 1 − δ 2 ) x + δ 2 T n x, ∀ x ∈ K . (3:10) Then, it follows that for x, y Î K and i Î F, S n x − q, j(x − q)≤||x − q|| 2 − ¯ λ||x − T n x|| 2 an d ||S n x − S n y || ≤ ¯ L||x − y ||, where ¯ λ = λδ 2 > 0 , ¯ L =1+ δ 2 L , λ := m i n 1 ≤ i ≤ N {λ i } and L := max 1 ≤ i ≤ N {L i } . The following Theorem is a consequence of Lemma 3.1. Theorem 3.4. Let E, K and {a n } be as in Lemma 3.1. Let T 1 , T 2 , , T N : K ® Kbe Lipschitzian demi-contractive mappings with T s demicompact for at least one s Î {1, 2, Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 8 of 10 , N}. Let {x n ] be a sequence generated by x 1 Î K x n+1 = α n+1 x n + ( 1 − α n+1 ) S n+1 x n , (3:11) where T n = T nmodN . Then,{x n } converges strongly to a common fixed point of the family {T i } N i = 1 . Proof. Following the line of proof of lemma 3.1 we immediat ely obtain lim n →∞ ||x n − q| | qk exists for any q Î F and lim inf n → ∞ ||T i x n − x n || = 0 , ∀i Î {1,2, . N}. Let { x n k } be a subsequence of {x n } such that lim k →∞ ||T i x n k − x n k || = lim inf n→∞ ||T i x n − x n || =0 . Therefore lim k →∞ ||T s x n k − x n k || =0 and, by demicompactness of T s , there exists a sub- sequence {x n k j } of {x n k } and q* Î K, such that x n k j → q ∗ as j ® ∞. Since, 0 = lim j→∞ ||T i x n k j − x n k j || = ||T i lim j→∞ x n k j − lim j→∞ x n k j | | = ||T i q ∗ − q ∗ ||, we obtain q* Î F.But, lim n →∞ ||x n − q ∗ | | exists, thus x n ® q* Î F and this completes the proof. □ The following corollaries follow from Theorem 3.4 Corollary 3.5. Let E, Kand{a n } be as in Th eorem 3.4. Let T 1 , T 2 , ,T N : K ® Kbe Lipschitzian demi-contractive mappings. Suppose there exists a comp act subset C in E such that N ∪ i =1 T i (K) ⊂ C . Let {x n } be defined by (3.11). Then,{x n } converges strongly to a common fixed point of the family {T i } N i = 1 . Corollary 3.6. Let E; K and {a n } be as in Th eorem 3.4. Let T 1 , T 2 , ,T N : K ® Kbe Lipschitzian demi-contractive mappings satisfying condition C . Let {x n } be defined by (3.11). Then,{x n } converges strongly to a common fixed point of the family {T i } N i = 1 . Proof. Following the line of proof of lemma 3.1, we obtain lim inf n → ∞ ||x n − T i x n || = 0 for all i Î {1,2,3, ,N}and||x n+1 - q|| 2 ≤ (1 + s n+1 )||x n - q|| 2 ,where σ n+1 =2 ( 1+ ¯ L ) 3 ( 1 − α n+1 ) 2 .Since ∞ n =1 (1 − σ n+1 ) 2 < ∞ , by lemma 2.2 lim n →∞ ||x n − p| | exists and consequently lim n →∞ d(x n , F ) exists. Let { x n k } be a subsequence of {x n }such that lim k →∞ ||x n k − T i x n k || = lim inf n→∞ ||x n − T i x n || = 0 . Then, by using condition C ,there exists s Î {1,2, ,N} such that 0 = lim k → ∞ ||x n k − T s x n k || ≥ lim k → ∞ f (d(x n k , F) ) and, using the property of f, we get that lim k →∞ d(x n k , F)= 0 , and since the limit lim n →∞ d(x n , F ) exists we have that lim n →∞ d(x n , F)=0 . We next s how that {x n } is Cauchy. Let ε >0begiven, then there exists p* Î F and n* Î N such that ∀n ≥ n*, ||x n − p ∗ || < ε 2 . Hence, for n ≥ n* and k Î N, we have | |x n+k − x n || ≤ ||x n+k − p ∗ || + ||x n − p ∗ | | <ε . Thus, {x n } is Cauchy and so x n ® q* Î K.Wenowshowthatq*isinF.Since lim n →∞ d(x n , F)=0 , there exists m 0 Î N large enough and p* Î F such that for all n ≥ m 0 , Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 9 of 10 and | |x n − p ∗ || < ε 6 ( 1+L ) .Hence, ||q ∗ − T l q ∗ || ≤ ||x n − q ∗ || + ||x n − p ∗ || + ||p ∗ − T l q ∗ | | ≤ ε 6(1 + L) + ε 6(1 + L) + L||p ∗ − q ∗ || < ε 6(1 + L) + ε 6(1 + L) + 3Lε 6(1 + L) < ε. Thus, q* Î F(T l )andsincel Î {1,2, ,N} is arbitrary, we have q* Î F.Thiscom- pletes the proof. □ Acknowledgements This study was conducted when the first author was visiting the AbdusSalam International Center for Theoretical Physics Trieste Italy as an Associate, and the hospitality and financial support provided by the centre is gratefully acknowledged. Authors’ contributions BA conceived the study, GCU carried out the computations for Theorem 3.4. BA Modified Theorem 3.4 to obtain Theorem 3.2. Both authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 6 March 2011 Accepted: 23 July 2011 Published: 23 July 2011 References 1. Belluce, LP, Kirk, WA: Fixed point theorem for families of contraction mappings. Pacific J Math. 18, 213–217 (1966) 2. Browder, FE: Nonexpansive nonlinear operators in Banach space. Proc Natl Acad Sci USA. 54, 1041–1044 (1965). doi:10.1073/pnas.54.4.1041 3. Bruck, RE: A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific J Math. 53, 59–71 (1974) 4. De Marr, R: Common fixed points for commuting contraction mappings. Pacific J Math. 13, 1139–1141 (1963) 5. Lim, TC: A fixed point theorem for families of nonexpansive mappings. Pacific J Math. 53, 487–493 (1974) 6. Suzuki, T: Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups. 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Chang, SS, Cho, YJ, Lee, HWJ, Chan, C: Strong convergence theorems for Lipschitzian demicontraction semigroups in Banach spaces, Fixed Point Theory Application. (2011) 13. Zhang, SS: Convergence theorem of common fixed points for Lipshitzian pseudocontraction semigroups in Banach spaces. Appl Math Mech. 30, 145–152 (2009). doi:10.1007/s10483-009-0202-y 14. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991). doi:10.1016/0362-546X(91) 90200-K 15. Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl. 305, 227–239 (2005). doi:10.1016/j.jmaa.2004.11.017 doi:10.1186/1687-1812-2011-18 Cite this article as: Ali and Ugwunnadi: Convergence theorem for finite family of lipschitzian demi-contractive semigroups. Fixed Point Theory and Applications 2011 2011:18. Ali and Ugwunnadi Fixed Point Theory and Applications 2011, 2011:18 http://www.fixedpointtheoryandapplications.com/content/2011/1/18 Page 10 of 10 . N. Strong convergence theorem for common fixed point for finite family {J i } N i = 1 is proved in a real Banch space. As an application, a new convergence theorem for finite family of Lipschitzian demi-contractive. F. Thepurposeinthisarticleistoproveastrong convergence theorem for common fixed point for finite families { J i } N i = 1 of demi-contractive semigroups in a real Banach space. As application, we al so prove convergence theorem for finite. Open Access Convergence theorem for finite family of lipschitzian demi-contractive semigroups Bashir Ali * and Godwin Chidi Ugwunnadi * Correspondence: bashiralik@yahoo.com Department of Mathematical Sciences,