Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 979261, 11 pages doi:10.1155/2011/979261 Research Article The Iterative Method of Generalized u 0 -Concave Operators Yanqiu Zhou, Jingxian Sun, and Jie Sun Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China Correspondence should be addressed to Jingxian Sun, jxsun7083@sohu.com Received 16 November 2010; Accepted 12 January 2011 Academic Editor: N. J. Huang Copyright q 2011 Yanqiu Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We define the concept of the generalized u 0 -concave operators, which generalize the definition of the u 0 -concave operators. By using the iterative method and the partial ordering method, we prove the existence and uniqueness of fixed points of this class of the operators. As an example of the application of our results, we show the existence and uniqueness of solutions to a class of the Hammerstein integral equations. 1. Introduction and Preliminary In 1, 2, C ollatz divided the typical problems in computation mathematics into five classes, and the first class is how to solve the operator equation Ax  x 1.1 by the iterative method, that is, construct successively the sequence x n1  Ax n 1.2 for some initial x 0 to solve 1.1. Let P be a cone in real Banach space E and the partial ordering ≤ defined by P ,that is, x ≤ y if and only if y − x ∈ P. The concept and properties of the cone can be found in 3– 5. People studied how to solve 1.1 by using the iterative method and the partial ordering method see 1–11. 2 Fixed Point Theory and Applications In 7, Krasnosel’ski ˘ ı gave the concept of u 0 -concave operators and studied the existence and uniqueness of the fixed point for the operator by the iterative method. The concept of u 0 -concave operators was defined by Krasnosel’ski ˘ ı as follows. Let operator A : P → P and u 0 >θ. Suppose that i for any x>θ, there exist α  αx > 0andβ  βx > 0, such that αu 0 ≤ Ax ≤ βu 0 ; 1.3 ii for any x ∈ P satisfying α 1 u 0 ≤ x ≤ β 1 u 0 α 1  α 1 x > 0, β 1  β 1 x > 0 and any 0 <t<1, there exists η  ηx, t > 0, such that A  tx  ≥  1  η  tAx. 1.4 Then A is called an u 0 -concave operator. In many papers, the authors studied u 0 -concave operators and obtained some results see 3–5, 8–15. In this paper, we generalize the concept of u 0 -concave operators, give a concept of the generalized u 0 -concave operators, and study the existence and uniqueness of fixed points for this class of operators by the iterative method. Our results generalize the results in 3, 4, 7, 15. 2. Main Result In this paper, we always let P be a cone in real Banach space E and the partial ordering ≤ defined by P .Givenw 0 ∈ E,letPw 0 {x ∈ E | x ≥ w 0 }. Definition 2.1. Let operator A : P w 0  → Pw 0  and u 0 >θ. Suppose that i for any x>w 0 , there exist α  αx > 0andβ  βx > 0, such that αu 0  w 0 ≤ Ax ≤ βu 0  w 0 ; 2.1 ii for any x ∈ Pw 0  satisfying α 1 u 0  w 0 ≤ x ≤ β 1 u 0  w 0 α 1  α 1 x > 0, β 1  β 1 x > 0 and any 0 <t<1, there exists η  ηx, t > 0, such that A  tx   1 − t  w 0  ≥  1  η  tAx   1 −  1  η  t  w 0 . 2.2 Then A is called a generalized u 0 -concave operator. Remark 2.2. In Definition 2.1,letw 0  θ, we get the definition of the u 0 -concave operator. Theorem 2.3. Let operator A : Pw 0  → Pw 0  be generalized u 0 -concave and increasing (i.e., x ≤ y ⇒ Ax ≤ Ay), then A has at most one fixed point in Pw 0  \{w 0 }. Fixed Point Theory and Applications 3 Proof. Let x 1 >w 0 , x 2 >w 0 be two different fixed points of A,thatis,Ax 1  x 1 , Ax 2  x 2 ,andx 1 /  x 2 .ByDefinition 2.1, there exist real numbers α 1  α 1 x 1  > 0, β 1  β 1 x 1  > 0, α 2  α 2 x 2  > 0, β 2  β 2 x 2  > 0, such that α 1 u 0  w 0 ≤ x 1 ≤ β 1 u 0  w 0 ,α 2 u 0  w 0 ≤ x 2 ≤ β 2 u 0  w 0 . 2.3 Hence α 1 /β 2 x 2 − w 0 w 0 ≤ α 1 u 0  w 0 ≤ x 1 ≤ β 1 u 0  w 0 ≤ β 1 /α 2 x 2 − w 0 w 0 . Let α  α 1 /β 2 , β  β 1 /α 2 ,wegetthatαx 2 − w 0 w 0 ≤ x 1 ≤ βx 2 − w 0 w 0 ,that is, αx 2 1 − αw 0 ≤ x 1 ≤ βx 2 1 − βw 0 .Let t 0  sup  t>0 | tx 2   1 − t  w 0 ≤ x 1 ≤ t −1 x 2   1 − t −1  w 0  , 2.4 hence 0 <t≤ t −1 ,thatis,0<t≤ 1, then t 0 ∈ 0, 1. Next we will show that t 0  1. Assume that t 0 < 1; by 2.2 and 2.4, there exists η 1  η 1 x 2 ,t 0  > 0, such that x 1  Ax 1 ≥ A  t 0 x 2   1 − t 0  w 0  ≥  1  η 1  t 0 Ax 2   1 −  1  η 1  t 0  w 0   1  η 1  t 0 x 2   1 −  1  η 1  t 0  w 0 . 2.5 By 2.2, there exists η 2  η 2 x 2 ,t 0  > 0, such that x 2  Ax 2  A  t 0  t −1 0 x 2   1 − t −1 0  w 0    1 − t 0  w 0  ≥  1  η 2  t 0 A  t −1 0 x 2   1 − t −1 0  w 0    1 −  1  η 2  t 0  w 0 , 2.6 hence, A  t −1 0 x 2   1 − t −1 0  w 0  ≤  1  η 2  −1 t −1 0 Ax 2   1 −  1  η 2  −1 t −1 0  w 0 . 2.7 Therefore, x 1  Ax 1 ≤ A  t −1 0 x 2   1 − t −1 0  w 0  ≤  1  η 2  −1 t −1 0 Ax 2   1 −  1  η 2  −1 t −1 0  w 0 ≤  1  η 2  −1 t −1 0 x 2   1 −  1  η 2  −1 t −1 0  w 0 . 2.8 Obviously, by 2.5 and 2.8,weget  1  η 1  t 0 x 2   1 −  1  η 1  t 0  w 0 ≤ x 1 ≤  1  η 2  −1 t −1 0 x 2   1 −  1  η 2  −1 t −1 0  w 0 . 2.9 4 Fixed Point Theory and Applications Let η  min{η 1 ,η 2 }, we have  1  η  t 0 x 2   1 −  1  η  t 0  w 0 ≤ x 1 ≤  1  η  −1 t −1 0 x 2   1 −  1  η  −1 t −1 0  w 0 , 2.10 in contradiction to the definition of t 0 . Therefore, t 0  1. By 2.4, x 1  x 2 . The proof is completed. To prove the following Theorem 2.4, we will use the definition of the u 0 -norm as follows. Given u 0 >θ,set E u 0  { x ∈ E | there exists a real number λ>0, such that − λu 0 ≤ x ≤ λu 0 } ,  x  u 0  inf { λ>0 |−λu 0 ≤ x ≤ λu 0 } , ∀x ∈ E u 0 . 2.11 It is easy to see that E u 0 becomes a normed linear space under t he norm · u 0 . x u 0 is called the u 0 - norm of the element x ∈ E u 0 see 3, 4. Theorem 2.4. Let operator A : Pw 0  → P w 0  be increasing and generalized u 0 -concave. Suppose that A has a fixed point x ∗ in Pw 0  \{w 0 }, then, constructing successively the sequence x n1  Ax n n  0, 1, 2,  for any initial x 0 ∈ Pw 0  \{w 0 }, we have x n − x ∗  u 0 → 0 n →∞. Proof. Suppose that {x n } is generated from x n1  Ax n n  0, 1, 2, . Take 0 <ε 0 < 1, such that ε 0 x ∗ 1−ε 0 w 0 ≤ x 1 ≤ ε −1 0 x ∗ 1−ε −1 0 w 0 .Lety 0  ε 0 x ∗ 1−ε 0 w 0 ,z 0  ε −1 0 x ∗ 1−ε −1 0 w 0 , and constructing successively the sequences y n1  Ay n , z n1  Az n n  0, 1, 2, . Since A is a generalized u 0 -concave operator, we know that there exists η 1  η 1 x ∗ ,ε 0  > 0, such that x ∗  Ax ∗  A  ε 0  ε −1 0 x ∗   1 − ε −1 0  w 0    1 − ε 0  w 0  ≥  1  η 1  ε 0 A  ε −1 0 x ∗   1 − ε −1 0  w 0    1 −  1  η 1  ε 0  w 0 , 2.12 hence, Aε −1 0 x ∗ 1 − ε −1 0 w 0  ≤ 1  η 1  −1 ε −1 0 Ax ∗ 1 − 1  η 1  −1 ε −1 0 w 0 , then z 1  A  z 0   A  ε −1 0 x ∗   1 − ε −1 0  w 0  ≤  1  η 1  −1 ε −1 0 Ax ∗   1 −  1  η 1  −1 ε −1 0  w 0   1  η 1  −1 ε −1 0  Ax ∗ − w 0   w 0 <ε −1 0  Ax ∗ − w 0   w 0  ε −1 0 Ax ∗   1 − ε −1 0  w 0  ε −1 0 x ∗   1 − ε −1 0  w 0  z 0 . 2.13 By 2.2, we can easily get y 1 >y 0 .Soitiseasytoshowthat y 0 ≤ y 1 ≤···≤y n ≤···≤ x ∗ ≤···≤ z n ≤···≤ z 1 ≤ z 0 . 2.14 Fixed Point Theory and Applications 5 Let t n  sup  t>0 | tx ∗   1 − t  w 0 ≤ y n ,z n ≤ t −1 x ∗   1 − t −1  w 0   n  0, 1, 2,  , 2.15 then, 0 ≤ t 0 ≤ t 1 ≤···≤ t n ≤···≤ 1, 2.16 which implies that the limit of {t n } exists. Let lim n →∞ t n  t ∗ , then 0 <t n ≤ t ∗ ≤ 1. Next we will show that t ∗  1. Suppose that 0 <t ∗ < 1. Since A is a generalized u 0 -concave operator, then there exists η 2  η 2 x ∗ ,t ∗  > 0, such that A  t ∗ x ∗   1 − t ∗  w 0  ≥  1  η 2  t ∗ Ax ∗   1 −  1  η 2  t ∗  w 0   1  η 2  t ∗ x ∗   1 −  1  η 2  t ∗  w 0 . 2.17 Moreover, x ∗  Ax ∗  A  t ∗   t ∗  −1 x ∗   1 −  t ∗  −1  w 0    1 − t ∗  w 0  ≥  1  η 2  t ∗ A   t ∗  −1 x ∗   1 −  t ∗  −1  w 0    1 −  1  η 2  t ∗  w 0 . 2.18 Therefore, A   t ∗  −1 x ∗   1 −  t ∗  −1  w 0  ≤  1  η 2  −1  t ∗  −1 x ∗   1 −  1  η 2  −1  t ∗  −1  w 0 . 2.19 By 2.17 and 2.19, for any 0 <t≤ t ∗ , there exists η 3  η 3 x ∗ ,t > 0, such that A  tx ∗   1 − t  w 0  ≥  1  η 3  tx ∗   1 −  1  η 3  t  w 0 , A  t −1 x ∗   1 − t −1  w 0  ≤  1  η 3  −1 t −1 x ∗   1 −  1  η 3  −1 t −1  w 0 . 2.20 Particularly, for any 0 <t n ≤ t ∗ n  0, 1, 2, , we have A  t n x ∗   1 − t n  w 0  ≥  1  η  t n x ∗   1 −  1  η  t n  w 0 , A  t −1 n x ∗   1 − t −1 n  w 0  ≤  1  η  −1 t −1 n x ∗   1 −  1  η  −1 t −1 n  w 0 , 2.21 where η  ηt n ,x ∗  > 0. Hence, y n1  Ay n ≥ A  t n x ∗   1 − t n  w 0  ≥  1  η  t n x ∗   1 −  1  η  t n  w 0 , z n1  Az n ≤ A  t −1 n x ∗   1 − t −1 n  w 0  ≤  1  η  −1 t −1 n x ∗   1 −  1  η  −1 t −1 n  w 0 . 2.22 6 Fixed Point Theory and Applications By 2.15,and2.22,wegett n1 ≥ 1  ηt n n  0, 1, 2,  therefore, t n1 ≥ 1  η n1 t 0 n  0, 1, 2, , in contradiction to 0 <t n ≤ 1 n  1, 2, . Hence, t ∗  1. 2.23 Since A is a generalized u 0 -concave operator, thus there exist real numbers α  αx ∗  > 0, β  βx ∗  > 0, such that αu 0  w 0 ≤ x ∗ ≤ βu 0  w 0 ,andt n x ∗ 1 − t n w 0 ≤ y n ≤ x n1 ≤ z n ≤ t −1 n x ∗ 1 − t −1 n w 0 n  0, 1, 2, , we have  t n − 1  x ∗   1 − t n  w 0 ≤ x n1 − x ∗ ≤  t −1 n − 1  x ∗   1 − t −1 n  w 0 . 2.24 Moreover  t n − 1  x ∗   1 − t n  w 0 ≥  t n − 1   βu 0  w 0    1 − t n  w 0   t n − 1  βu 0 ,  t −1 n − 1  x ∗   1 − t −1 n  w 0 ≤  t −1 n − 1   βu 0  w 0    1 − t −1 n  w 0   t −1 n − 1  βu 0 . 2.25 Hence,  1 − t −1 n  βu 0 ≤  t n − 1  βu 0 ≤ x n1 − x ∗ ≤  t −1 n − 1  βu 0  n  0, 1, 2,  . 2.26 Consequently, by 2.23,wegetx n − x ∗  u 0 → 0 n →∞. To prove the following Theorem 2.5, we will use the definition of the normal cone as follows. Let P be a cone in E. Suppose that there exist constants N>0, such that θ ≤ x ≤ y ⇒  x  ≤ N   y   , 2.27 then P is said to be normal, and the smallest N is called the normal constant of P see 3–5. Theorem 2.5. vLetP be a normal cone of E. If operator A : Pw 0  −→ Pw 0  is increasing and generalized u 0 -concave, and η  ηt, x is irrelevant to x in 2.2,thenA has the only one fixed point x ∗ ∈ Pw 0  \{w 0 }. Moreover, constructing successively the sequence x n1  Ax n n  0, 1, 2,  for any initial x 0 >w 0 , we have x n − x ∗ →0 n →∞. Proof. Since A is a generalized u 0 -concave operator, hence there exist real numbers β>α>0, such that αu 0  w 0 ≤ Au 0  w 0  ≤ βu 0  w 0 . Take t 0 ∈ 0, 1 small enough, then t 0 u 0  w 0 ≤ Au 0  w 0  ≤ 1/t 0 u 0  w 0 . Therefore, t n1 ≥ t n ,thatis,{t n } is an increasing sequence and 0 <t n ≤ 1, hence, the limit of {t n } exists. Set lim n →∞ t n  t ∗ , then 0 <t ∗ ≤ 1. Fixed Point Theory and Applications 7 Let γt1  ηtt, where ηt which is irrelevant to x is ηt, x in 2.2,andA is increasing, so t<γt ≤ 1,Atx 1 − tw 0  ≥ γtAx 1 − γtw 0 , for all t ∈ 0, 1.By γt 0 /t 0 > 1, we can choose a natural number k>0 big enough, such that  γ  t 0  t 0  k > 1 t 0 . 2.28 Let y 0  t k 0 u 0  w 0 ,z 0  1 t k 0 u 0  w 0 ; y n  Ay n−1 ,z n  Az n−1  n  1, 2,  . 2.29 Obviously, y 0 ,z 0 ∈ Pw 0 ,y 0 <z 0 . Since A is increasing, we have y 1  Ay 0  A  t k 0 u 0  w 0   A  t 0  t k−1 0 u 0  w 0    1 − t 0  w 0  ≥ γ  t 0  A  t k−1 0 u 0  w 0    1 − γ  t 0   w 0  γ  t 0  A  t 0  t k−2 0 u 0  w 0    1 − t 0  w 0    1 − γ  t 0   w 0 ≥ γ  t 0   γ  t 0  A  t k−2 0 u 0  w 0    1 − γ  t 0   w 0    1 − γ  t 0   w 0  γ 2  t 0  A  t k−2 0 u 0  w 0    1 − γ 2  t 0   w 0 ≥···≥ γ k  t 0  A  u 0  w 0    1 − γ k  t 0   w 0 >t k−1 0  t 0 u 0  w 0    1 − t k−1 0  w 0  t k 0 u 0  w 0  y 0 . 2.30 Since Ax  A{t 0 t −1 0 x 1 − t −1 0 w 0 1 − t 0 w 0 }≥γt 0 At −1 0 x 1 − t −1 0 w 0 1 − γt 0 w 0 , we get At −1 0 x 1 − t −1 0 w 0  ≤ 1/γt 0 Ax 1 − 1/γt 0 w 0 . Hence z 1  A  1 t k 0 u 0  w 0   A  1 t 0  1 t k−1 0 u 0  w 0    1 − 1 t 0  w 0  ≤ 1 γ  t 0  A  1 t k−1 0 u 0  w 0    1 − 1 γ  t 0   w 0 ≤···≤ 1 γ k  t 0  A  u 0  w 0    1 − 1 γ k  t 0   w 0 ≤ 1 t 0 γ k  t 0  u 0  w 0 < 1 t k 0 u 0  w 0  z 0 , 2.31 then y 0 ≤ y 1 ≤ z 1 ≤ z 0 .Itiseasytosee y 0 ≤ y 1 ≤···≤ y n ≤···≤ z n ≤···≤ z 1 ≤ z 0 . 2.32 8 Fixed Point Theory and Applications Let t n  sup  t>0 | y n ≥ tz n   1 − t  w 0  . 2.33 Obviously, y n ≥ t n z n 1 − t n w 0 .Soy n1 ≥ y n ≥ t n z n 1 − t n w 0 ≥ t n z n1 1 − t n w 0 . Therefore, t n1 ≥ t n ,thatis,{t n } is an increasing sequence and 0 <t n ≤ 1, hence, the limit of {t n } exists. Set lim n →∞ t n  t ∗ , then 0 <t ∗ ≤ 1. Next we will show that t ∗  1. Suppose that 0 <t ∗ < 1, we have the following. i If for any natural number n, t n <t ∗ < 1, then y n1  Ay n ≥ A  t n z n   1 − t n  w 0   A  t n t ∗  t ∗ z n   1 − t ∗  w 0    1 − t n t ∗  w 0  ≥ γ  t n t ∗  A  t ∗ z n   1 − t ∗  w 0    1 − γ  t n t ∗  w 0 ≥ γ  t n t ∗   γ  t ∗  Az n   1 − γ  t ∗   w 0    1 − γ  t n t ∗  w 0  γ  t n t ∗  γ  t ∗  Az n   1 − γ  t n t ∗  γ  t ∗   w 0  γ  t n t ∗  γ  t ∗  z n1   1 − γ  t n t ∗  γ  t ∗   w 0 , 2.34 hence, t n1 ≥ γ  t n t ∗  γ  t ∗    1  η  t n t ∗  t n t ∗  1  η  t ∗   t ∗ ≥ t n  1  η  t ∗   . 2.35 Taking limits, we have t ∗ ≥ t ∗ 1  ηt ∗  >t ∗ , a contradiction. ii Suppose that there exists a natural number N>0, such that t n  t ∗ n>N. When n>N, so we have y n1  Ay n ≥ A  t n z n   1 − t n  w 0   A  t ∗ z n   1 − t ∗  w 0  ≥ γ  t ∗  Az n   1 − γ  t ∗   w 0  γ  t ∗  z n1   1 − γ  t ∗   w 0 , 2.36 then t ∗  t n1 ≥ γt ∗ 1  ηt ∗ t ∗ >t ∗ , a contradiction. Therefore, t ∗  1. For any natural numbers n, p, we have θ ≤ y np − y n ≤ z np − y n ≤ z n − y n ≤ z n −  t n z n   1 − t n  w 0    1 − t n  z n − w 0  . 2.37 Similarly, θ ≤ z n − z np ≤ z n − y n ≤ 1 − t n z n − w 0 . By the normality of P and lim n →∞ t n  1, we get    y np − w 0  −  y n − w 0       y np − y n   ≤ N  1 − t n   z n − w 0  → 0  n →∞  ,    z np − w 0  −  z n − w 0       z n − z np   ≤ N  1 − t n   z n − w 0  → 0  n →∞  , 2.38 Fixed Point Theory and Applications 9 where N is the normal constant of P. Hence the limits of {y n } and {z n } exist. Let lim n →∞ y n  y ∗ , and let lim n →∞ z n  z ∗ , then y n ≤ y ∗ ≤ z ∗ ≤ z n n  0, 1, 2, , hence, θ ≤ z ∗ − y ∗ ≤ z n − y n ≤  1 − t n  z n − w 0  → θ  n →∞  . 2.39 That is, y ∗  z ∗ .Letx ∗  y ∗  z ∗ , then y n1  Ay n ≤ Ax ∗ ≤ Az n  z n1 . Taking limits, we get x ∗ ≤ Ax ∗ ≤ x ∗ . Hence Ax ∗  x ∗ ,thatis,x ∗ ∈ P w 0 \{w 0 } is a fixed point of A.ByTheorem 2.4, the conclusions of Theorem 2.5 hold. The proof is completed. 3. Examples Example 3.1. Let I 0, 1,let CI{xt : I → R | xt is continuous},letx  sup{|xt||t ∈ I},letP  {x ∈ CI | xt ≥ 0, ∀t ∈ I}, then CI is a real Banach space and P is a normal and solid cone in CIP is called solid if it contains interior points, i.e., ◦ P /  ∅. Take a<0, let w 0  w 0 t ≡ a, Pw 0 {x ∈ CI | xt ≥ w 0 , ∀t ∈ I},and ◦ Pw 0 {x  w 0 ∈ Pw 0  | x ∈ ◦ P}. Considering the Hammerstein integral equation x  t    1 0 k  t, s  f  s, x  s  ds, t ∈  0, 1  , 3.1 where kt, s : I × I → 0, ∞ is continuous, fs, u : I × a, ∞ → R is increasing for u. Suppose that 1 there exist real numbers 0 ≤ m ≤ M ≤ 1, such that m ≤ kt, s ≤ M, for all t, s ∈ I × I,andfs, u ≥ a/M, for alls, u ∈ I × a, ∞ , 2 for any λ ∈ 0, 1 and u ∈ a, ∞, there exists η  ηλ > 0, such that mf  s, λu   1 − λ  a  ≥  1  η  λmf  s, u    1 −  1  η  λ  a. 3.2 Then 3.1 has the only one solution x ∗ ∈ P w 0  \{w 0 }. Moreover, constructing successively the sequence: x n  t    1 0 k  t, s  f  s, x n−1  s  ds, ∀t ∈ I, n  1, 2, 3.3 for any initial x 0 ∈ Pw 0  \{w 0 }, we have x n − x ∗ →0 n →∞. Proof. Considering the operator Ax  t    1 0 k  t, s  f  s, x  s  ds, t ∈ I. 3.4 10 Fixed Point Theory and Applications Obviously, A : Pw 0 \{w 0 } → ◦ Pw 0  is increasing. Therefore, i of Definition 2.1 is satisfied. For any x ∈ ◦ Pw 0 ,by3.2, we have A  λx  t    1 − λ  w 0    1 0 k  t, s  f  s, λx  s    1 − λ  w 0  ds   1 0 1 m k  t, s  mf  s, λx  s    1 − λ  w 0  ds ≥  1  η  λ  1 0 1 m k  t, s  mf  s, x  s  ds   1 −  1  η  λ  w 0  1 0 1 m k  t, s  ds ≥  1  η  λAx  t    1 −  1  η  λ  w 0 . 3.5 Therefore, ii of Definition 2.1 is satisfied. Hence the operator A : Pw 0  → P w 0  is generalized u 0 -concave. Consequently, operator A satisfies all conditions of Theorem 2.5,thus the conclusion of Example 3.1 holds. Example 3.2. Let R be a real numbers set, and let P  {x | x ≥ 0, x ∈ R}, then R is a real Banach space and P is a normal and solid cone in R.LetAx x2 1/2 −2. Considering the equation: x  Ax. Obviously, A is a generalized u 0 -concave operator and satisfies all the conditions of Theorem 2.5. Hence A has the only one fixed point x ∗ ∈ P−2 \{−2} −2, ∞. 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By using the iterative method and the partial ordering method, we prove the. Krasnosel’ski ˘ ı gave the concept of u 0 -concave operators and studied the existence and uniqueness of the fixed point for the operator by the iterative method. The concept of u 0 -concave operators. generalize the concept of u 0 -concave operators, give a concept of the generalized u 0 -concave operators, and study the existence and uniqueness of fixed points for this class of operators by the iterative