Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications

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1 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 R E S E A R C H Open Access Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications Wataru Takahashi,, Ngai-Ching Wong,3 and Jen-Chih Yao 3,4* * Correspondence: yaojc@kmu.edu.tw 3 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 8070, Taiwan 4 Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah, 589, Saudi Arabia Full list of author information is available at the end of the article Abstract Recently, Kawasaki and Takahashi J. Nonlinear Convex Anal. 4:7-87, 03) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings Kocourek et al. in Taiwan. J. Math. 4:497-5, 00) and strict pseudo-contractive mappings Browder and Petryshyn in J. Math. Anal. Appl. 0:97-8, 967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo et al. Fixed Point Theory :3-6, 0) and Kawasaki and Takahashi fixed point theorems J. Nonlinear Convex Anal. 4:7-87, 03). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem J. Math. Anal. Appl. 0:97-8, 967) for strict pseudo-contractive non-self mappings and the Kocourek et al. fixed point theorem Taiwan. J. Math. 4:497-5, 00) for super hybrid non-self mappings. In particular, we solve a fixed point problem. MSC: Primary 47H0; secondary 47H05 Keywords: Hilbert space; nonexpansive mapping; nonspreading mapping; hybrid mapping; fixed point; non-self mapping Introduction Let R bethereallineandlet[0, π ] be a bounded, closed and convex subset of R. Consider a mapping T :[0, π ] R defined by Tx = + ) x cos x x for all x [0, π ]. Such a mapping T has a unique fixed point z [0, π ]suchthatcos z = z. What kind of fixed point theorems can we use to find such a unique fixed point z of T? Let H be a real Hilbert space and let C be a non-empty subset of H. Kocourek,Takahashi and Yao [] introduced a class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings [] and hybrid mappings [3]. A mapping T : C H is said to be generalized hybrid if there exist α, β R such that α Tx Ty + α) x Ty β Tx y + β) x y.) for all x, y C. We call such a mapping an α, β)-generalized hybrid mapping. An α, β)-generalized hybrid mapping is nonexpansive for α =andβ =0,i.e., Tx Ty x y 03 Takahashi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page of 4 for all x, y C. It is nonspreading for α =andβ =,i.e., Tx Ty x Ty + y Tx for all x, y C.Furthermore,itishybridforα = 3 and β =, i.e., 3 Tx Ty x Ty + y Tx + y x for all x, y C. They proved fixed point theorems and nonlinear ergodic theorems of Baillon type [4] for generalized hybrid mappings; see also Kohsaka and Takahashi [5] and Iemoto and Takahashi [6].Very recently,kawasaki and Takahashi [7] introduced a broader class of nonlinear mappings than the class of generalized hybrid mappings in a Hilbert space. A mapping T from C into H is called widely more generalized hybrid if there exist α, β, γ, δ, ε, ζ, η R such that α Tx Ty + β x Ty + γ Tx y + δ x y + ε x Tx + ζ y Ty + η x Tx) y Ty) 0.) for all x, y C. Such a mapping T is called an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping. In particular, an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping is generalized hybrid in the sense of Kocourek, Takahashi and Yao []ifα + β = γ δ = and ε = ζ = η =0.Anα, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping is strict pseudo-contractive in the sense of Browder and Petryshyn [8] ifα =,β = γ =0,δ =, ε = ζ =0,η = k, where0 k <.Ageneralizedhybridmappingwithafixedpointis quasi-nonexpansive. However, a widely more generalized hybrid mapping is not quasinonexpansive in general even if it has a fixed point. In [7], Kawasaki and Takahashi proved fixed point theorems and nonlinear ergodic theorems of Baillon type [4] for such widely more generalized hybrid mappings in a Hilbert space. In particular, they proved directly the Browder and Petryshyn fixed point theorem [8] for strict pseudo-contractive mappings and the Kocourek, Takahashi and Yao fixed point theorem [] for super hybrid mappings by using their fixed point theorems. However, we cannot use Kawasaki and Takahashi fixed point theorems to solve the above problem. For a nice synthesis on metric fixed point theory, see Kirk [9]. In this paper, motivated by such a problem, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo, Takahashi and Yao [0] and Kawasaki and Takahashi fixedpoint theorems [7]. Using these fixed point theorems for non-self mappings, we prove the Browder and Petryshyn fixed point theorem [8] for strict pseudo-contractive non-self mappings and the Kocourek, Takahashi and Yao fixed point theorem [] for super hybrid non-self mappings. In particular, we solve the above problem by using one of our fixed point theorems. Preliminaries Throughout this paper, we denote by N the set of positive integers. Let H be a real) Hilbert space with the inner product, and the norm,respectively.from[], we know the following basic equality: For x, y H and λ R,wehave λx + λ)y = λ x + λ) y λ λ) x y..)

3 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 3 of 4 Furthermore, we know that for x, y, u, v H, x y, u v = x v + y u x u y v..) Let C be a non-empty, closed and convex subset of H and let T be a mapping from C into H. ThenwedenotebyFT) thesetoffixedpointsoft. A mapping S : C H is called super hybrid [, ]ifthereexistα, β, γ R such that α Sx Sy + α + γ ) x Sy β +β α)γ ) Sx y + β β α )γ ) x y +α β)γ x Sx + γ y Sy.3) for all x, y C. We call such a mapping an α, β, γ )-super hybrid mapping. An α, β,0)- super hybrid mapping is α, β)-generalized hybrid. Thus the class of super hybrid mappings contains generalized hybrid mappings. The following theorem was proved in []; see also []. Theorem. []) Let C be a non-empty subset of a Hilbert space H and let α, β and γ be real numbers with γ.let S and T be mappings of C into H such that T = +γ S + γ +γ I. Then S is α, β, γ )-super hybrid if and only if T is α, β)-generalized hybrid. In this case, FS)=FT). In particular, let C be a nonempty, closed and convex subset of H and let α, β and γ be real numbers with γ 0. If a mapping S : C Cisα, β, γ )-super hybrid, then the mapping T = +γ S + γ Iisanα, β)-generalized hybrid mapping of C into itself. +γ In [], Kocourek, Takahashi and Yao also proved the following fixed point theorem for super hybrid mappings in a Hilbert space. Theorem. []) Let C be a non-empty, bounded, closed and convex subset of a Hilbert space H and let α, β and γ be real numbers with γ 0. Let S : C Cbeanα, β, γ )-super hybrid mapping. Then S has a fixed point in C. In particular, if S : C C is an α, β)-generalized hybrid mapping, then S has a fixed point in C. A super hybrid mapping is not quasi-nonexpansive in general even if it has a fixed point. There exists a class of nonlinear mappings in a Hilbert space defined by Kawasaki and Takahashi [3] which covers contractive mappings and generalized hybrid mappings. A mapping T from C into H is said to be widely generalized hybrid if there exist α, β, γ, δ, ε, ζ R such that α Tx Ty + β x Ty + γ Tx y + δ x y + max { ε x Tx, ζ y Ty } 0 for any x, y C. Such a mapping T is called α, β, γ, δ, ε, ζ )-widely generalized hybrid. Kawasaki and Takahashi [3] proved the following fixed point theorem. Theorem.3 [3]) Let H be a Hilbert space, let C be a non-empty, closed and convex subset of H and let T be an α, β, γ, δ, ε, ζ )-widely generalized hybrid mapping from C into itself which satisfies the following conditions ) and ):

4 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 4 of 4 ) α + β + γ + δ 0; ) ε + α + γ >0, or ζ + α + β >0. Then T has a fixed point if and only if there exists z Csuchthat{T n z n =0,,...} is bounded. In particular, a fixed point of T is unique in the case of α + β + γ + δ >0under the condition ). Veryrecently, KawasakiandTakahashi[7] also proved the following fixed point theorem which will be used in the proofs of our main theorems in this paper. Theorem.4 [7]) Let H be a Hilbert space, let C be a non-empty, closed and convex subset of H and let T be an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping from C into itself, i.e., there exist α, β, γ, δ, ε, ζ, η R such that α Tx Ty + β x Ty + γ Tx y + δ x y + ε x Tx + ζ y Ty + η x Tx) y Ty) 0 for all x, y C. Suppose that it satisfies the following condition ) or ): ) α + β + γ + δ 0, α + γ + ε + η >0and ζ + η 0; ) α + β + γ + δ 0, α + β + ζ + η >0and ε + η 0. Then T has a fixed point if and only if there exists z Csuchthat{T n z n =0,,...} is bounded. In particular, a fixed point of T is unique in the case of α + β + γ + δ >0under the conditions ) and ). In particular, we have the following theorem from Theorem.4. Theorem.5 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping from C into itself which satisfies the following condition ) or ): ) α + β + γ + δ 0, α + γ + ε + η >0and ζ + η 0; ) α + β + γ + δ 0, α + β + ζ + η >0and ε + η 0. Then T has a fixed point. In particular, a fixed point of T is unique in the case of α + β + γ + δ >0under the conditions ) and ). 3 Fixed point theorems for non-self mappings In this section, using the fixed point theorem Theorem.5), we first prove the following fixed point theorem for widely more generalized hybrid non-self mappings in a Hilbert space. Theorem 3. Let C be a non-empty, bounded, closed and convex subset of a Hilbert space Handletα, β, γ, δ, ε, ζ, η R. Let T : C Hbeanα, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping. Suppose that it satisfies the following condition )or ): ) α + β + γ + δ 0, α + γ + ε + η >0, α + β + ζ + η 0 and ζ + η 0; ) α + β + γ + δ 0, α + β + ζ + η >0, α + γ + ε + η 0 and ε + η 0. Assume that there exists a positive number m >such that for any x C, Tx = x + ty x)

5 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 5 of 4 for some y Candtwith0<t m. Then T has a fixed point in C. In particular, afixed point of T is unique in the case of α + β + γ + δ >0under the conditions ) and ). Proof We give the proof for the case of ). By the assumption, we have that for any x C, there exist y C and t with 0 < t m such that Tx = x + ty x). From this, we have Tx = ty + t)x and hence y = t t Tx + x. t Define Ux C as follows: Ux = t )x + tm m y = t ) x + t ) t Tx + x = m Tx + m m t t m m x. Taking λ >0withm =+λ,wehavethat Ux = +λ Tx + λ +λ x and hence T =+λ)u λi. 3.) Since T : C H is an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping, we have from 3.)and.) that for any x, y C, α + λ)ux λx + λ)uy λy ) + β x + λ)uy λy ) + γ + λ)ux λx y + δ x y + ε x + λ)ux λx ) + ζ + λ)uy λy y + η x + λ)ux λx ) y + λ)uy λy )) = α + λ)ux Uy) λx y) + β + λ)x Uy) λx y) + γ + λ)ux y) λx y) + δ x y + ε + λ)x Ux) + ζ + λ)y Uy) + η + λ)x Ux) +λ)y Uy) = α + λ) Ux Uy αλ x y + αλ + λ) x y Ux Uy) + β + λ) x Uy βλ x y + βλ + λ) y Uy + γ + λ) Ux y γλ x y + γλ + λ) x Ux + δ x y + ε + λ) x Ux + ζ + λ) y Uy + η + λ) x Ux y Uy) = α + λ) Ux Uy + β + λ) x Uy + γ + λ) Ux y + αλ βλ γλ+ δ) x y

6 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 6 of 4 +γλ+ ελ + ε) + λ) x Ux +βλ+ ζλ+ ζ ) + λ) y Uy +αλ + ηλ + η) + λ) x y Ux Uy) 0. This implies that U is widely more generalized hybrid. Since α +β +γ +δ 0, α +γ +ε +η > 0, α + β + ζ + η 0andζ + η 0, we obtain that α + λ)+β + λ)+γ + λ) αλ βλ γλ+ δ = α + β + γ + δ 0, α + λ)+γ + λ)+γλ+ ελ + ε) + λ)+αλ + ηλ + η) + λ) =+λ) α + γ + ε + η + λγ + ε + α + η) ) =+λ) α + γ + ε + η)>0, βλ+ ζλ+ ζ ) + λ)+αλ + ηλ + η) + λ) = α + β + ζ + η)λ + ζ + η ) + λ) 0. By Theorem.5, weobtainthatfu). Therefore, we have from FU) =FT) that FT).Supposethatα + β + γ + δ >0.Letp and p be fixed points of T.Wehavethat α Tp Tp + β p Tp + γ Tp p + δ p p + ε p Tp + ζ p Tp + η p Tp ) p Tp ) =α + β + γ + δ) p p 0 and hence p = p. Therefore, a fixed point of T is unique. Similarly, we can obtain the desired result for the case when α + β + γ + δ 0, α + β + ζ + η >0,α + γ + ε + η 0andε + η 0. This completes the proof. The following theorem is a useful extension of Theorem 3.. Theorem 3. Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping from C into H which satisfies the following condition ) or ): ) α + β + γ + δ 0, α + γ + ε + η >0, α + β + ζ + η 0 and [0, ) {λ α + β)λ + ζ + η 0} ; ) α + β + γ + δ 0, α + β + ζ + η >0, α + γ + ε + η 0 and [0, ) {λ α + γ )λ + ε + η 0}. Assume that there exists m >such that for any x C, Tx = x + ty x) for some y Candtwith0<t m. Then T has a fixed point. In particular, afixedpoint of T is unique in the case of α + β + γ + δ >0under the conditions ) and ). Proof Let λ [0, ) {λ α + β)λ + ζ + η 0} and define S = λ)t + λi. ThenS is a mapping from C into H. Sinceλ,weobtainthatFS)=FT). Moreover, from T =

7 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 7 of 4 λ S λ I and.), we have that λ α λ Sx λ ) λ x λ Sy λ ) λ y + β x λ Sy λ ) λ y + γ λ Sx λ ) λ x y + δ x y + ε x λ Sx λ ) λ x + ζ y λ Sy λ ) λ y + η x λ Sx λ )) λ x y λ Sy λ )) λ y = α λ Sx Sy) x y) λ λ + β λ x Sy) x y) λ λ + γ λ Sx y) x y) λ λ + δ x y + ε x Sx) λ + ζ y Sy) λ + η x Sx) y Sy) λ λ = α λ Sx Sy + β + γ λ Sx y + x Sy λ λ α + β + γ )+δ λ + ε + γλ λ) x Sx + ζ + βλ y Sy λ) + η + αλ λ) x Sx) y Sy) 0. ) x y Therefore S is an α λ, β λ, γ λ, λ ε+γλ α + β + γ )+δ,, ζ +βλ, η+αλ )-widely more generalized hybrid mapping. Furthermore, we obtain λ λ) λ) λ) that α λ + β λ + γ λ λ α + β + γ )+δ = α + β + γ + δ 0, λ α λ + γ λ + ε + γλ λ) + η + αλ λ) = α + γ + ε + η >0, λ) α λ + β λ + ζ + βλ λ) + η + αλ λ) = α + β + ζ + η 0, λ) ζ + βλ λ) + η + αλ α + β)λ + ζ + η = 0. λ) λ) Furthermore, from the assumption, there exists m > such that for any x C, Sx = λ)tx + λx = λ) x + ty x) ) + λx = t λ)y x)+x,

8 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 8 of 4 where y C and 0 < t m. From0 λ <,wehave0<t λ) m. Putting s = t λ), we have that there exists m > such that for any x C, Sx = x + sy x) for some y C and s with0<s m. Therefore, we obtain from Theorem 3. that FS). Since FS)= FT), we obtain that FT). Next, suppose that α + β + γ + δ >0.Letp and p be fixed points of T.Asintheproof of Theorem 3.,wehavep = p. Therefore a fixed point of T is unique. In the case of α + β + γ + δ 0, α + β + ζ + η >0,α + γ + ε + η 0and[0,) {λ α + γ )λ + ε + η 0}, we can obtain the desired result by replacing the variables x and y. Remark We can also prove Theorems 3. and 3. by using the condition β δ + ε + η >0, or γ δ + ε + η >0 instead of the condition α + γ + ε + η >0, or α + β + ζ + η >0, respectively. In fact, in the case of the condition β δ + ε + η >0,weobtainfromα + β + γ + δ 0that 0< β δ + ε + η α + γ + ε + η. Thus we obtain the desired results by Theorems 3. and 3.. Similarly, in the case of γ δ + ε + η > 0, we can obtain the results by using the case of α + β + ζ + η >0. 4 Fixed point theorems for well-known mappings Using Theorem 3., we first show the following fixed point theorem for generalized hybrid non-self mappings in a Hilbert space; see also Kocourek, Takahashi and Yao []. Theorem 4. Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a generalized hybrid mapping from C into H, i.e., there exist α, β R such that α Tx Ty + α) x Ty β Tx y + β) x y for any x, y C. Suppose α β 0 and assume that there exists m >such that for any x C, Tx = x + ty x) for some y Candtwith0<t m. Then T has a fixed point. Proof An α, β)-generalized hybrid mapping T from C into H is an α, α, β, β), 0, 0, 0)-widely more generalized hybrid mapping. Furthermore, α + α) β β)=

9 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 9 of 4 0, α + α)+0+0=>0,α β +0+0=α β 0 and = 0, that is, it satisfies the condition ) in Theorem 3.. Furthermore, since there exists m such that for any x C, Tx = x + ty x) for some y C and t with 0 < t m, we obtain the desired result from Theorem 3.. Using Theorem 3., we can alsoshow the following fixedpoint theoremforwidely generalized hybrid non-self mappings in a Hilbert space; see Kawasaki and Takahashi [3]. Theorem 4. Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be an α, β, γ, δ, ε, ζ )-widely generalized hybrid mapping from C into H which satisfies the following condition ) or ): ) α + β + γ + δ 0, α + γ + ε >0and α + β 0; ) α + β + γ + δ 0, α + β + ζ >0and α + γ 0. Assume that there exists m >such that for any x C, Tx = x + ty x) for some y Candt R with 0<t m. Then T has a fixed point. In particular, afixed point of T is unique in the case of α + β + γ + δ >0under the conditions ) and ). Proof Since T is α, β, γ, δ, ε, ζ )-widely generalized hybrid, we obtain that α Tx Ty + β x Ty + γ Tx y + δ x y + max { ε x Tx, ζ y Ty } 0 for any x, y C.Inthecaseofα + γ + ε >0,from ε x Tx max { ε x Tx, ζ y Ty }, we obtain that α Tx Ty + β x Ty + γ Tx y + δ x y + ε x Tx 0, that is, it is an α, β, γ, δ, ε, 0, 0)-widely more generalized hybrid mapping. Furthermore, we have that α + β + γ + δ 0, α + γ + ε +0=α + γ + ε >0,α + β +0+0=α + β 0 and = 0, that is, it satisfies the condition ) in Theorem 3.. Furthermore, since there exists m such that for any x C, Tx = x + ty x) for some y C and t with 0 < t m, we obtain the desired result from Theorem 3.. In thecaseofα + β + γ + δ 0, α + β + ζ >0andα + γ 0,wecanobtainthedesiredresult by replacing the variables x and y.

10 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 0 of 4 We know that an α, β, γ, δ, ε, ζ, η)-widely more generalized hybrid mapping with α =, β = γ = ε = ζ =0,δ = andη = k, 0] is a strict pseudo-contractive mapping in the sense of Browder and Petryshyn [8]. We also define the following mapping: T : C H is called a generalized strict pseudo-contractive mapping if there exist r, k R with 0 r and 0 k <suchthat Tx Ty r x y + k x Tx) y Ty) for any x, y C. UsingTheorem3., we can show the following fixed point theorem for generalized strict pseudo-contractive non-self mappings in a Hilbert space. Theorem 4.3 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a generalized strict pseudo-contractive mapping from C into H, that is, there exist r, k R with 0 r and 0 k <such that Tx Ty r x y + k x Tx) y Ty) for all x, y C. Assume that there exists m >such that for any x C, Tx = x + ty x) for some y Candt R with 0<t m. Then T has a fixed point. In particular, if 0 r <, then T has a unique fixed point. Proof A generalized strict pseudo-contractive mapping T from C into H is a, 0, 0, r,0, 0, k)-widely more generalized hybrid mapping. Furthermore, r) 0, k)= k >0,+0+0+ k)= k >0and[0,) {λ +0)λ+0 k 0} =[k,), that is, it satisfies the condition ) in Theorem 3.. Furthermore, since there exists m such that for any x C, Tx = x + ty x) for some y C and t with 0 < t m, we obtain the desired result from Theorem 3.. In particular, if 0 r <,then+0+0+ r)>0.wehavefromtheorem3. that T has a unique fixed point. Let us consider the problem in the Introduction. A mapping T :[0, π ] R was defined as follows: Tx = + ) x cos x x 4.) for all x [0, π ]. We have that Tx = + ) x cos x x + xtx + x + xx = cos x.

11 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page of 4 Thus we have that for any x [0, π ], + x +π + xtx + x + x and hence = + x +π cos x + + x +π ) + + x +π ) x, +π Tx + π +π x = + x +π cos x + π x +π x. ) x Using this, we also have from.) that for any x, y [0, π ], +π Tx + π = and hence +π x + x +π cos x + π x +π x +π Ty + π +π y ) + y +π cos y + π y +π y +π Tx Ty + π +π x π y x y Tx Ty) + π) + = x +π cos x + π x + +π x y +π cos y + π y ) +π y. 4.) Define a function f :[0, π ] R as follows: f x)= + x +π cos x + π x +π x for all x [0, π ]. Then we have and Since f x)= +π cos x + x +π sin x + π +π x +π f x)= +π sin x + x +π cos x +π. f 0) = + π +π, f π ) = + 4 π +π and f x) < 0 for all x [0, π ], we have from the mean value theorem that there exists a positive number r with 0 < r <suchthat ) + x +π cos x + π x + +π x y +π cos y + π y ) +π y r x y

12 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page of 4 for all x, y [0, π ]. Therefore, we have from 4.)that +π Tx Ty + π +π x y r x y π + x y Tx Ty) + π) for all x, y [0, π ]. Furthermore, we have from 4.)that Tx = + ) x cos x x)+x for all x [0, π ]. Take m =+π and let t =+ x and y = cos x for all x [0, π ]. Then we have that [ Tx = ty x)+x, y = cos x 0, π ] and 0 < t =+ x +π. Using Theorem 3.,wehavethatT has a unique fixed point z [0, π ]. We also know that z = Tz is equivalent to cos z = z.infact, z = Tz z = + ) z cos z z)+z 0= + ) z cos z z) 0=cos z z. Using Theorem 3., we canalsoshow thefollowingfixedpointtheoremforsuper hybrid non-self mappings in a Hilbert space; see []. Theorem 4.4 Let H be a Hilbert space, let C be a non-empty, bounded, closed and convex subset of H and let T be a super hybrid mapping from C into H, that is, there exist α, β, γ R such that α Tx Ty + α + γ ) x Ty β +β α)γ ) Tx y + β β α )γ ) x y +α β)γ x Tx + γ y Ty for all x, y C. Assume that there exists m >such that for any x C, Tx = x + ty x) for some y Candtwith0<t m. Suppose that α β 0 or γ 0. Then T has a fixed point. Proof An α, β, γ )-super hybrid mapping T from C into H is an α, α + γ, β β α)γ, + β +β α )γ, α β)γ, γ, 0)-widely more generalized hybrid mapping. Furthermore, α + α + γ )+ β β α)γ )+ +β +β α )γ)=0,α + α + γ )+ γ)+0=>0andα β β α)γ α β)γ +0=α β 0, that is, it satisfies the

13 Takahashi et al. Fixed Point Theory and Applications 03, 03:6 Page 3 of 4 conditions α + β + γ + δ 0, α + β + ζ + η >0andα + γ + ε + η 0in)ofTheorem3.. Moreover, we have that [0, ) { λ α + β β α)γ )) λ + α β)γ ) +0 0 } =[0,) { λ α β) + γ )λ γ ) 0 }. If α β >0,then [0, ) { λ α β) + γ )λ γ ) 0 } =[0,) { λ + γ )λ γ 0 } [0, ) if γ <0, =,) ifγ 0, [ γ +γ that is, it satisfies the condition [0, ) {λ α + γ )λ + ε + η 0} in ) of Theorem 3.. If α β =0,then [0, ) { λ α β) + γ )λ γ ) 0 } =[0,), that is, it satisfies the condition [0, ) {λ α + γ )λ + ε + η 0} in ) of Theorem 3.. If α β <0andγ 0, then [0, ) { λ α β) + γ )λ γ ) 0 } =[0,) { λ + γ )λ γ 0 } = [ 0, γ +γ ], that is, it again satisfies the condition [0, ) {λ α + γ )λ + ε + η 0} in ) of Theorem 3.. Then we obtain the desired result from Theorem 3.. Similarly, we obtain the desired result from Theorem 3. in the case of ). We remark that some recent results related to this paper have been obtained in [4 7]. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to this work. All authors read and approved the final manuscript. Author details Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 5-855, Japan. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 8044, Taiwan. 3 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 8070, Taiwan. 4 Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah, 589, Saudi Arabia. Acknowledgements Dedicated to Professor Hari M Srivastava. The first author was partially supported by Grant-in-Aid for Scientific Research No from Japan Society for the Promotion of Science. The second and the third authors were partially supported by the grant NSC 99-5-M MY3 and the grant NSC 99-5-M MY3, respectively. Received: 30 January 03 Accepted: 5 April 03 Published: 9 April 03

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