Báo cáo sinh học: "Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type"

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Fixed Point Theory and Applications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type Fixed Point Theory and Applications 2011, 2011:102 doi:10.1186/1687-1812-2011-102 Manuel De la Sen (wepdepam@lg.ehu.es) Ravi P Agarwal (Agarwal@tamuk.edu) ISSN 1687-1812 Article type Research Submission date 12 September 2011 Acceptance date 21 December 2011 Publication date 21 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/102 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 De la Sen and Agarwal ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution 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.
Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type Manuel De la Sen*1 and Ravi P Agarwal2 1 Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia) – Aptdo. 644-Bilbao, 48080-Bilbao, Spain 2 Department of Mathematics, Texas A&M University - Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA *Corresponding author: manuel.delasen@ehu.es Email address: RPA: Agarwal@tamuk.edu Abstract This article discusses three weak φ-contractive conditions of rational type for a class of 2- cyclic self-mappings defined on the union of two non-empty subsets of a metric space to itself. If the space is uniformly convex and the subsets are non-empty, closed, and convex, then the iterates of points obtained through the self-mapping converge to unique best proximity points in each of the subsets. 1. Introduction A general contractive condition has been proposed in [1, 2] for mappings on a partially ordered metric space. Some results about the existence of a fixed point and then its uniqueness under supplementary conditions are proved in those articles. The rational 1
contractive condition proposed in [3] includes as particular cases several of the previously proposed ones [1, 4–12], including Banach principle [5] and Kannan fixed point theorems [4, 8, 9, 11]. The rational contractive conditions of [1, 2] are applicable only on distinct points of the considered metric spaces. In particular, the fixed point theory for Kannan mappings is extended in [4] by the use of a non-increasing function affecting the contractive condition and the best constant to ensure a fixed point is also obtained. Three fixed point theorems which extended the fixed point theory for Kannan mappings were stated and proved in [11]. More attention has been paid to the investigation of standard contractive and Meir-Keeler-type contractive 2-cyclic self- mappings T : A ∪ B → A ∪ B defined on subsets A , B ⊆ X and, in general, p-cyclic self- mappings defined on any number of subsets , T : U i∈ p A i → U i∈ p A i Ai ⊂ X i ∈ p := {1 , 2 , ... , p } , where ( X , d ) is a metric space (see, for instance [13–22]). More recent investigation about cyclic self-mappings is being devoted to its characterization in partially ordered spaces and also to the formal extension of the contractive condition through the use of more general strictly increasing functions of the distance between adjacent subsets. In particular, the uniqueness of the best proximity points to which all the sequences of iterates of composed self-mappings T 2 : A ∪ B → A ∪ B converge is proven in [14] for the extension of the contractive principle for cyclic self-mappings in uniformly convex Banach spaces (then being strictly convex and reflexive [23]) if the subsets A, B ⊂ X in the metric space ( X , d ) , or in the Banach space (X , ) , where the 2- cyclic self-mappings are defined, are both non-empty, convex and closed. The research in [14] is centred on the case of the cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in [15] for Meir-Keeler cyclic contraction maps and, in general, for the self-mapping T : U i∈ p A i → U i∈ p A i be a p (≥ 2) - cyclic self-mapping being defined on any number of subsets of the metric space with 2
p : = {1 , 2 , ... , p } . Also, the concept of best proximity points of (in general) non-self- mappings S ,T : A → B relating non-empty subsets of metric spaces in the case that such maps do not have common fixed points has recently been investigated in [24, 25]. Such an approach is extended in [26] to a mapping structure being referred to as K-cyclic mapping with contractive constant k < 1 / 2 . In [27], the basic properties of cyclic self- mappings under a rational-type of contractive condition weighted by point-to-point- dependent continuous functions are investigated. On the other hand, some extensions of Krasnoselskii-type theorems and general rational contractive conditions to cyclic self- mappings have recently been given in [28, 29] while the study of stability through fixed point theory of Caputo linear fractional systems has been provided in [30]. Finally, promising results are being obtained concerning fixed point theory for multivalued maps (see, for instance [31–33]). This manuscript is devoted to the investigation of several modifications of rational type of the φ-contractive condition of [21, 22] for a class of 2-cyclic self-mappings on non- empty convex and closed subsets A , B ⊂ X . The contractive modification is of rational type and includes the nondecreasing function associated with the ϕ -contractions. The existence and uniqueness of two best proximity points, one in each of the subsets , of 2-cyclic self-mappings defined on the union of two non- A,B ⊂ X T : A∪ B → A∪ B empty, closed, and convex subsets of a uniformly convex Banach spaces, is proven. The convergence of the sequences of iterates through to one of such best T : A∪ B → A∪ B proximity points is also proven. In the case that A and B intersect, both the best proximity points coincide with the unique fixed point in the intersection of both the sets. 3
2. Basic properties of some modified constraints of 2-cyclic - ϕ contractions Let ( X , d ) be a metric space and consider two non-empty subsets and of . Let A B X T : A ∪ B → A ∪ B be a 2-cyclic self-mapping, i.e., T ( A) ⊆ B and T (B ) ⊆ A . Suppose, in addition, that T : A ∪ B → A ∪ B is a 2-cyclic modified weak ϕ -contraction (see [21, 22]) for some non-decreasing function ϕ : R0 + → R0 + subject to the rational modified ϕ - contractive constraint:  d (x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty )  d (Tx ,Ty ) ≤ α   + β ( d ( x , y ) − ϕ ( d (x , y )) ) + ϕ (D ) ; −ϕ    d (x , y ) d (x , y )      ∀x , y (≠ x ) ∈ A ∪ B (2.1) where D : = dist ( A, B ) : = inf { d ( x , y ) : x ∈ A , y ∈ B } (2.2) ( )   1 − k n (1 − k ) ( ) D ≤ lim sup d T n +1 x ,T n x ≤ lim  k n d (x ,Tx ) + ϕ (D ) = ϕ (D ) ; ∀x ∈ A ∪ B (2.3) n →∞   1− k   n →∞ Note that (2.1) is, in particular, a so-called 2-cyclic ϕ -contraction if α = 0 and ϕ (t ) = (1 − α ) t for some real constant α ∈ [ 0 , 1 ) since ϕ : R0 + → R0 + is strictly increasing [1]. We refer to “modified weak ϕ -contraction” for (2.1) in the particular case α ≥ 0 , β ≥ 0 , α + β < 1 , and ϕ : R0 + → R0 + being non-decreasing as counterpart to the term ϕ -contraction (or via an abuse of terminology “modified strong ϕ -contraction”) for the case of ϕ : R0 + → R0 + in (2.1) being strictly increasing. There are important background results on the properties of weak contractive mappings (see, for instance, [1, 2, 34] and references therein). The so-called “ ϕ -contraction”, [1, 2], involves the particular contractive 4
condition obtained from (2.1) with α = 0 , β = 1 , and ϕ : R0 + → R0 + being strictly increasing, that is, d (Tx ,Ty ) ≤ d ( x , y ) − ϕ ( d (x , y )) + ϕ (D ) , ∀x ∈ A ∪ B In the following, we refer to 2-cyclic self-maps T : A ∪ B → A ∪ B simply as cyclic self- maps. The following result holds: Lemma 2.1. Assume that T : A ∪ B → A ∪ B is a modified weak ϕ -contraction, that is, a cyclic self-map satisfying the contractive condition (2.1) subject to the constraints min (α , β ) ≥ 0 and α + β < 1 with ϕ : R0 + → R0 + being non-decreasing. Then, the following properties hold: (i) Assume that ϕ (D ) ≥ D ( ) D ≤ d T n +1 x, T n x ≤ kd (Tx , x ) + (1 − k ) ϕ (D ) ; ∀n ∈ N 0 : = N ∪ { 0 } , ∀x ∈ A ∪ B (2.4) ( ) ( ) D ≤ lim inf d T n + m +1 x ,T n + m x ≤ lim sup d T n + m +1 x ,T n + m x ≤ ϕ (D ) ; ∀x ∈ A ∪ B ∀m ∈ N 0 (2.5) n→∞ n→∞ ( ) lim sup d T n + m +1 x ,T n + m x ≤ ϕ (D ) ϕ (D ) = D = 0 and if D ≠ 0. If then n→∞ ( ) ∃ lim d T n + m +1x ,T n + m x = 0 ; ∀x ∈ A ∪ B , ∀m ∈ N 0 . n→∞ (ii) Assume that d (x, Tx ) ≤ m(x ) for any given x ∈ A ∪ B . Then k m(x ) 1 − k ( ) ϕ (D ) ; ∀x ∈ A ∪ B , ∀ n ∈ N d T nx , x ≤ (2.6) + 1− k k 5
If d (x, Tx ) is finite and, in particular, if x and Tx in are finite then the sequences A∪ B {T x} and {T n +1 x}n∈ N 0 are bounded sequences where T n x ∈ A and T n +1 x ∈ B if x ∈ A n n∈ N 0 and n is even, T n x ∈ B and T n +1 x ∈ B if x ∈ B and n is even. Proof: Take y = Tx so that Ty = T 2 x . Since ϕ : R0 + → R0 + is non-decreasing ϕ (x ) ≥ ϕ (D ) for x ≥ D , one gets for any x ∈ A and any Tx ∈ B or for any x ∈ B and any Tx ∈ A : (1 − α )d (T 2 x ,Tx ) ≤ −αϕ ( d (Tx ,T 2 x ) ) + β [d (x ,Tx ) − ϕ ( d (x ,Tx )) ] + ϕ (D ) = β d (x ,Tx ) + ϕ (D ) − αϕ ( d (Tx ,T 2 x ) ) − βϕ ( d (x ,Tx ) ) ; ∀x ∈ A ∪ B 1−α − β ( ) ⇔ d T 2 x ,Tx ≤ k d (x ,Tx ) + ϕ (D ) = k d ( x ,Tx ) + (1 − k ) ϕ (D ) ; ∀x ∈ A ∪ B (2.7) 1−α if Tx ≠ x where k := β < 1 , since T : A ∪ B → A ∪ B is cyclic, d (x ,Tx ) ≥ D and ϕ : R0 + → R0 + 1−α is increasing. Then ( ) ( ) d T n +1 x ,T n x ≤ k n d ( x ,Tx ) + 1 − k n ϕ (D ) ; ∀x ∈ A ∪ B ; ∀n ∈ N (2.8) ϕ (D ) ≥ D ≠ 0 since min (α , β ) ≥ 0 and α + β < 1 . Proceeding recursively from (2.8), one gets for any m ∈ N :  n −1 i  ( ) ( ) (2.9a) ∑ D ≤ d T n +1 x ,T n x ≤ k n d (Tx , x ) + ϕ (D )(1 − k ) k  ≤ kd (Tx , x ) + ϕ (D ) 1 − k n    i =0  ≤ kd (Tx , x ) + (1 − k )ϕ (D ) ≤ kd (Tx , x ) + ϕ (D ) < d (Tx , x ) + ϕ (D ) ; ∀x ∈ A ∪ B (2.9b)   n + m −1   ( ) D ≤ lim sup d T n + m +1x, T n + m x ≤ lim  k n + m d (Tx , x ) + ϕ (D )(1 − k ) ∑ k i       i =0  n →∞  n→∞ 6
 1 − k n+ m  ≤ ϕ (D )(1 − k ) lim   = ϕ (D ) ; ∀x ∈ A ∪ B (2.10) n → ∞ 1 − k    ( ) ϕ (D ) ≥ D ≠ 0 and if ϕ (D ) = D = 0 then the ∃ lim d T n + m +1x ,T n + m x = 0 ; ∀x ∈ A ∪ B . Hence, n→∞ Property (i) follows from (2.9) and (2.10) since ϕ (D ) ≥ D and d (x, Tx ) ≥ D ; ∀x ∈ A ∪ B , since T : A ∪ B → A ∪ B is a 2-cyclic self-mapping and ϕ : R0 + → R0 + is non-decreasing. Now, it follows from triangle inequality for distances and (2.9a) that: ( )∑ ( )∑ ∑ (1 − k )  d T i +1 x, T i x ≤  k  d (x ,Tx ) + ϕ (D ) n −1 n −1 i n −1 d T n x, x ≤ i         i =1 i =1 i =1 ( ) ( ) ( ) k 1 − k n −1 (1 − k ) 1 − (1 − k ) n −1 ϕ (D ) k 1 − k n −1 ( ) −1 d (x ,Tx ) + d ( x ,Tx ) + ϕ (D ) ∑in=1 (1 − k ) i ≤ ≤ 1− k 1− k k 1− k k d ( x ,Tx ) + ϕ (D ) < ∞ , ∀x ∈ A ∪ B , ∀ n ∈ N (2.11) ≤ 1− k k {} and {T n +1 x}n∈ N 0 being bounded which leads directly to Property (ii) with T n x n∈ N 0 sequences for any finite x ∈ A ∪ B . □ Concerning the case that A and B intersect, we have the following existence and uniqueness result of fixed points: Theorem 2.2. If ϕ (D ) = D = 0 (i.e., A0 ∩ B 0 ≠ ∅ ) then ∃ lim d (T n + m +1 x ,T n + m x ) = 0 and n →∞ k d ( x ,Tx ) ( ) ; ∀x ∈ A ∪ B . Furthermore, if ( X , d ) is complete and A and B are non- d T nx , x ≤ 1− k empty closed and convex then there is a unique fixed point of T : A ∪ B → A ∪ B z∈ A∩ B 7
{T x} n to which all the sequences , which are Cauchy sequences, converge; n∈N 0 ∀x ∈ A ∪ B . Proof: It follows from Lemma 2.1(i)–(ii) for ϕ (D ) = D = 0 . It also follows that ( ) ( )( ) lim d T n + m +1 x, T n + m x = lim k n d T m + 2 x, T m +1 x = 0 ; ∀x ∈ A ∪ B , what implies ∀ m ∈N0 n→∞ n→∞ ( ) {} lim d T n + m +1 x, T n + m x = 0 so that T n x is a Cauchy sequence, ∀x ∈ A ∪ B , then being n∈ N 0 n ,m → ∞ bounded and also convergent in A ∩ B as n → ∞ since ( X , d ) is complete and A and B are lim T n x = z ∈ A ∩ B non-empty, closed, and convex. Thus, and n →∞ z = lim T n +1 x = T  lim T n +1 x  = Tz , since the iterate composed self-mapping    n →∞  n →∞ T n : A ∪ B → A ∪ B , ∀n ∈ N 0 is continuous for any initial point x ∈ A ∪ B (since it is contractive, then Lipschitz continuous in view of (2.9a) with associate Lipschitz constant for ϕ (D ) = D = 0 ). Thus, z ∈ A ∩ B is a fixed point of T : A ∪ B → A ∪ B . Its 0 ≤ k 0 since z ≠ y . Then, d (Tz , Ty ) ≤ β d (x , y ) ≤ β d (x , y ) < d ( z , y ) what leads to the ( ) contradiction lim d T n z , T n y = 0 = d (z , y ) > 0 . Thus, z = y . Hence, the theorem. □ n→∞ Now, the contractive condition (2.1) is modified as follows:  d (x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty )  d (Tx ,Ty ) ≤ α 0   + β 0 ( d ( x , y ) − ϕ ( d (x , y )) ) + ϕ (D ) −ϕ  (2.12)   d (x , y ) d (x , y )    8
for x , y (≠ x ) ∈ X , where min (α 0 , β 0 ) ≥ 0 , min (α 0 , β 0 ) > 0 , and α 0 + β 0 ≤ 1 . Note that in the former contractive condition (2.1), α + β < 1 . Thus, for any non-negative real constants α ≤ α 0 and β ≤ β 0 , (2.12) can be rewritten as  d (x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty )  d (Tx ,Ty ) ≤ α   + β ( d (x , y ) − ϕ ( d ( x , y )) ) + ϕ (D ) −ϕ    d (x , y ) d (x , y )     d (x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty )  + (α 0 − α )   + (β 0 − β )( d (x , y ) − ϕ ( d ( x , y )) ) ; ∀x , y ∈ A ∪ B . −ϕ  (2.13)   d (x , y ) d (x , y )    The following two results extend Lemma 2.1 and Theorem 2.2 by using constants α 0 and in (2.1) whose sum can equalize unity α 0 + β 0 = 1 . β 0 Lemma 2.3. Assume that T : A ∪ B → A ∪ B is a cyclic self-map satisfying the contractive condition (2.13) with min (α 0 , β 0 ) ≥ 0 , α 0 + β 0 ≤ 1 , and ϕ : R0 + → R0 + is non-decreasing. Assume also that 1−α ϕ (d (Tx , x )) ≥ d (Tx , x ) − M 0 ; ∀x ∈ A ∪ B (2.14) 1−α − β For some non-negative real constants M 0 ≤ 1 − α − β D , α ≤ α 0 and β ≤ β 0 with α + β < 1 . 1−α Then, the following properties hold: ( ) (i) D ≤ lim sup d T n + m +1 x ,T n + m x ≤ ϕ (D ) + (α 0 + β 0 − α − β ) D ; ∀x ∈ A ∪ B , ∀m ∈ N 0 (2.15) n→∞ for any arbitrarily small ε ∈R + . 9
( ) (ii) If ϕ (D ) = (1 + α + β − α 0 − β 0 ) D then ∃ lim d T n + m +1 x ,T n + m x = D ; ∀x ∈ A ∪ B , ∀m ∈ N 0 . n→∞ (iii) If d (x, Tx ) is finite and, in particular, if x and Tx are finite then the sequence {T x} and {T n +1 x}n∈ N 0 are bounded sequences, where T n x ∈ A and T n +1 x ∈ B if x ∈ A n n∈ N 0 and n is even and T n x ∈ B and T n +1 x ∈ B if x ∈ B and n is even. Proof: Since ϕ : R0 + → R0 + is non-decreasing then ϕ (x ) ≥ ϕ (D ) for x(∈ R 0 + ) ≥ D . Note also 1−α − β implies the necessary condition ϕ (d (Tx , x )) ≥ 0 and (2.14) implies that that M 0 ≤ D 1−α 0 ≤ ϕ (D ) ≤ D . Note also for y = Tx and Ty = Tx 2 and (2.14), since ϕ (x ) > ϕ (D ) for x > D , that for x∈ A ∪ B , one gets from (2.14): (( )) ( ) 1−α ϕ d T 2 x, Tx ≥ d T 2 x, Tx − M 0 ; ∀x∈ A ∪ B (2.16) 1−α − β leading from (2.14) to (α 0 − α ) [d (T 2 x ,Tx ) − ϕ ( d (T 2 x ,Tx ) ) ] + (β 0 − β )( d (Tx , x ) − ϕ ( d (Tx , x ) ) ) ≤ M : = (α 0 + β 0 − α − β ) 1−α M0 1−α − β (2.17) and M ≤ (α 0 + β 0 − α − β ) D since M 0 ≤ 1 − α − β D . One gets from (2.13) and (2.17) the 1−α following modifications of (2.9) and (2.10) by taking y = Tx , Ty = T 2 x , and successive iterates by composition of the self-mapping T : A ∪ B → A ∪ B :  n−1 i  ( ) ( ) ∑ D ≤ d T n+1 x ,T n x ≤ k n d (Tx , x ) + (ϕ (D ) + M )(1 − k ) k  ≤ k n d (Tx , x ) + 1 − k n (ϕ (D ) + M )    i =0  ≤ kd (Tx , x ) + ϕ (D ) + M ; ∀x ∈ A ∪ B , ∀n ∈ N 0 := N ∪ { 0 } (2.18) 10
( ) D ≤ lim sup d T n +1 x, T n x ≤ ϕ (D ) + M ≤ ϕ (D ) + (α 0 + β 0 − α − β ) D ; ∀x ∈ A ∪ B , ∀m ∈ N 0 n→∞ (2.19) ( ) D ≤ lim sup d T n + m +1 x, T n + m x n →∞   n + m −1   ≤ lim  k n + m d (Tx , x ) + (ϕ (D ) + (α − α − β ) D )(1 − k ) ∑ k i   +β   n→∞  0 0  i =0   ≤ ϕ (D ) + (α − α − β )D ; ∀x ∈ A ∪ B , ∀m ∈ N 0 (2.20) +β 0 0 and Property (i) has been proven. Property (ii) follows from (2.20) directly by replacing ϕ (D ) = (1 + α + β − α 0 − β ) D in (2.15). To prove Property (iii), note from (2.18) that 0 ( )∑ ( )∑ ∑ (1 − k )  d T i +1 x, T i x ≤  n −1 i  k  d (x ,Tx ) + (ϕ (D ) + M ) n −1 n −1 d T n x, x ≤ i        i =1 i =1 i =1 ( ) k 1 − k n −1 ( ) −1 d (x ,Tx ) + (ϕ (D ) + M ) ∑ in=1 (1 − k ) i ≤ 1− k ( ) ( ) k 1 − k n −1 (1 − k ) 1 − (1 − k ) n −1 (ϕ (D ) + M ) d (x ,Tx ) + ≤ 1− k k 1− k k d (x ,Tx ) + (ϕ (D ) + M ) < ∞ ; ∀x ∈ A ∪ B , ∀ n ∈ N . ≤ 1− k k Hence, {T n x}n∈ N 0 and {T n +1 x} n∈ N 0 are bounded for any finite x ∈ A ∪ B . Property (iii) has been proven. Hence, the lemma. □ Theorem 2.4. If ϕ (D ) = D = 0 then ∃ lim d (T n + m +1 x ,T n + m x ) = 0 ; ∀x ∈ A ∪ B . Furthermore, if n →∞ ( X ,d ) is complete and both A and B are non-empty, closed, and convex then there is a 11
{} of T : A ∪ B → A ∪ B to which all the sequences T n x n∈N 0 , unique fixed point z∈ A∩ B which are Cauchy sequences, converge; ∀x ∈ A ∪ B . Proof guideline: It is identical to that of Theorem 2.2 by using ϕ (D ) = D = M 0 = M = 0 and the fact that from (2.17) α 0 = α and β 0 = β with 0 ≤ α + β < 1 if there is a pair ( ) (( )); (x ,Tx ) ∈ A × B ∪ B × A such that d (Tx , x ) = ϕ (d (Tx , x )) ; d T 2 x ,Tx = ϕ d T 2 x ,Tx ∀x ∈ A ∪ B . Hence, the theorem. □ Remark 2.5. Note that Lemma 2.2 (ii) for ϕ (D ) ≤ D ( ϕ (D ) < D if α + β ≤ α 0 + β 0 ≤ 1 ) leads to an identical result as Lemma 2.1 (i) for ϕ (D ) = D and α + β < 1 consisting in proving that ( ) ∃ lim d T n +1 x ,T n x = D . This result is similar to a parallel obtained for standard 2-cyclic n→∞ contractions [2, 5, 8]. □ Remark 2.6. Note from (2.7) that Lemma 2.1 is subject to the necessary condition ( ) D ≤ ϕ (D ) since d T 2 x , Tx ≥ D and d (T x , x ) ≥ D ; ∀x ∈ A ∪ B . On the other hand, note from 1−α Lemma 2.2, Equation (2.14) that ϕ (D ) ≥ D − M0 , and one also gets from (2.18) for n 1−α − β 1−α = 1 the dominant lower-bound ϕ (D ) ≥ D − M ≥ D − M 0 (α 0 + β 0 − α − β ) , that is, 1−α − β 1−α D ≤ ϕ (D ) + M 0 (α 0 + β 0 − α − β ) which coincides with the parallel constraint obtained 1−α − β from Lemma 2.1 if α 0 + β 0 = α + β . □ 12
Remark 2.7. Note that Lemmas 2.2 and 2.3 apply for non-decreasing functions ϕ : R0 + → R0 + . The case of ϕ : R0 + → R0 + being monotone increasing, then unbounded, is also included as it is the case of ϕ : R0 + → R0 + being bounded non-decreasing. □ Now, modify the modified cyclic φ-contractive constraint (2.1) as follows:  d ( x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty ) + β ( d ( x , y ) − ϕ ( d (x , y )) ) + (1 − α )ϕ   + ϕ (D ) ; ∀x ∈ A ∪ B d (Tx ,Ty ) ≤ α   d (x , y ) d (x , y )   (2.21) Thus, the following parallel result to Lemmas 2.1 and 2.2 result holds under a more restrictive modified weak φ-contraction Assume that T : A ∪ B → A ∪ B is modified weak φ- ϕ (D ) contraction subject to ϕ : R0 + → R0 + subject to the constraint lim sup (x − ϕ (x )) > and 1− α − β x → +∞ having a finite limit: Lemma 2.8. Assume that T : A ∪ B → A ∪ B is a cyclic self-map satisfying the contractive condition (2.21) with min (α , β ) ≥ 0 , α + β < 1 , and ϕ : R0 + → R0 + is non-decreasing having a finite limit lim ϕ (x ) = ϕ and subject to ϕ (0 ) = 0 . Assume also that ϕ : R0 + → R0 + satisfies x→∞ ϕ (D ) lim sup (x − ϕ ( x )) > . Then, the following properties hold: 1− α − β x → +∞ (i) The following relations are fulfilled: 13
ϕ (D ) ( ) 1− α − β 2 −α − β ϕ (D ) ≤ D ≤ d T n +1 x ,T n x ≤ ϕ < ∞ ; ∀n ∈ N , ∀x ∈ A ∪ B (2.22) +ϕ ≤ 2 −α − β 1− α − β 1−α − β ϕ (D ) 1−α − β 2 −α − β ( ) ϕ (D ) ≤ D ≤ lim sup d T n+1 x ,T n x ≤ ϕ < ∞ ; ∀x ∈ A ∪ B (2.23) +ϕ ≤ 2 −α − β 1−α − β 1−α − β n→∞ (ii) If, furthermore, ϕ : R0 + → R0 + is, in addition, sub-additive and d (x, Tx ) is finite (in {} and {T n +1 x} n∈ N 0 are both particular, if x and Tx are finite) then the sequences T n x n∈ N 0 bounded, where T n x ∈ A and T n +1 x ∈ B if x ∈ A and n is even and T n x ∈ B and T n +1 x ∈ A if and n is even. If ϕ : R0 + → R0 + is identically zero then ∃ lim d (T n + m +1 x ,T n + m x ) = 0 ; x∈B n →∞ ∀x ∈ A ∪ B . Proof: One gets directly from (2.21): (1 − α )(d (T 2 x ,Tx )− ϕ ( d (T 2 x ,Tx ) ) ) ( ) (( ) )+ ϕ (D ) ; ∀x ∈ A ∪ B ≤ β d T 2 x ,Tx − ϕ d T 2 x ,Tx (2.24) or, equivalently, one gets for k = β < 1 that 1−α ϕ (D ) ( ) (( )) ( ) (( )) d T 2 x ,Tx − ϕ d T 2 x ,Tx ≤ k d T 2 x ,Tx − ϕ d T 2 x ,Tx + ; ∀x ∈ A ∪ B (2.25) 1−α leading to (( ) (( ))) 0 ≤ D − ϕ (D ) ≤ lim inf d T n +1 x ,T n x − ϕ d T n +1 x ,T n x n →∞ ϕ (D ) ϕ (D ) (( ) (( ) ) )≤ ≤ lim sup d T n +1 x ,T n x − ϕ d T n +1 x ,T n x ; ∀x ∈ A ∪ B (2.26) = (1 − α )(1 − k ) 1 − α − β n →∞ 14
1−α − β 2 −α − β D what implies the necessary condition ϕ (D ) ≥ leading to if = >1 D ϕ (D ) 1 − α − β 2 −α − β (( ) (( ) ) )≥ D − ϕ (D ) ≥ 0 ; ∀x ∈ A ∪ B . Also, since and then lim inf d T n +1 x ,T n x − ϕ d T n +1 x ,T n x D≠0 n→∞ ϕ (D ) ( ) lim sup (x − ϕ ( x )) > ; ∀ x ∈ R+ , by construction, then d T n +1 x ,T n x is bounded; ∀n ∈ N 1− α − β x → +∞ since, otherwise, a contradiction to (2.24) holds. Since ϕ : R 0 + → R0 + is non-decreasing and has a finite limit ϕ ≥ ϕ (x ) ≥ 0 ; ∀ x ∈ R0 + ( ϕ = 0 if and only if ϕ : R0 + → R0 + is identically zero), thus ϕ ≥ ϕ (D ) ≥ 0 . Then, (2.22)–(2.23) hold and Property (i) has been proven. On the other hand, one gets from (2.25), since ϕ : R0 + → R0 + is sub-additive and nondecreasing and has a finite limit, that: ( ) (( )) ∑ ( ) (( )) d T n x, x − ϕ d T n x , x ≤  d T i +1 x, T i x − ϕ d T i +1 x ,T i x  n −1     i =1 ϕ (D )  ∑ (1 − k )  ≤ n −1 i  ∑ n −1 k  (d (x ,Tx ) − ϕ ( d (x ,Tx ) )) + i      1−α   i =1 i =1 ( ) (d (x ,Tx ) − ϕ ( d (x ,Tx ) )) + ϕ (D )  k 1 − k n −1 (1 − k )  i 1−α ∑ n −1   ≤  1− k i =1 ( ) ( ) (1 − k ) 1 − (1 − k ) n −1 ϕ (D ) k 1 − k n −1 d (x ,Tx ) + ≤ 1−α 1− k k (d (x ,Tx ) − ϕ ( d (x ,Tx ) )) + 1 − k ϕ (D ) < ∞ ; ∀x ∈ A ∪ B , ∀ n ∈ N k (2.27) ≤ k 1−α 1− k (d (x ,Tx ) − ϕ ( d (x ,Tx ) )) + 1 − k ϕ (D ) + ϕ < ∞ ; ∀x ∈ A ∪ B ( ) k lim sup d T n x, x ≤ (2.28) k 1−α 1− k n→∞ 15
{} and {T n +1 x} n∈ N 0 are both bounded for any x ∈ A ∪ B . Then the sequences T n x n∈ N 0 Hence, the first part of Property (ii). If ϕ : R0 + → R0 + is identically zero then ( ) ϕ ≡ ϕ (x ,Tx ) = 0 ; ∀x ∈ A ∪ B so that ∃ lim d T n + m +1 x ,T n + m x = 0 from (2.23). Hence, the n →∞ lemma. □ The existence and uniqueness of a fixed point in A ∩ B if A and B are non-empty, closed, and convex and ( X , d ) is complete follows in the subsequent result as its counterpart in Theorem 2.2 modified cyclic φ-contractive constraint (2.21): Theorem 2.9. if ( X , d ) is complete and A and B intersect and are non-empty, closed, and convex then there is a unique fixed point z ∈ A ∩ B of T : A ∪ B → A ∪ B to which all the {} sequences T n x n∈N 0 , which are Cauchy sequences, converge; ∀x ∈ A ∪ B . □ Remark 2.7. Note that the nondecreasing function ϕ : R0 + → R0 + of the contractive condition (2.21) is not monotone increasing under Lemma 2.5 since it possesses a finite limit and it is then bounded. □ 2.8. The case of being a φ-contraction, namely, Remark T : A∪ B → A∪ B d (Tx ,Ty ) ≤ d ( x , y ) − ϕ ( d (x , y )) + ϕ (D ) with strictly increasing ϕ : R0 + → R0 + ; ∀x ∈ A ∪ B , [1, 2] implies , since ϕ (x ) = 0 if and only if x = 0 , implies the relation d (Tx ,Ty ) ≤ β 1d (x , y ) + ϕ (D ) < d (x , y ) + ϕ (D ) ; ∀x , y (≠ x ) ∈ A ∪ B (2.29) 16
for some real constant 0 ≤ β1 = β1 (x , y ) < 1 ; ∀x , y (≠ x ) ∈ A ∪ B so that proceeding recursively: ]d (Tx , x ) + ϕ (D )(∑ nj =1 ∏ l = j +1 [β l ] ) ≤ d (Tx , x ) + Lϕ (D ) , ( ) d T n +1 x ,T n x ≤ ∏ in=1 [β n ∀x ∈ A ∪ B i (2.30) γ ϕ (D ) ( ) D ≤ lim sup d T n +1x ,T n x ≤ ; ∀x ∈ A ∪ B (2.31) 1− β n →∞ 1 ( [ β i ]) ( ) n and ∃ lim d T n +1 x ,T n x = 0 ; ∀x ∈ A ∪ B if ϕ (D ) = D = 0 , and ∏in=1 where β : = lim 0 since 0 ≤ α + β < 1 . (1 − α − β )ϕ (D ) ≤ (1 − α − β ) D what implies However, such a constraint in Lemma 2.3 and Theorem 3.4 implies that (1 − α 0 − β 0 )ϕ (D ) ≤ (1 − α 0 − β 0 )D . □ 3. Properties for the case that A and B do not intersect This section considers the contractive conditions (2.1) and (2.21) for the case A ∩ B = ∅ . For such a case, Lemmas 2.1, 2.3, and 2.8 still hold. However, Theorems 2.2, 2.4, and 2.9 do not further hold since fixed points in A ∩ B cannot exist. Thus, the investigation is centred in the existence of best proximity points. It has been proven in [1] that if is a cyclic φ-contraction with A and B being weakly closed subsets of a T : A∪ B → A∪ B reflexive Banach space (X , ) then, ∃ (x , y ) ∈ A × B such that D = d (x , y ) = x − y where 17
is a norm-induced metric, i.e., x and y are best proximity points. Also, if d : R 0+ → R 0+ is a cyclic contraction ∃ (x , y ) ∈ A × B such that D = d (x , y ) if A is compact T : A∪ B → A∪ B and B is approximatively compact with respect to A with both A and B being subsets of a ( ) metric space ( X , d ) (i.e., if lim d T 2n x , y = d (B , y ) : = inf d (z , y ) for some y ∈ A and x ∈ B n→∞ z∈ B { } then the sequence T 2n x has a convergent subsequence [14]). Theorem 2.2 extends n∈N 0 via Lemma 2.1 as follows for the case when A and B do not intersect, in general: Theorem 3.1. Assume that T : A ∪ B → A ∪ B is a modified weak φ-contraction, that is, a cyclic self-map satisfying the contractive condition (2.1) subject to the constraints min (α , β ) ≥ 0 and α + β < 1 with ϕ : R0 + → R0 + being nondecreasing with ϕ (D ) = D . Assume also that A and B are non-empty closed and convex subsets of a uniformly convex Banach (X , ) . space Then, there exist two unique best proximity points z ∈ A , y ∈ B of such that Tz = y , Ty = z to which all the sequences generated by T : A∪ B → A∪ B iterations of T : A ∪ B → A ∪ B converge for any x ∈ A ∪ B as follows. The sequences {T x} { } and T 2n+1 x 2n converge to z and y for all x ∈ A , respectively, to y and z for n∈ N 0 n∈N 0 all x ∈ B . If A ∩ B ≠ ∅ then z = y ∈ A ∩ B is the unique fixed point of T : A ∪ B → A ∪ B . Proof: If D = 0 , i.e., A and B intersect then this result reduces to Theorem 2.2 with the best proximity points being coincident and equal to the unique fixed point. Consider the case that A and B do not intersect, that is, D > 0 and take x ∈ A ∪ B . Assume with no loss in generality that x ∈ A . It follows, since A and B are non-empty and closed, A is convex and Lemma 3.1 (i) that: 18
[ d (T ] ⇒ d (T ( ) ( ) ) 2 n+ p) 2n +1 x , T 2n x → D ; d T 2n +1x , T 2n + 2 x → D x , T 2n x → 0 as n → ∞ (3.1) (proven in Lemma 3.8 [14]). The same conclusion arises if x ∈ B since B is convex. Thus, {T x} 2n is bounded [Lemma 2.1 (ii)] and converges to some point z = z (x ) , being n∈ N 0 potentially dependently on the initial point x, which is in A if x ∈ A , since A is closed, and in B if x ∈ B since B is closed. Take with no loss in generality the norm-induced metric and consider the associate metric space ( X , d ) which can be identified with (X , ) in this context. It is now proven by contradiction that for every ε ∈ R+ , there exists n 0 ∈ N 0 such ( ) that d T 2m x, T 2n +1 x ≤ D + ε for all m > n ≥ n0 . Assume the contrary, that is, given some ( ) ε ∈ R+ , there exists n 0 ∈ N 0 such that d T 2mk x ,T 2nk +1 x > D + ε for all mk > nk ≥ n0 ∀k ∈ N 0 . Then, by using the triangle inequality for distances: ( )( )( ) as n → ∞ D + ε < d T 2mk x, T 2n k +1 x ≤ d T 2 mk x, T 2 mk + 2 x + d T 2mk + 2 x, T 2 nk +1 x (3.2) One gets from (3.1) and (3.2) that (( )( )) ( ) lim inf d T 2 mk x, T 2m k + 2 x + d T 2 mk + 2 x, T 2nk +1 x = lim inf d T 2 m k + 2 x, T 2n k +1 x > D + ε (3.3) k →∞ k →∞ Now, one gets from (3.1), (3.3), ϕ (D ) ≥ D , and Lemma 2.1 (i) the following contradiction: ( ) ( ) ( ) D + ε < lim sup d T 2 mk + 2 x, T 2 nk +1 x ≤ lim sup d T 2 nk + 2 x, T 2n k +1 x + lim sup d T 2 mk + 2 x, T 2 nk + 2 x k →∞ nk → ∞ k →∞ ( ) = lim sup d T 2 nk + 2 x, T 2nk +1 x = D (3.4) nk → ∞ 19