Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 815637, 25 pages doi:10.1155/2009/815637 Research Article Some Combined Relations between Contractive Mappings, Kannan Mappings, Reasonable Expansive Mappings, and T-Stability M. De la Sen Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus de Leioa (Bizkaia), Apertado 644 de Bilbao, 48080 Bilbao, Spain Correspondence should be addressed to M. De la Sen, manuel.delasen@ehu.es Received 13 May 2009; Accepted 31 August 2009 Recommended by Andrzej Szulkin In recent literature concerning fixed point theory for self-mappings T : X → X in metric spaces X, d, there are some new concepts which can be mutually related so that the inherent properties of each one might be combined for such self-mappings. Self-mappings T : X → X can be referred to, for instance, as Kannan-mappings, reasonable expansive mappings, and Picard T- stable mappings. Some relations between such concepts subject either to sufficient, necessary, or necessary and sufficient conditions are obtained so that in certain self-mappings can exhibit combined properties being inherent to each of its various characterizations. Copyright q 2009 M. De la Sen. 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. 1. Introduction As it is wellknown fixed point theory and related techniques are of increasing interest for solving a wide class of mathematical problems where convergence of a trajectory or sequence to some equilibrium set is essential, see, e.g., 1–7. Some of the specific topics recently covered in the field of fixed point theory are, for instance as follows. 1 The properties of the so-called n-times reasonably expansive mapping are investigated in 1 in complete metric spaces X, d as those fulfilling the property that dx, T n x ≥ βdx, Tx for some real constant β>1. The conditions for the existence of fixed points in such mappings are investigated. 2 Strong convergence of the wellknown Halpern’s iteration and variants is investi- gated in 2, 8 and several the references therein. 3 Fixed point techniques have been recently used in 4 for the investigation of global stability of a wide class of time-delay dynamic systems which are modeled by functional equations. 2 Fixed Point Theory and Applications 4 Generalized contractive mappings have been investigated in 5 and references therein, weakly contractive and nonexpansive mappings are investigated in 6 and references therein. 5 The existence of fixed points of Liptchitzian semigroups has been investigated, for instance, in 3. 6 Picard’s T-stability is discussed in 9 related to the convergence of perturbed iterations to the same fixed points as the nominal iteration under certain conditions in a complete metric space. 7 The so-called Kannan mappings in 10 are recently investigated in 11, 12 and references therein. Let X, d be a metric space. Consider a self-mapping T : X → X. The basic concepts used through the manuscript are the subsequent ones: 1 T : X → X is k-contractive, following the contraction Banach’s principle, if there exists a real constant k ∈ 0, 1 such that d  Tx,Ty  ≤ kd  x, y  ; ∀x, y ∈ X, 1.1  2 T : X → X is α-Kannan, 10–12, if there exists a real constant α ∈ 0, 1/2 such that d  Tx,Ty  ≤ α  d  x, Tx   d  y, Ty  ; ∀x, y ∈ X, 1.2 3 T : X → X is n Z   n ≥ 2-times reasonable expansive self-mapping if there exists a real constant β>1 such that dx, T n x ≥ βdx, Tx; ∀x ∈ X, Z   n ≥ 2, 1, 4 Picard’s T-stability means that if X, d is a complete metric space and Picard’s iteration x k1  Tx k satisfies dy k1 ,Ty k  → 0ask →∞for {y k }⊂X then lim k →∞ y k  lim k →∞ x k  q ∈ FT,thatis,q is a fixed point of T, 9.Itis proven in 9 that, if the self-mapping T satisfies a property, referred to through this manuscript as the L, m property for some real constants L ∈ R 0 and m ∈ 0, 1 ∈ R 0 see Definition 1.2 in what follows, then Picard’s iteration is T-stable if lim k →∞ dy k1 ,Ty k 0. The following result is direct. Proposition 1.1. If a self-mapping T : X → X is k-contractive, then it is also k  -contractive; ∀k  ∈ k, 1. If a self-mapping T : X → X is α-Kannan, then it is also α  -Kannan; ∀α  ∈ α, 1/2. The so- called t he L, m-property is defined as follows. Definition 1.2. A self-mapping T : X → X with F T /  ∅ possesses the L, m-property for some real constants L ∈ R 0 and m ∈ 0, 1 ∈ R 0 if dTx,q ≤ Ldx, Txmdx, q; ∀q ∈ FT, ∀x ∈ X. The above property has been introduced in [9] to discuss the T-stability of Picard’s iteration. If the L, m-property is fulfilled in a complete metric space and, furthermore, lim k →∞ dy k1 ,Ty k 0, then Picard’s iteration x k1  Tx k is T-stable defined as dx k1 ,Ty k  → 0 as k →∞⇒x k → q ∈ FT as k →∞. The main results obtained in this paper rely on the following features. Fixed Point Theory and Applications 3 1 In fact k-contractive mappings T : X → X are α-Kannan self-mappings and vice-versa under certain mutual constraints between the constants k and α,[10–12]. A necessary and sufficient condition for both properties to hold is given. Some of s uch constraints are obtained in the manuscript. The existence of fixed points and their potential uniqueness is discussed accordingly under completeness of the metric space, [1–4, 8–10 , 13]. 2 If T : X → X is n (Z   n ≥ 2)-times reasonable expansive self-mapping then it cannot be contractive as expected but it is α-Kannan under certain constraints. The converse is also true under certain constraints. Some of such constraints referred to are obtained explicitly in the manuscript. The existence of fixed points is also discussed for two types of n (Z   n ≥ 2)-times reasonable expansive self-mappings proposed in [1]. 3 The L, m-property guaranteeing Picard’s T-stability of iterative schemes, under the added condition lim k →∞ dy k1 ,Ty k 0, is compatible with both contractive self-mappings and α-Kannan ones under certain constraints. A sufficient condition that as self-mapping possessing the L, m-property is α-Kannan is also given. It may be also fulfilled by n (Z   n ≥ 2)-times reasonable expansive self-mappings. 1.1. Notation Assume that Z and R are the sets of integer and real numbers, Z  : {z ∈ Z : z>0}, Z 0 : {z ∈ Z : z ≥ 0}, R  : {r ∈ R : r>0}, R 0 : {r ∈ R : r ≥ 0}. If T : X → X is a self mapping in a metric space X, d, then FT denotes the set of fixed points of T. 2. Combined Compatible Relations of k-Contractive Mappings, α-Kannan Mappings, and the L − m-Property It is of interest to establish when a k-contractive mapping is also α-Kannan and viceversa. Theorem 2.1. The following properties hold: i if T : X → X is k-contractive with k ∈ 0, 1/3 then it is α-Kannan with α  k/1 − k, ii T : X → X is k-contractive and α-Kannan if and only if d  Tx,Ty  ≤ min  kd  x, y  ,α  d  x, Tx   d  y, Ty   k min  d  x, y  , α k  d  x, Tx   d  y, Ty    α min  k α d  x, y  ,  d  x, Tx   d  y, Ty   , 2.1 iii if T : X → X is k-contractive and α-Kannan with k /  0 and α /  0 then the inequality α  d  x, Tx   d  y, Ty  ≤ kd  x, y  2.2 cannot hold for all x,yinX, 4 Fixed Point Theory and Applications iv if T : X → X is k-contractive and α-Kannan with k /  0, and 0 /  α<kthen the inequalities: d  x, y  ≤ α k  d  x, Tx   d  y, Ty  , d  x, y  ≤ α k − α min  d  x, Tx   d  x, Ty  ,d  y, Tx   d  y, Ty  , d  Tx,Ty  ≤ min  kα k − α min  d  x, Tx   d  x, Ty  ,d  y, Tx   d  y, Ty  , k 1 − k  d  x, Tx   d  Ty,y   2.3 are feasible for all x,yinX. Proof. i Since T : X → X is k-contractive, then d  Tx,Ty  ≤ kd  x, y  ≤ k  d  x, Tx   d  Tx,Ty   d  Ty,y  ; ∀x, y ∈ X, 2.4 from the triangle inequality property of the distance in metric spaces. Since k ∈ 0, 1, then d  Tx,Ty  ≤ k 1 − k  d  x, Tx   d  Ty,y  ; ∀x, y ∈ X, 2.5 so that T : X → X is α-Kannan with α  k/1 − k provided that k/1 − k < 1/2 ⇔ k<1/3. As a result, if T : X → X is k-contractive with k ∈ 0, 1/3, then it is also k/1 − k-Kannan. ii It is direct if T : X → X is k-contractive and α-Kannan with k /  0andα /  0. For α  k  0, the result holds trivially. iii Proceed by contradiction. Assume that the inequality holds for x, y ∈ X ∩ FT with x  y where FT is the empty or nonempty set of fixed points of T. Since x  y, the inequality leads to 2αdx, Tx0. This implies that dx, Tx0sinceα /  0. However, dx, Tx > 0; ∀x / ∈ FT, what is a contradiction. Therefore, the inequality cannot cold in X. iv The first inequality can potentially hold even for the set of fixed points. Furthermore, one gets from the triangle inequality for the distance dx, Tx ≤ dx, y dy, Tx, ∀x, y ∈ X: kd  x, y  ≤ α  d  x, Tx   d  y, Ty  ≤ αd  x, y   α  d  y, Tx   d  y, Ty  ⇒ d  x, y  ≤ α k − α  d  y, Tx   d  y, Ty  2.6 for all x, y ∈ X since α<k. Also, by using dy,Ty ≤ dx, ydx, Tx,onegetsdx, y ≤ α/k − αdx, Txdx, Ty. As a result, the second inequality follows by combining both partial results. The third inequality follows from the second one and Property i. Property iv has been proven. Theorem 2.1ii leads to the subsequent result. Fixed Point Theory and Applications 5 Corollary 2.2. If T : X → X is k-contractive and α-Kannan, then d  Tx,T 2 x  ≤ min  k, α 1 − α  d  x, Tx  ; ∀x ∈ X. 2.7 Proof. One gets from Theorem 2.1ii for y  Tx that dTx,T 2 x ≤ kdx, Tx; ∀x ∈ X and dTx,T 2 x ≤ αdx, TxdTx,T 2 x ⇒ dTx,T 2 x ≤ α/1 − αdx, Tx; ∀x ∈ X.Both inequalities together yield the result. The following two results follows directly from Theorem 2.1iii for y  Tx. Corollary 2.3. If T : X → X is k-contractive and α-Kannan with k>α /  0, then the inequality dTx,T 2 x ≤ k − α/αdx, Tx cannot hold ∀x ∈ X. Corollary 2.4. If T : X → X is k-contractive and α-Kannan with α>k /  0, then the inequality α − kdx, TxαdTx,T 2 x ≤ 0 cannot hold for x ∈ X ∩ FT. The following three results follows directly from Theorem 2.1iv for y  Tx. Corollary 2.5. If T : X → X is k-contractive and α-Kannan with k>α /  0, then the inequality dx, Tx ≤ α/k − αdTx,T 2 x is feasible ∀x ∈ X. Proof. The proof follows since d  x, Tx  ≤ α k  d  x, Tx   d  Tx,T 2 x  ⇒ d  Tx,T 2 x  ≤  1 − α k  −1 α k d  Tx,T 2 x  2.8 is feasible from the first feasible inequality in Theorem 2.1ii ∀x ∈ X and y  Tx. Corollary 2.6. If T : X → X is k-contractive and α-Kannan with k>2α /  0, then the inequality dx, Tx ≤ α/k − αk − 2αdx, T 2 x is feasible ∀x ∈ X. Proof. The proof follows since d  x, Tx  ≤ α k − α  d  x, Tx   d  x, T 2 x  ≤  1 − α k − α  −1 α k − α d  Tx,T 2 x  2.9 is feasible from the second feasible inequality in Theorem 2.1ii ∀x ∈ X and y  Tx. Corollary 2.7. If T : X → X is k-contractive and α-Kannan with k>2α /  0, then the inequality dx, Tx ≤ α/k − αk − 2αdx, T 2 x is feasible ∀x ∈ X. Proof. The proof follows directly since d  Tx,T 2 x  ≤ kα k − α  d  x, Tx   d  x, T 2 x  , d  Tx,T 2 x  ≤  1 − k 2 α  1 − k  k − α   −1  d  x, Tx   d  x, T 2 x  2.10 are feasible from the third feasible inequality in Theorem 2.1ii ∀x ∈ X and y  Tx. 6 Fixed Point Theory and Applications Remark 2.8. It turns out from Definition 1.2 that if T : X → X has the L, m property for some real constants L ∈ R 0 and m ∈ 0, 1 ∈ R 0 , then it has also the L 0 ,m 0 ; ∀L 0 ∈ L, ∞, ∀m 0 ∈ m, 1. The subsequent result is concerned with some joint L, m, α-Kannan and k-contractiveness of a self- mapping T : X → X. Theorem 2.9. The following properties hold: i T : X → X is α-Kannan if it has the L, m-property for any real constants L and m which satisfy the constraints α L  m/1 − m, 0 ≤ L<1 − 3m/2, 0 ≤ m<1/3, ii assume that T : X → X is k-contractive. Then, it is also k/1 − k-Kannan and it possesses the k − m/1 − k,m-property for any real constant m which satisfies 0 ≤ m ≤ k<1/3, iii assume that T : X → X is α-Kannan and FT /  ∅.ThenT : X → X has the L, m- property with L  α  2/1 − α and ∀m ∈ 0, 1 ∩ R, iv assume that T : X → X is k-contractive with k ∈ 0, 1/3 ∩ R and FT /  ∅.Then T : X → X is k/1 − k-Kannan and it has the L, m-property with L 2 − 3k/1 − k1 − 2k and ∀m ∈ 0, 1 ∩ R. Proof. i If T : X → X has the L, m-property, one has from the triangle inequality for distances d  Tx,q  ≤  L  m  d  x, Tx   md  Tx,q  ⇒ d  Tx,q  ≤ L  m 1 − m d  x, Tx  ; ∀q ∈ F  T  , ∀x ∈ X, 2.11 since m<1. The above inequality together with the triangle inequality leads to d  Tx,Ty  ≤  Tx,q   d  Ty,q  ≤ L  m 1 − m  d  x, Tx   d  y, Ty  ; ∀q ∈ F  T  , ∀x ∈ X. 2.12 Thus, T : X → X is α-Kannan with α :L  m/1 − m < 1/2 which holds if 0 ≤ L< 1−3m/2and0≤ m<1/3. Property i is proven. Furthermore, if T : X → X is k-contractive then it is also α-Kannan if α  k/1 − k with k<1/3fromTheorem 2.1ii . Then, T : X → X is k-contractive, α-Kannan, and it has the L, m-property if α :L m/1 −mk/1− k < 1/2 which holds for k :Lm/L1α/1α < 1/3if0≤ L :k−m/1−k < 1−3m/2 and 0 ≤ m ≤ k<1/3 which is already fulfilled since T : X → X is α-Kannan with the k − m/1 − k,m-property. Property ii has been proven. iii By using the triangle inequality for distances and taking x ∈ X and q ∈ FT,one gets d  Tx,q  ≤ d  Tx,T 2 x   d  x, T 2 x   d  x, q  ≤ 2d  Tx,T 2 x   d  x, Tx   d  x, q  ≤ 1  α 1 − α d  x, Tx   d  x, q  , 2.13 Fixed Point Theory and Applications 7 for any real constant m ∈ 0, 1 after using the subsequent relation: d  Tx,T 2 x  ≤ α  d  x, Tx   d  Tx,T 2 x  ⇒ d  Tx,T 2 x  ≤ α 1 − α d  x, Tx  ; ∀x ∈ X, 2.14 which follows directly from the α-Kannan property. Furthermore, since q  T 2 q ∈ FT,the relation 2.14 leads to d  Tx,q   d  Tx,T 2 q  ⇒ d  Tx,q  ≤ α  d  x, Tx   d  Tq,T 2 q  ≤ αd  x, Tx  ; ∀x ∈ X, 2.15 d  x, q  ≤ d  x, Tx   d  Tx,q  ≤  1  α  d  x, Tx   md  x, q  ; ∀x ∈ X, ∀m ∈  0, 1  ∩ R. 2.16 Then, the substitution of 2.16 into 2.13 yields d  Tx,q  ≤  α  2 1 − α  d  x, Tx   md  x, q  ; ∀x ∈ X, ∀m ∈  0, 1  ∩ R 2.17 which proves Property iii. Property iv is a direct consequence of Properties ii-iii since T : X → X is α-Kannan with α  k/1 − k. Further results concerning α-Kannan mappings follow below. Theorem 2.10. Assume that T : X → X is α-Kannan. Then, the following properties hold: i dTx,T n1 x ≤  n i1 α/1 − α  i dx, Tx ≤ 1 − α/1 − 2αdx, Tx; ∀x ∈ X, ∀n ∈ Z  , ii if T : X → X is α-Kannan and k-contractive, then ii.1 dTx,T 2 x ≤ mink, α/1 − αdx, Tx; ∀x ∈ X, ii.2 dT j x, T nj1 x ≤ k m−1 dTx,T n1 x ≤  n i1 α/1 − α i k j−1 dx, Tx ≤ k j−1 1 − α/1 − 2αdx, Tx ∀x ∈ X, ∀n ∈ Z  , ∀j ≥ 2 ∈ Z  , ii.3 lim j →∞ T nj x  z  zx ∈ cl X; ∀x ∈ X, ∀n ∈ Z  , iii if T : X → X is k-contractive for some k ∈ 0, 1/3,then d  T j x, T nj1 x  ≤ k m−1 d  Tx,T n1 x  ≤ n  i1  α 1 − α  i k m−1 d  x, Tx  ≤ k m−1  1 − 2k  1 − 3k d  x, Tx  2.18 ∀x ∈ X, ∀n ∈ Z  , ∀m≥ 2 ∈ Z  , also, lim j →∞ T nj x  z  zx ∈ cl X; ∀x ∈ X, ∀n ∈ Z  , iv if X, d is a complete metric space and T : X → X is k-contractive for some k ∈ 0, 1/3 or if it is α-Kannan and k-contractive, then z  lim j →∞ T nj x ∈ X is independent of x; ∀x ∈ X, ∀n ∈ Z  so that FT{z} consists of a unique fixed point. 8 Fixed Point Theory and Applications Proof. Proceed by complete induction by assuming that dTx,T j1 x ≤  j i1 α/ 1 − α i dx, Tx; ∀x ∈ X, ∀j ∈ n − 1: { 1, 2, ,n− 1}. Since T : X → X is α-Kannan, take y  T n x so that one gets from the triangle inequality for distances and the above assumption for j ∈ n − 1that d  Tx,T n1 x  ≤ α  d  x, Tx   d  T n x, T n1 x  ≤ α  d  x, Tx   d  T n x, Tx   d  Tx,T n1 x  ⇒ d  Tx,T n1 x  ≤ α 1 − α  d  x, Tx   d  T n x, Tx  ≤ α 1 − α d  x, Tx    α 1 − α  n−1  i1  α 1 − α  i d  x, Tx   n  i1  α 1 − α  i d  x, Tx  ; ∀x ∈ X, ∀n ∈ Z  . 2.19 Since α/1 − α < 1; ∀α ∈ 0, 1/2, then  n i1 α/1 − α i ≤  ∞ i1 α/1 − α i  1/1 − α/1 − α1 − α/1 − 2α so that dTx,T n1 x ≤ 1 − α/1 − 2αdx, Tx; ∀x ∈ X and the proof of Property i is complete. Property ii.1 follows from Property i,sinceT is α-Kannan, by taking into account that it is k-contractive Property ii.2 follows directly from Property i and Theorem 2.1i. Property ii.3 follows from 0 ≤ d  T j x, T nj1 x  ≤ 1 − α 1 − 2α d  x, Tx   lim sup j →∞ k j−1   0 ⇒ lim j →∞ T j x  lim j →∞ T nj x, ∀x ∈ X, ∀n ∈ Z  . 2.20 Property iii follows again directly from Property i and Theorem 2.1i and the first part of Property ii for m →∞. Property iv follows directly from Properties ii and iii from the uniqueness of the fixed point Banach’s contraction mapping principle since T is a strict contraction. Proposition 2.11. If T : X → X is α-Kannan, then dTx,x ≤ 1 − α/1 − 2αdTx,T 2 x; ∀x ∈ X. If, in addition, T : X → X is k-contractive, then 1 − 2α/1 − α ≤ k<1. Proof. It holds that d  Tx,x  ≤ d  Tx,T 2 x   d  T 2 x, x  ≤ α 1 − α d  Tx,x   d  T 2 x, x  ; ∀x ∈ X ⇒ d  Tx,x  ≤ 1 − α 1 − 2α d  T 2 x, x  ; ∀x ∈ X, 2.21 Fixed Point Theory and Applications 9 for all x ∈ X by using the triangle property of distances and Theorem 2.10i. The first part of the result has been proven. The second part of the result follows since d  Tx,x  ≤ 1 − α 1 − 2α d  Tx,T 2 x  ≤  1 − α  k 1 − 2α d  Tx,x  ; ∀x ∈ X so that k ≥  1 − 2α  1 − α , 2.22 if T : X → X is k-contractive. Remark 2.12. If T : X → X is k-contractive and α-Kannan, it follows from Corollary 2.2 and Proposition 2.11 that 1 > mink, α/1 − α ≥ 1 − α/1 − α,α/1 − α; ∀x ∈ X. Proposition 2.13. If T : X → X is α-Kannan then dTx,T n1 x ≤ 1 − α/1− 2α 2 dTx,T 2 x; ∀x ∈ X. Proof. It follows from Proposition 2.11 and Theorem 2.10i since d  Tx,T n1 x  ≤ 1 − α 1 − 2α d  x, Tx  ≤  1 − α 1 − 2α  2 d  x, T 2 x  ; ∀x ∈ X, ∀n ∈ Z  . 2.23 Proposition 2.14. If T : X → X is α-Kannan for some α ∈ 1/3, 1/2,then 1 − α α d  Tx,T 2 x  ≤ d  Tx,x  ≤ 1 − α 1 − 2α d  Tx,T 2 x  ; ∀x ∈ X. 2.24 . Proof. The upper-bound for dTx,x has been obtained in Proposition 2.11. Its lower-bound 1−α/αdTx,T 2 x follows from Theorem 2.10i subject to 1−α/α ≤ 1−α/1 −2α which holds ∀x ∈ X if and only if α ≥ 1/3. The proof is complete. 3. Combined Compatible Results about the L, m-Property, α-Kannan-Mappings, and a Class of Expansive Mappings Definition 3.1 see 1. Let X, d be a complete metric space. Also, T : X → X is said to be an n (Z   n ≥ 2)-times reasonable expansive self-mapping if there exists a real constant β>1 such that d  x, T n x  ≥ βd  x, Tx  ; ∀x ∈ X, Z   n ≥ 2. 3.1 . Theorem 3.2. Let X, d be a complete metric space. Assume that T : X → X is a continuous surjective self-mapping which is continuous everywhere in X and α-Kannan while it also satisfies dT n−1 x, T n x ≥ βdx, Tx for some real constant β>1,somen ≥ 2 ∈ Z  , ∀x ∈ X (i.e., T : X → X is n (Z   n ≥ 2) times reasonable expansive self-mapping). Then, the following properties hold if 10 Fixed Point Theory and Applications β>1/1 − α: i dx, Tx ≤ β1 − α − α/β1 − αdT n−1 x, T n x; ∀x ∈ X, ii T : X → X has a unique fixed point in X, iii T : X → X has a fixed point in X even if it is not α-Kannan. Proof. Since T : X → X is α-Kannan and it satisfies dT n−1 x, T n x ≥ βdx, Tx; some real constant β>1, some n ≥ 2 ∈ Z  , ∀x ∈ X, then α  d  T n−2 x, T n−1 x   d  T n−1 x, T n x  − d  x, Tx  ≥ d  T n−1 x, T n x  − d  x, Tx  ≥  β − 1  d  x, Tx  ⇒ d  x, Tx  ≤ 1 β − 1  d  T n−1 x, T n x  − d  x, Tx   ; ∀x ∈ X. 3.2 Since α ∈ 0, 1/2  and β>1, then d  x, Tx  ≤ β  1 − α  − α β  1 − α  d  T n−1 x, T n x  ; ∀x ∈ X, 3.3 and Property i has been proven. Also, d  x, Tx  ≤ min  1 β − 1 , β  1 − α  − α β  1 − α    d  T n−1 x, T n x  − d  x, Tx    β  1 − α  − α β  1 − α   d  T n−1 x, T n x  − d  x, Tx   . 3.4 The last expression can be rewritten as d  f  x  ,g  x   ≤ ϕ  f  x   − ϕ  g  x   ; ∀x ∈ X, 3.5 where g : X → X is the identity mapping on X;thatis,gxx; ∀x ∈ X, f : X → X is defined by fxTx  Tgx; ∀x ∈ X and then it is a surjective mapping since T is surjective and the functional ϕ :Im T ⊂ X → R 0 is defined as ϕxβ1−α−α/β 1− α  n−2 j0 dT j x, T j1 x. It turns out that ϕ :ImT ⊂ X → R 0 is continuous everywhere on its definition domain and then lower semicontinuous bounded from below as a result since the distance mapping d : X×X → R 0 is continuous on X. Then, T : X → X has a fixed point in X in 1, Lemma 2.4,evenifT : X → X is not α-Kannan, since f is surjective on X, g is the identity mapping on X,andϕ is lower semicontinuous bounded from below. The fixed point is unique since X, d is a complete metric space. Properties ii-iii have been proven. The subsequent result gives necessary conditions for Theorem 3.2 to hold as well as a sufficient condition for such a necessary condition to hold. [...]... vol 2009, Article ID 314581, 19 pages, 2009 9 Y Qing and B E Rhoades, “T -stability of Picard iteration in metric spaces,” Fixed Point Theory and Applications, vol 2008, Article ID 418971, 4 pages, 2008 10 R Kannan, Some results on fixed points II,” The American Mathematical Monthly, vol 76, no 4, pp 405–408, 1969 11 M Kikkawa and T Suzuki, Some similarity between contractions and Kannan mappings, ... in X for the case when T : X → X is n ≥ 2 -times reasonable expansive self-mapping Note that simultaneously α -Kannan and n Z Theorem 3.2 is based on the fulfilment of the inequality d T n−1 x, T n x ≥ βd x, T x ; ∀x ∈ X, for some β > 1 for some real constant β > 1 while Theorem 3.11 is based on d T n x, T n 1 x ≥ γd x, T n x ≥ γβd x, T x ; ∀x ∈ X for some real constants β > 1, γ > 1 It is also of interest... 2.1 On the other hand, the L, m -property of contractive Kannan self-mappings can be tested for this example according to the formula xi 1 2 d T xi , 0 ≤ Ld xi 1 , xi 1−β xi β−m md xi, , 0 1 − xi 2 m xi 2 , ∀i ∈ Z 4.12 from Theorem 2.9 with α β/ 1 − β L m / 1−m ⇔ L 1−β / β−m with β ∈ 0, 1/3 , m ∈ 0, β since T : Rn × Z0 → Rn × Z0 is β/ 1 − β -Kannan and β -contractive Note that 1−β xi β−m 1 − xi 2 m xi... comments References 1 C F Chen and C X Zhu, “Fixed point theorems for n times reasonable expansive mapping,” Fixed Point Theory and Applications, vol 2008, Article ID 302617, 6 pages, 2008 2 L.-G Hu, “Strong convergence of a modified Halpern’s iteration for nonexpansive mappings, Fixed Point Theory and Applications, vol 2008, Article ID 649162, 9 pages, 2008 3 S Saeidi, “Approximating common fixed points... Theory and Applications, vol 2008, Article ID 363257, 17 pages, 2008 4 M De la Sen, “About robust stability of dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol 2008, Article ID 480187, 20 pages, 2008 5 A Latif and A A N Abdou, “Fixed points of generalized contractive maps,” Fixed Point Theory and Applications, vol 2009, Article ID 487161, 9 pages, 2009... 2009 6 J.-Z Xiao and X.-H Zhu, “Common fixed point theorems on weakly contractive and nonexpansive mappings, Fixed Point Theory and Applications, vol 2008, Article ID 469357, 8 pages, 2008 7 S Karpagam and S Agrawal, “Best proximity point theorems for p-cyclic Meir-Keeler contractions,” Fixed Point Theory and Applications, vol 2009, Article ID 197308, 9 pages, 2009 8 M De la Sen, “Stability and convergence... that by combining the above three relations: d T xi , T yi ≤ |t| |xi | yi t d xi , T xi 1−t d yi , T yi 4.3 Fixed Point Theory and Applications 21 and T : R → R is α -Kannan if 0 ≤ α : |t/ 1 − t | < 1/2 which is guaranteed for |t| < 1 if |t|/ 1 − |t| < 1/2 ⇒ |t| ≤ k < 1/3 which is the condition of Theorem 2.1(i) guaranteeing that if T : R → R is k -contractive, it is also α -Kannan Example 4.2 Now consider... Rn × Z0 → Rn × Z0 is β -contractive with β β1 f with q 0 ∈ Rn being its unique stable equilibrium point and its unique fixed point provided that 0 ≤ f < 1− β1 and 0 ≤ β1 < 1 The time-varying system is globally asymptotically stable 2 If β1 β2 < β1 f < 1/3, that is f ∈ 0, 1/3 − β1 and β1 ∈ 0, 1/3 then the β -contractive self-mapping T : Rn × Z0 → Rn × Z0 is furthermore β/ 1 − β -Kannan Those results still... surjective selfmapping which is continuous everywhere in X which satisfies d T n−1 x, T n x ≥ βd x, T x for some real constant β > 1, some n ≥ 2 ∈ Z , ∀x ∈ X The following holds i The following zero limit exists lim d T j n x, T jn 1 x − d T j n−1 j →∞ x, T j n x ∀x ∈ X 0; 3.6 ii If T : X → X is α -Kannan then a sufficient condition for Property (i) to hold is: βd x, T x ≥ α d T n−1 x, T n x d T n x, T n... βγd x, T x ; d T n x, T n 1 x md T n−1 x, q ; for all q ∈ ∀x ∈ X, some real constants β, γ > 1 3.34 ii assume that X, d is a nonempty complete metric space and that T : X → X is a continuous surjective n (Z n ≥ 2) times reasonable expansive self-mapping which satisfies Theorem 3.2 Then, T : X → X also possesses the L, m -property for some real constants L ∈ R0 and m ∈ 0, 1 ∈ R0 if and only if d T j . Applications Volume 2009, Article ID 815637, 25 pages doi:10.1155/2009/815637 Research Article Some Combined Relations between Contractive Mappings, Kannan Mappings, Reasonable Expansive Mappings, and T-Stability M points of T. 2. Combined Compatible Relations of k -Contractive Mappings, α -Kannan Mappings, and the L − m-Property It is of interest to establish when a k -contractive mapping is also α -Kannan and. Self-mappings T : X → X can be referred to, for instance, as Kannan -mappings, reasonable expansive mappings, and Picard T- stable mappings. Some relations between such concepts subject either to sufficient,