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 Article type 1687-1812 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 Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia) – Aptdo 644-Bilbao, 48080-Bilbao, Spain 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 2cyclic 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 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 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 selfmappings T : A ∪ B → A ∪ B defined on subsets A , B ⊆ X and, in general, p-cyclic selfmappings T : U i∈ p A i → U i∈ p A i i ∈ p := {1 , , , p } , defined on any number of subsets Ai ⊂ X , 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 : 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 p := {1 , , , 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 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 < / In [27], the basic properties of cyclic selfmappings under a rational-type of contractive condition weighted by point-to-pointdependent continuous functions are investigated On the other hand, some extensions of Krasnoselskii-type theorems and general rational contractive conditions to cyclic selfmappings 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 nonempty 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 A,B ⊂ X , of 2-cyclic self-mappings T : A∪ B → A∪ B defined on the union of two non- empty, closed, and convex subsets of a uniformly convex Banach spaces, is proven The convergence of the sequences of iterates through T : A∪ B → A∪ B to one of such best 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 Basic properties of some modified constraints of 2-cyclic ϕ - contractions Let ( X , d ) be a metric space and consider two non-empty subsets A and B of X Let 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 )   + β ( d ( x , y ) − ϕ ( d (x , y )) ) + ϕ (D ) ; d (Tx ,Ty ) ≤ α  −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) ( )   − 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  1− k n →∞  n →∞   ( ) (2.3) Note that (2.1) is, in particular, a so-called 2-cyclic ϕ -contraction if α = and ϕ (t ) = (1 − α ) t for some real constant α ∈ [ , ) since ϕ : R0 + → R0 + is strictly increasing [1] We refer to “modified weak ϕ -contraction” for (2.1) in the particular case α ≥ , β ≥ , α + β < , 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 condition obtained from (2.1) with α = , β = , 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 selfmaps 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 (α , β ) ≥ and α + β < 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 : = N ∪ { } , ∀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 n→∞ ( ) lim sup d T n + m +1 x ,T n + m x ≤ ϕ (D ) and if D ≠ If ϕ (D ) = D = n→∞ ( ) ∃ lim d T n + m +1x ,T n + m x = ; ∀x ∈ A ∪ B , ∀m ∈ N n→∞ (ii) Assume that d (x, Tx ) ≤ m(x ) for any given x ∈ A ∪ B Then ( ) (2.5) n→∞ d T nx , x ≤ k m(x ) − k + ϕ (D ) ; ∀x ∈ A ∪ B , ∀n ∈ N 1− k k (2.6) then If d (x, Tx ) is finite and, in particular, if x and Tx in {T x} n n∈ N A∪ B are finite then the sequences and {T n +1 x}n∈ N are bounded sequences where T n x ∈ A and T n +1 x ∈ B if x ∈ A 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 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 x ,Tx ) ≤ −αϕ ( d (Tx ,T x ) ) + β [d (x ,Tx ) − ϕ ( d (x ,Tx )) ] + ϕ (D ) = β d (x ,Tx ) + ϕ (D ) − αϕ ( d (Tx ,T x ) ) − βϕ ( d (x ,Tx ) ) ; ∀x ∈ A ∪ B ( ) ⇔ d T x ,Tx ≤ k d (x ,Tx ) + 1−α − β ϕ (D ) = k d ( x ,Tx ) + (1 − k ) ϕ (D ) ; ∀x ∈ A ∪ B 1−α (2.7) if Tx ≠ x where k := β < , 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 ) + − k n ϕ (D ) ; ∀x ∈ A ∪ B ; ∀n ∈ N (2.8) ϕ (D ) ≥ D ≠ since (α , β ) ≥ and α + β < Proceeding recursively from (2.8), one gets for any m ∈ N :  n −1 i  D ≤ d T n +1 x ,T n x ≤ k n d (Tx , x ) + ϕ (D )(1 − k ) k  ≤ kd (Tx , x ) + ϕ (D ) − k n    i =0  ( ) ( ∑ ) (2.9a) ≤ kd (Tx , x ) + (1 − k )ϕ (D ) ≤ kd (Tx , x ) + ϕ (D ) < d (Tx , x ) + ϕ (D ) ; ∀x ∈ A ∪ B   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      n →∞  n→∞  i =0  ( ) (2.9b)  − k n+ m   = ϕ (D ) ; ∀x ∈ A ∪ B ≤ ϕ (D )(1 − k ) lim  n → ∞ − k    (2.10) ( ) ϕ (D ) ≥ D ≠ and if ϕ (D ) = D = then the ∃ lim d T n + m +1x ,T n + m x = ; ∀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: ( ) ∑ d T n x, x ≤ ( n −1 i =1 ( ) ∑ d T i +1 x, T i x ≤    ) k  d (x ,Tx ) + ϕ (D )     n −1 i i =1 (  ∑ (1 − k )   n −1 i i =1 ) ≤ k − k n −1 −1 d ( x ,Tx ) + ϕ (D ) ∑in=1 (1 − k ) 1− k ≤ ( k 1− k ϕ (D ) < ∞ , ∀x ∈ A ∪ B , ∀n ∈ N d ( x ,Tx ) + 1− k k ( ) i ≤ ) k − k n −1 (1 − k ) − (1 − k ) n −1 ϕ (D ) d (x ,Tx ) + 1− k k { } which leads directly to Property (ii) with T n x (2.11) n∈ N and {T n +1 x}n∈ N being bounded 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 = (i.e., A0 ∩ B ≠ ∅ ) then ∃ lim d (T n + m +1 x ,T n + m x ) = and n →∞ ( ) d T nx , x ≤ k d ( x ,Tx ) 1− k ; ∀x ∈ A ∪ B Furthermore, if ( X , d ) is complete and A and B 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 x} n n∈N , which are Cauchy sequences, converge; ∀x ∈ A ∪ B Proof: It follows from Lemma 2.1(i)–(ii) for ϕ (D ) = D = It also follows that ( ) ( )( ) lim d T n + m +1 x, T n + m x = lim k n d T m + x, T m +1 x = ; n→∞ ( n→∞ ) { } lim d T n + m +1 x, T n + m x = so that T n x n ,m → ∞ n∈ N ∀x ∈ A ∪ B , ∀ m ∈N0 what implies is a Cauchy sequence, ∀x ∈ A ∪ B , then being bounded and also convergent in A ∩ B as n → ∞ since ( X , d ) is complete and A and B are non-empty, closed, and z = lim T n +1 x = T  lim T n +1 x  = Tz ,   n →∞  n →∞  convex since the Thus, iterate lim T n x = z ∈ A ∩ B and n →∞ composed self-mapping T n : A ∪ B → A ∪ B , ∀n ∈ N 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 ≤ k or d (Tz ,Ty ) = d (z , y ) = what contradicts 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 = = d (z , y ) > 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 )   + β ( d ( x , y ) − ϕ ( d (x , y )) ) + ϕ (D ) d (Tx ,Ty ) ≤ α  −ϕ    d (x , y ) d (x , y )    (2.12) □ for x , y (≠ x ) ∈ X , where (α , β ) ≥ , (α , β ) > , and α + β ≤ Note that in the former contractive condition (2.1), α + β < Thus, for any non-negative real constants α ≤ α and β ≤ β , (2.12) can be rewritten as  d (x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty )   + β ( d (x , y ) − ϕ ( d ( x , y )) ) + ϕ (D ) d (Tx ,Ty ) ≤ α  −ϕ    d (x , y ) d (x , y )     d (x ,Tx )d ( y ,Ty )  d (x ,Tx )d ( y ,Ty )   + (β − β )( d (x , y ) − ϕ ( d ( x , y )) ) ; ∀x , y ∈ A ∪ B + (α − α )  −ϕ    d (x , y ) d (x , y )    (2.13) The following two results extend Lemma 2.1 and Theorem 2.2 by using constants α and β in (2.1) whose sum can equalize unity α + β = Lemma 2.3 Assume that T : A ∪ B → A ∪ B is a cyclic self-map satisfying the contractive condition (2.13) with (α , β ) ≥ , α + β ≤ , and ϕ : R0 + → R0 + is non-decreasing Assume also that ϕ (d (Tx , x )) ≥ d (Tx , x ) − 1−α M ; ∀x ∈ A ∪ B 1−α − β (2.14) For some non-negative real constants M ≤ − α − β D , α ≤ α and β ≤ β with α + β < 1−α Then, the following properties hold: ( ) (i) D ≤ lim sup d T n + m +1 x ,T n + m x ≤ ϕ (D ) + (α + β − α − β ) D ; ∀x ∈ A ∪ B , ∀m ∈ N n→∞ for any arbitrarily small ε ∈R + (2.15) ( ) As a result, d T 2m x, T 2n +1 x ≤ D + ε for every given ε ∈ R+ and all m > n ≥ n0 for some existing n ∈ N This leads by a choice of arbitrarily small ε to ( ) ( ) D ≤ lim sup d T 2m x, T 2n +1 x ≤ D ⇒ ∃ lim d T m x, T n +1 x = D n→∞ { } But T 2n x y = T2y (3.5) n→∞ n∈ N is a Cauchy sequence with a limit z = T z in A (respectively, with a limit in B) if x ∈ A (respectively, if x ∈ B ) such that D = Tz − z = d (z , Tz ) (Proposition { } 3.2 [14]) Assume on the contrary that x ∈ A and T 2n x T z − Tz = z − Tz ≠ z − y n∈N → so that since A is convex and (X , ) z ≠ T z as n → ∞ so that is uniformly convex Banach space, then strictly convex, one has  T 2z + z  T z − Tz z − Tz D = d (z , Tz ) = d  − Tz  = +   2   ≤ T z − Tz z − Tz D D + < + =D 2 2 (3.6) which is a contradiction so that z = T z is a best approximation point in A of T : A∪ B → A∪ B T2y = y∈ B {T x} 2n n∈ N is a Cauchy sequence with a limit which is a best approximation point in B of T : A ∪ B → A ∪ B if x ∈ B since B is convex and (X , that In the same way, y ≠ Tz ) is strictly convex We prove now that y = T y ,Tz = T z ∈ B , with d (Tz , z ) = d (Ty , y ) = D , y = Tz Assume, on the contrary d (z , y ) > D , z = T 2z ∈ A , d (Tz ,Ty ) ≥ D , and ϕ (D ) = D One gets from (2.1) since ϕ : R0 + → R0 + is non- decreasing the following contradiction: ( )( ) ( )(  d T z ,Tz d T y ,Ty  d T z ,Tz d T y ,Ty D < d ( z , y ) = d T z ,T y ≤ α  −ϕ  d (Tz ,Ty ) d (Tz ,Ty )    ( ) (α + β ) D + (1 − α − β )D = D (3.7) 20 ) + β (d (Tz ,Ty ) − ϕ ( d (Tz ,Ty ) ) ) + D   z = Ty = T z = T y Thus, T : A∪ B → A∪ B and y = Tz = T y = T z are the best proximity points of in A and B Finally, we prove that the best proximity points z ∈ A and y ∈ B are unique Assume that z1 (≠ z ) ∈ A are two distinct best proximity points of T : A∪ B → A∪ B in A Thus, Tz1 (≠ Tz ) ∈ B are two distinct best proximity points in B Otherwise, Tz1 = Tz ⇒ T z1 = T z2 ⇒ z1 = z2 , since z1 and z1 are best proximity points, contradicts ( z1 ≠ z ) ( One gets ) from d Tz1 , T z = d Tz , T z1 = d (z1 , Tz ) = d (z , Tz1 ) = D Lemma 2.1(i) and Through a similar argument to that concluding with (3.6) with the convexity of A and the strict convexity of (X , ), guaranteed by its uniform convexity, one gets the contradiction: ( D = d T z1 , Tz )≤ T z1 − Tz1 z − Tz2 D D + < + =D 2 2 (3.8) since T z1 − Tz1 ≠ Tz1 − z1 Thus, z1 is the unique best proximity point in A while Tz1 is the unique best proximity point in B □ In a similar way, Theorem 2.4 extends via Lemma 2.3 as follows from the modification (2.12) of the contractive condition (2.1): Theorem 3.2 Assume the following hypotheses: (1) T : A∪ B → A∪ B is a modified weak ϕ-contraction, that is, a cyclic self-map satisfying the contractive condition (2.12) subject to the constraints (α , β ) ≥ , (α , β ) > , and α + β ≤ 21 (2) ϕ : R0 + → R0 + ∀x ∈ A ∪ B M0 ≤ (3) is non-decreasing subject to and ϕ (D ) = (1 + α + β − α − β ) D 1−α − β D , ≤ α ≤α0 1−α ϕ (d (Tx , x )) ≥ d (Tx , x ) − 1−α M0 ; 1−α − β for some non-negative real constants and ≤ β ≤ β with α + β < A and B are non-empty closed and convex subsets of a uniformly convex Banach space (X , ) Then, there exist two unique best proximity points z ∈ A , y ∈ B of T : A ∪ B → A ∪ B such that Tz = y , Ty = z to which all the sequences generated by iterations of T : A ∪ B → A ∪ B converge for any x ∈ A ∪ B as follows The sequences {T x} 2n n∈ N and {T n +1 x } n∈N converge to z and y for all x ∈ A , respectively, to y and z for all x ∈ B If A ∩ B ≠ ∅ then z = y ∈ A ∩ B is the unique fixed point of T : A ∪ B → A ∪ B Outline of proof: It is similar to that of Theorem 3.1 since (3.1) to (3.3) still hold, (3.4) and (3.5) still hold as well from Lemma 2.3(ii) as well as the results from the contradictions (3.6)–(3.8) □ The following result may be proven using identical arguments to those used in the proof of Theorem 3.1 by using Lemma 2.8 starting with its proven convergence property (2.23) for distances: 22 Theorem 3.3 Assume that T : A ∪ B → A ∪ B is a cyclic self-map satisfying the contractive condition (2.21) with (α , β ) ≥ , α + β < , and ϕ : R0 + → R0 + is non-decreasing having a finite limit lim ϕ (x ) = ϕ and subject to ϕ (0 ) = Assume also that ϕ : R0 + → R0 + satisfies x→∞ lim sup (x − ϕ ( x )) > x → +∞ ϕ (D ) Finally, assume that A and B are non-empty closed and convex 1− α − β subsets of a uniformly convex Banach space (X , ) Then, there exist two unique best proximity points z ∈ A , y ∈ B of T : A ∪ B → A ∪ B such that Tz = y , Ty = z to which all the sequences generated by iterations of T : A ∪ B → A ∪ B converge for any x ∈ A ∪ B as { follows The sequences T 2n x } n∈ N { and T 2n +1 x } n∈N converge to z and y for all x ∈ A , respectively, to y and z for all x ∈ B If A ∩ B ≠ ∅ then z = y ∈ A ∩ B is the unique fixed point of T : A ∪ B → A ∪ B □ Example 3.4 The first contractive condition (2.1) is equivalent to ( ) d Tx ,T x ≤ ( (( )) β d (x ,Tx ) + ϕ (D ) − α ϕ d Tx ,T x − β ϕ (d (x ,Tx )) 1−α ) (3.9) To fix ideas, we first consider the trivial particular case ϕ (x ) ≡ (⇒ ϕ (D ) = 0) ; ∀x ∈ R 0+ This figures out that T : A ∪ B → A ∪ B is a strict contraction if A ∩ B is non-empty and closed, (α , β ) ≥ , and α + β < Then, it is known from the contraction principle that there is a unique fixed point in A ∩ B Note that in this case ϕ : R0+ → If α + β = then ( ) T : A ∪ B → A ∪ B is non-expansive fulfilling d T p +1 x , T p x = d (x, Tx ) ; ∀x ∈ A ∪ B , ∀p ∈ Z 0+ The convergence to fixed points cannot be proven It is of interest to see if T : A ∪ B → A ∪ B being a weak contraction with ϕ : R0+ → R0+ being non-decreasing guarantees the convergence to a fixed point if α + β = and ϕ (0) = D = according to the 23 modified contractive condition (2.12) In this case, if ϕ (x ) > ; ∀x ∈ R+ then convergence to a fixed point is still potentially achievable since ( ) d Tx ,T x ≤ d (x ,Tx ) − ( (( )) ) α ϕ d Tx ,T x + β ϕ (d (x ,Tx )) < d (x ,Tx ) 1−α if x ≠ Tx (3.10) Now, consider the discrete scalar dynamic difference equation of respective state and control real sequences {xk }k∈Z 0+ and {u k }k∈Z 0+ and dynamics and control parametrical real sequences {ak }k∈Z 0+ and {bk ≠ 0}k∈Z 0+ , respectively: xk +1 = ak xk + bk u k + η k ; ∀k ∈ Z 0+ , x0 ∈ R where {xk }k∈Z 0+ (3.11) , of general term defined by xk := (x0 , x1 , , xk ) , is a sequence of real kth tuples built with state values up till the kth sampled value such that the real sequence {η k }k∈Z0+ with η k = η k ( xk ) is related to non-perfectly modeled effects which can include, for instance, contributions of unmodeled dynamics (if the real order of the difference equation is larger than one), parametrical errors (for instance, the sequences of parameters are not exactly known), and external disturbances It is assumed that upper- { } and lower-bounding real sequences {η k }k∈Z 0+ and η η k = η k (x k ) ≥ η k ≥ η k0 = η k0 ( xk ) ; ∀k ∈ Z 0+ k k∈Z + are known which satisfy Define a 2-cyclic self-mapping T : A ∪ B → A ∪ B with T ( A) ⊆ B and T (B ) ⊆ A for some sets A ⊂ R0+ := {z ∈ R : z ≥ 0} and B ⊂ R0− := {z ∈ R : z ≤ 0} being non-empty bounded connected sets containing {0 }, so that D = , such that T xk = xk +1 ; ∀k ∈ Z 0+ for the control sequence {u k }k∈Z 0+ lying in some appropriate class to be specified later on Note from (3.11) that xk + = ak xk +1 + bk +1 u k +1 + η k +1 24 = ak +1a k xk + a k +1bk u k + bk +1u k +1 + a k +1η k + η k +1 ; ∀k ∈ Z 0+ , x0 ∈ A ∪ B (3.12) An equivalent expression to (3.9) if ϕ (D ) = D = is by using the Euclidean distance: α ϕ ( xk +1 + xk + ) + βϕ ( xk + xk +1 ) ≤ β ( xk + xk +1 ) − (1 − α ) ( xk +1 + xk + ) ; ∀k ∈ Z 0+ (3.13) Consider different cases as follows by assuming with no loss in generality that the parametrical sequences {ak }k∈Z 0+ and {bk }k∈Z 0+ are positive: (a) D = Then xk + = ak +1ak xk + ak +1bk u k + bk +1u k +1 + ak +1η k +η k +1 ; ∀k ∈ Z 0+ , x0 ∈ A ∪ B (3.14) Note that if xk ≥ then xk +1 ≤ and xk + ≥ if uk ≤ − ak xk + η k a (η − ak xk − bk u k ) + η k +1 ≤ ; u k +1 ≥ k +1 k ; ∀k ∈ Z 0+ bk bk +1 (3.15) If xk ≤ then xk +1 ≥ and xk + ≤ if uk ≥ η k − ak xk bk ; u k +1 ≤ − ak +1 (η k + ak xk + bk u k ) + η k +1 bk +1 ; ∀k ∈ Z 0+ (3.16) Thus, if x0 ≥ then the control law is u 2k ≤ − a2 k x2k + η k a (η − a2k x2k − b2k u2k ) + η 2k +1 ≤ ; u k +1 ≥ 2k +1 k b2k b2 k +1 and if x0 < then 25 ; ∀k ∈ Z 0+ (3.17) u2k ≥ η k − a k x2 k b2 k ; u 2k +1 ≤ − a2 k +1 (η 2k + a2 k x2k + b2k u k ) + η k +1 b2 k +1 ; ∀k ∈ Z 0+ (3.18) The stabilization and convergence of the state sequence to zero is achieved by using a control sequence that makes compatible (3.16) and (3.17) with (3.13) First, assume x0 ≥ and rewrite the controls (3.17) in equivalent equality form as: u2k = − a2k x2 k + η 2k + ε k b2 k ; u 2k +1 = a2 k +1 (η k − a2 k x2 k − b2 k u k ) + η 2k +1 + ε 2k +1 b2k +1 ; ∀k ∈ Z 0+ (3.19) for any non-negative real sequence {ε k }k∈Z 0+ to be defined so that (3.13) holds Then (3.11) and (3.14) lead to: 0 − ε 2k = η 2k − η 2k − ε 2k ≤ x2k +1 = a2k x2k + b2k u 2k + η 2k = η 2k − η 2k − ε 2k ≤ −ε 2k ≤ 0 ⇒ ε k ≥ x2 k +1 ≥ ε 2k ; ∀k ∈ Z 0+ (3.20) ε 2k +1 = 2(a 2k +1 η k + η 2k +1 ) + ε k +1 ≥ x2k + = a2k +1a2k x2k + a 2k +1b2k u 2k + b2k +1u 2k +1 + a2k +1η 2k +η 2k +1 = a2 k +1 (η k + η k ) + η k +1 +η k +1 + ε k +1 ≥ ε k +1 ≥ 0 ⇒ ε k +1 ≥ x2k + ≥ ε 2k +1 ; ∀k ∈ Z 0+ (3.21) for the given controls (3.19) Then, (3.13) becomes for x0 ∈ A : α ϕ ( x2k +1 + x2k + ) + βϕ ( x2k + x2k +1 ) + (1 − α ) ( x2k +1 + x2k + ) ≤ β ( x2k + x2k +1 ) ; (3.22) 26 ∀k ∈ Z 0+ which is guaranteed from (3.20) and (3.21), without a need for directly testing the solution of the difference equation, if the sequence {ε k }k∈Z 0+ can be chosen to have zero limit while satisfying: ( ) ( ) ) ) ( ( 0 0 0 α ϕ ε k + ε k +1 + βϕ ε 2k −1 + ε k + (1 − α ) ε k + ε k +1 ≤ β ε 2k −1 + ε 2k ; ∀k ∈ Z 0+ { } for some upper-bounding sequence ε k k∈Z + (3.23) satisfying (3.20) and (3.21) and some given non-decreasing function ϕ : R0+ → R0+ This implies that xk → as k → ∞ , which is the unique fixed point of T : A ∪ B → A ∪ B , by using the proposed control law (3.19) Note the following: (1) Even, although {ε k }k∈Z 0+ converges to zero, it is not required for the contribution of the non-perfectly modeled part of the model to converge to zero It can suffice, for instance, η 2k → η 2k ; η 2k +1 → −(η 2k +1 + 2a2k +1η 2k ) as k → ∞ It is not necessary that {η k }k∈Z0+ be convergent fulfilling ˆ η k → η k → η < ∞ as k → ∞ for some non-negative ˆ ˆ real η = η (xk ) However, there are particular cases in this framework as, for instance, ˆ η k → η k → as k → ∞ or η k → η > ; a k +1 → as k → ∞ (2) The constraints (3.23) imply ϕ (x ) = for x ∈ [0, x0 ] and some x0 ∈ R0+ but not that ϕ : R0+ → R0+ is strictly increasing or that ϕ (x ) = if and only if x = If x0 ∈ B , then x0 < , take u0 ≥ ˆ a x0 + η b0 leading to x ≥ The above stabilization/convergence condition (3.23) still holds with the replacement k → k − for any k ∈ Z + 27 (b) Now, assume that D > , A := {z ∈ R+ : z ≥ D / 2} , and B := {z ∈ R+ : z ≤ − D / 2} are bounded subsets of R and reconsider the above Case b modified so that T : A ∪ B → A ∪ B the sequence {ε k }k∈Z 0+ is subject to ε 2k ≥ D / , ε 2k → D / as k → ∞ and ϕ (D ) = D = dist ( A , B ) Also, the stabilization constraints (3.22) and (3.23) become modified as follows: α ϕ ( x k +1 + x k + ) + βϕ ( x k + x k +1 ) + (1 − α ) ( x k +1 + x k + ; ∀k ∈ Z 0+ ( ) ≤ D + β ( x2 k + x2 k +1 ) (3.24) ) ( ) ( ) ( ) 0 0 0 α ϕ ε k + ε k +1 + βϕ ε 2k −1 + ε k + (1 − α ) ε k + ε k +1 ≤ D + β ε 2k −1 + ε 2k , ∀k ∈ Z 0+ (3.25) the second one being a sufficient condition for the first one to hold Note that x2k and x2 k +1 both converge to best proximity points as k → ∞ If x0 ≥ D / then x2k → D / and x2k +1 → − D / as k → ∞ and if x0 ≤ − D / then x2k → − D / and x2k +1 → D / Note that Case a is a particular version of Case b for D = (c) The conditions (3.23) and (3.25) can be generalized to the nonlinear potentially non- perfectly modeled difference equation: xk +1 = ak g (xk ) + bk u k + η k ; ∀k ∈ Z 0+ , x0 ∈ R n (3.26) for some function g : R → R leading to the nonlinear real sequence {g k = g (xk )}k∈Z 0+ Proceed by replacing the controls (3.19) by their counterparts obtained correspondingly with right-hand side replacements xk → g k = g (xk ) by choosing the sequence {ε k }k∈Z 0+ with ϕ : [D , ∞ ) → [D , ∞ ) satisfying ϕ (x ) = D for x ∈ [D , D + x0 ] and some x0 ∈ R0+ so that (3.25) holds 28 (d) Consider the nth-order nonlinear dynamic system: xk +1 = Ak xk + Bk u k ; ∀k ∈ Z + , x0 ∈ R n (3.27) for some matrix function sequences sampling point-wise defined by Ak = Ak (xk ) and Bk = Bk ( x k ) of images in R n×n and R n×m , respectively; ∀k ∈ Z 0+ Proceeding recursively with (3.27) over n consecutive samples, one gets x(k +1)n = Φ k xkn + Γ k u kn ; ∀k ∈ Z 0+ , x0 ∈ R n (3.28) with Φ k = Φ k (xkn ) and Γ k = Γ k (xkn ) as: +1) Φ k := ∏(ik +1)n −1 [Ai ] ; Γ k :=  B(k +1)n−1 M A(k +1)n−1B(k +1)n− M L M ∏(jk= knn −1 A j Bkn  +1 = kn     [ ] (3.29) with the extended nm control real vector sequence over n consecutive samples being defined by ukn = u kn (xk ) := (u(k +1)n−1 ,u(k +1)n−2 , ,u kn ) T Consider solutions of (3.28) lying alternately in a non-empty closed bounded connected subset A of the first closed orthant of R n and in B = − A for each couple of subsequent samples for some extended control sequence {u k }k∈Z 0+ in R m , for some integer ≤ m ≤ n , assumed to exist A unique such a control sequence exist, if for instance, the controllability condition rank Γ k = n ; ∀k ∈ Z 0+ holds for each matrix sequence Λk = Λk (xk ) by achieving: x(k +1)n = Φ k xkn + Γ k u kn = −Λk xkn ; ∀k ∈ Z 0+ , x0 ∈ A 29 (3.30) with Λk = Λk (xkn ) ; ∀k ∈ Z 0+ defining some prefixed positive real matrix sequence taking values in R n×n with at least a non-zero entry per row The closed-loop control objective (3.30) is achievable by the feedback control sequence: u kn = −Γ T k (Γ k ) Γ kT (Φ k + Λk ) xk n , ∀k ∈ Z 0+ ; x0 ∈ A (3.31) Thus, a modified constraint of the type (3.22), or (3.23), ensures that the solution of (3.28), subject to the extended control (3.31), lies alternately in A and B for each two consecutive samples for x0 ∈ A and converges to zero, while a modification of (3.24), or (3.25), ensures that the solution lies alternately in B and A and converges to zero, provided that Λk x0 ∈ A ∪ ( − A) ; ∀x0 ∈ A ∪ (− A) , ∀k ∈ Z 0+ , i.e., A ∪ ( − A) is Λ k -invariant, ∀k ∈ Z 0+ Furthermore, A and B are both Λ 2k -invariant Such a modifications are got directly by replacing x(.) → Λ(.) x(.) , ε (.) → Λ(.)ε (.) Note that the constraints (3.22), (3.23), (3.24), and (3.25) now become n-vector constraints The Euclidean distances are now replaced by any Minkowski distance of order p (p-norm-induced distance for some real p ≥ ) in R n as for instance, 1-norm-induced distance d1 ( x , y ) = ∑in=1 xi − yi , 2-norm1/ 2 induced (i.e., Euclidean) distance d (x , y ) =  ∑in=1 xi − yi      d p (x , y ) =  ∑in=1 xi − yi  1/ p p    d ∞ ( x , y ) = lim  ∑in=1 xi − yi p →∞  , 1/ p p   or  , p-norm-induced distance infinity-norm-induced = max ( xi − yi ) 1≤ i ≤ n Competing interests The authors declare that they have no competing interests 30 distance Authors’ contributions Both the authors contributed equally and significantly in writing this paper All authors read and approved the final manuscript Acknowledgments The authors are grateful to the Spanish Ministry of Education for its partial support of this study through Grant DPI2009-07197 They are also grateful to the Basque Government for 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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 2cyclic. .. modifications of rational type of the φ -contractive condition of [21, 22] for a class of 2 -cyclic self-mappings on nonempty convex and closed subsets A , B ⊂ X The contractive modification is of rational