Fixed points and best proximity points in contractive cyclic self-maps satisfying constraints in closed integral form with some applications

Applied Mathematics and Computation 219 (2013) 5410–5426

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Fixed points and best proximity points in contractive cyclic self-maps satisfying constraints in closed integral form with some applications M. De la Sen a,⇑, A. Ibeas b a b

Institute of Research and Development of Processes, Campus of Lejona (Bizkaia), P.O. Box 644-Bilbao, 48080 Bilbao, Spain Department of Telecommunications and Systems Engineering, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

a r t i c l e

i n f o

Keywords: Uniform space p-Cyclic self-maps Fixed points Best proximity points

a b s t r a c t This paper investigates the existence of best proximity points and the potential fixed points of p-cyclic self-maps which are defined on the union of a finite set of subsets of a certain uniform metric space under several integral-type contractive conditions. The corresponding properties of the associated restricted composed maps from any of the subsets to itself are also dealt with in a formal context. The above subsets of the uniform metric space are not assumed to necessarily intersect but the case when they intersect leads to particular results of the general case in the sense that best proximity points become fixed points. Common fixed points and best proximity points of pairs of self-maps on the above subsets of the uniform metric space are also investigated. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Fixed point theory is of an intrinsic theoretical interest but it is also a useful tool for studying a wide class of practical problems. In particular, there is an exhaustive variety of results concerning fixed point theory in both Banach spaces and metric spaces which are subject to different types of contractive conditions including some ones associated with the socalled contractive Kannan maps and some ones being associated with Meir–Keeler contractions. The theory is also relevant to derive related results concerning the existence of fixed points in retractions of self-mappings. See, for instance, [10– 15,22,23]. There is also a rich background literature available concerning the use of contractive conditions in integral form using altering distances, Lebesgue-integrable test functions and comparison functions, [16–18]. See also [38–42] for recent use of appropriate formalisms for the study of fixed points under certain integral-type conditions. On the other hand, the socalled reasonable expansive mappings have been investigated in [9] and conditions for the existence of fixed points have been given. It has been used, for instance, for the study of the Lyapunov stability of delay-free dynamic systems and also for that of dynamic systems subject to delays and then described by functional differential equations. See [2,7] concerning a related fixed point background for those systems and [3–5,7] concerning some related background for stability. The fixed point theory is also useful in many other applications as in certain physical problems, like for instance, those related to scattering, [24], some problems involving functions of bounded variation, [25] and some ones related to fractional calculus or to variational equations, [26,28]. On the other hand, a nonlinear generalization of the Gaussian integral transformation on separable real Hilbert spaces defined as a continuous map is proposed in [30] and a family of fixed points is investigated and fixed point theory has been used to compute the real zeros of hypergeometric functions [31]. Also, Jensen-type equations and hemicontractive operators have been investigated by using fixed point theory, [32,33]. Fixed point theory has also been proven to useful for investigating the stability of hybrid systems consisting of coupled continuous-time and discrete-time or ⇑ Corresponding author. E-mail addresses: [email protected] (M. De la Sen), [email protected] (A. Ibeas). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.016

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M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

digital dynamic subsystems, [1], and, in general, to investigate the convergence of many iterative calculation procedures. See, for instance, [34,35] and references therein. Fixed point theory is also of important relevance in stability analysis concerning the presence of nonlinear uncertainties and perturbations of models or in problems of control of industrial factories involving estimation and control of large amounts of state variables, [36,37]. Recent investigation of existence of fixed points of self-mappings based on the study of integral constraints has been performed in some papers. See, for instance, [16,43]. On the other hand, some recent research relating the study of fixed points with the use of numerical and iterative methods is available in the literature. See, for instance, [44–46] and references therein. This paper considers p-cyclic self-maps in a uniform space ðX; UÞ, where X is a nonempty set equipped with a nonempty family U of subsets of X  X satisfying certain properties of uniformity and the subsets of the uniform space are not necessarily assumed to be intersected. The family U is called the uniform structure of X and its elements are called entourages, neighborhoods or surroundings. The uniform space ðX; UÞ is assumed to be endowed with an A-distance or and E-distance. The existence of fixed points and best proximity points in restricted p-cyclic individual and pairs of self-maps S S F; G : jð i2p Ai Þ : X ! Xjð i2p Ai Þ of X, [17], subject to the constraint FðAi Þ # Aiþ1 for each pair of adjacent subsets Ai of X;  8i 2 p :¼ f1; 2; . . . ; pg, p P 2, under the cyclic condition Apþi ¼ Ai ; 8i 2 Zþ , is investigated separately under two groups of integral-type contractive conditions. One of such groups involves a positive integrand test function while the other combines a positive integrand with a comparison function. Some properties of the composed restricted self-maps on each of the subsets are also investigated. If the sets do not intersect then it is proven that g ¼ g i  g iiþ1 :¼ distðAi ; Aiþ1 Þ ¼ dðzi ; Fzi Þ if (X, d) is a  and zi is said to be a best proximity point. metric space endowed with a distance map d : X  X ! R0þ , some zi 2 Ai ; 8i 2 p The formulation is applicable to cyclic self-mappings either on non-intersecting or intersecting sets of uniform metric spaces so being useful to discuss both best proximity and fixed points. Also, some further results about common fixed and best proximity points of two cyclic self-mappings are investigated under the same above formalism. Finally, some application examples are discussed to illustrate the method. The paper is organized as follows. Section 2 is devoted to describe the necessary preliminary formalism and also to obtain some basic preliminary results on A-distances, E-distances and V-closeness then used for obtaining the results. Section 3 studies and characterized the asymptotic properties of distances of nonexpansive, reasonably nonexpansive and contractive p-cyclic self-maps in uniform metric spaces. It also gives and proves the main results on fixed points and best proximity points of such p-cyclic self-maps under a number of different integral-type contractive conditions. Section 4 extends some of the main results of Section 3 to the study of common fixed points and best proximity points of a pair of p-self-maps in uniform metric spaces under the same basic formalism. Section 4 describes applications examples and, finally, conclusions end the paper. 2. Basic results about A-distances, E-distances and V-closeness It is now defined the nonempty family U of subsets of X  X being of the form U :¼ ð  :¼ f1; 2; . . . ; pg. Note by construction that Uij :¼ Ai  Aj ; 8i; j 2 p

2 Þ _ V 2 U ) 4ðV 2 Ai  Aj ; 8i; j 2 p

V2

[ i2px

! Ai

0 @

[

S

 i;j2p

Uij Þ with

1 Aj A

i2py

  : for some nonempty finite subsets of positive integers px ; py # p The following definitions of V-closeness and A and E-distances are then used throughout the manuscript: Definition 2.1 ([16, 18]). If V 2 U and ðx; yÞ 2 V and ðy; xÞ 2 V then x and y are said to be V-close. A sequence fxn g1 0  X is a Cauchy sequence for U if for any V 2 U, there exists N P 1 such that xn and xm are V-close for n; m P N.

Definition 2.2. A function d : X  X ! R0þ is said to be an A-distance if (1) dðx; yÞ ¼ 0 () x ¼ y for 8x; y 2 X; (2) For each V 2 U, 9 d ¼ dðVÞ > 0 such that maxðdðz; xÞ; dðz; yÞÞ 6 d for some z 2 X ) ðx; yÞ 2 V. . Definition 2.2 generalizes slightly that of [16] by admitting d to depend on V since it is being used on distinct sets Ai ; i 2 p Note that V 2 U is symmetrical, that is, V ¼ V 1 ¼ fðy; xÞ : ðx; yÞ 2 Vg then ðx; yÞ 2 V () ðx; yÞ 2 V so that x and y are V-close under Definition 2.2. Definition 2.3 ([16]). A function d : X  X ! R0þ is said to be an E-distance if it is an A-distance and dðx; yÞ 6 dðx; zÞ þ dðz; yÞ;

8x; y; z.

Assertion 2.4. Assume that any Z 2 W  X  X is symmetrical, that is, Z ¼ Z 1 ¼ fðy; xÞ : jðx; yÞ 2 Zg. Then, d : X  X ! R0þ is an A-distance if and only if x and y are Z-close for all Z 2 W provided that maxðdðz; xÞ; dðz; yÞÞ < d for some z 2 X and some d > 0.

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M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

Proof. It follows from the symmetry of all Z 2 W and Definition 2.3 by a simple contradiction argument. Take a pair ðx; yÞ 2 Z from Definition 2.3 since d : X  X ! R0þ is an A-distance fulfilling maxðdðz; xÞ; dðz; yÞÞ < d for some z 2 X and some d > 0. Such a pair always exists for any Z 2 W. Since V is symmetrical then ðy; xÞ 2 Z. Since ðx; yÞ 2 Z if and only if ðy; xÞ 2 Z then x and y are V-close. h Assertions 2.5. S 1. Ui j and U :¼ ð i;j2p Uij Þ are symmetrical. 2. If V 2 U is of the form V X  V X then V is symmetrical. S  with px \ py ¼ ; then V is not symmetrical. If, in addition, Ai \ Aj ¼ ;; 8i; j 2 p  3. If V # i2px ;j2py Uij is nonempty and px ; py  p then there are no x, y in U being V-close.

Proof. S is symmetrical. ðx; yÞ 2 U () x; y 2 iep Ai 1. ½ðx; yÞ 2 Uij ¼ Ai  Aj () ðx 2 Ai y 2 Aj Þ () ðy; xÞ 2 Uji ¼ Aj  Ay  () Uij f ðy; xÞ 2 U () U is symmetrical. Assertion 2.5.1 has been proven. S S S S  ) V is symmetrical. 2. ðx; yÞ 2 V 2 U ) ððx; yÞ 2 ð iepx Ai Þ  ð iepx Ai Þ \ VÞ () ðy; xÞ 2 ð Ai Þ  ð iepx Ai Þ \ V for some px # p iepx Assertion 5.2 is proven. 3. 3. Proceed by contradiction: V symmetrical

2

() 4ðx; yÞ 2 V ) ððx; yÞ 2

[ i2px ;j2py

Uij ¼

[ i2px

Ai 

[ j2py

Aj 

[

Ai 

i2py

[

3

Aj ðy; xÞ ( ðy; xÞ 2 VÞ5 ) px \ py –;

j2px

what contradicts px \ py ¼ ;. h Assertion 2.5 states that some, but not all, nonempty subsets V of ; are symmetrical. For instance, if V ¼ ðA1 [ A2 Þ  ðA3 [ A4 Þ then V is not symmetrical since there are ðx; yÞ 2 V such that ðy; xÞ are not in V; i.e., there are pairs x, y which are not V-close. If furthermore the sets Að:Þ are disjoint then there is no pair in U being V-close (Assertion 2.5.3). Note that under symmetry of V, the second property of an A-distance can be rewritten in an equivalent form by replacing ðx; yÞ 2 V with (x, y) being V-close. The subsequent result states that, contrarily to results in former studies related to A and E-distances, [16,18], the second property guaranteeing an A-distance necessarily involves d  values exceeding distances . between the various subsets Ai ; i 2 p . If ðx; yÞ 2 V Lemma 2.6. Assume that d : X  X ! R0þ is an A-distance and V # Uij 2 U for some i; j 2 p maxðdðz; xÞ; dðz; yÞÞ < dij for some z 2 X and some dij > distðAi ; Aj Þ.

then

Proof. Assume that ðx; yÞ 2 V, so that x 2 Ai and y 2 Aj , and maxðdðz; xÞ; dðz; yÞÞ < distðAi ; Aj Þ for some z 2 X. The following cases can occur: (1) If z ¼ x 2 Ai , and since y 2 Aj , then 0 6 distðAi ; Aj Þ 6 dðz; yÞ ¼ maxðdðx; xÞ; dðz; yÞÞ < distðAi ; Aj Þ which leads to the contradiction distðAi ; Aj Þ < distðAi ; Aj Þ. (2) If zð–xÞ 2 Ai , and since y 2 Aj , then 0 6 distðAi ; Aj Þ 6 maxðdðz; xÞ; distðAi ; Aj ÞÞ 6 maxðdðz; xÞ; dðz; yÞÞ < distðAi ; Aj Þ which leads to the same contradiction as in Case 1. (3) If zð¼ yÞ 2 Aj and if zð–yÞ 2 Aj the above contradiction of Cases 1 and 2 are also obtained by replacing Ai $ Aj . (4) If z 2 Ai [ Aj \ X then 0 6 distðAi ; Aj Þ 6 maxðdðz; xÞ; distðAi ; Aj ÞÞ 6 maxðdðz; xÞ; dðz; yÞÞ < distðAi ; Aj Þ which leads to the same contradiction as that of Case 1. h The following lemma is a direct consequence of Lemma 2.6: Lemma 2.7. Assume that d : X  X ! R0þ is an A-distance and V # maxðdðz; xÞ; dðz; yÞÞ < d for some z 2 X and some d > maxi2px ;j2py distðAi ; Aj Þ.

S

i2px ;j2py Uij

. If ðx; yÞ 2 V for px ; py  p

then

3. Main results about fixed points and best proximity points for a single self-map of X S S  :¼ f1; 2; :::; pg. Note that, although pConsider p-cyclic self-maps F : Xjð i2p Ai Þ ! Xjð i2p Ai Þ subject to FðAi Þ # Aiþ1 ; 8i 2 p cyclic self-maps consider the case of at least two subsets in the metric space (i.e., p P 2), it is direct to obtain the corresponding particular results on fixed points for the case p ¼ 1, that is, for self-maps on one single subset of the metric space. In the following, it is not assumed that the cyclic self-map F is necessarily continuous. Note that although continuity assumptions

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

5413

are very usual in Fixed point theory, there are results available where this assumption is removed. See, for instance, [15,21]. In particular, Brouwer´s fixed point theorem is extended in [21] to discontinuous mappings which allows a wide range of new applications of the fixed point formalism as, for instance, in problems related to Economy. The objective is to first investigate if each of them has a fixed point and then if they have a common fixed point through contraction conditions on Lebesgue integrals and use of comparison functions. Without loss in generality, we discuss the fixed points of self-map F of X. Consider Re a Lebesgue-integrable map u : R0þ ! R0þ which satisfies the integral constraint 0 uðtÞdt > 0, 8e 2 Rþ such that 8x 2 Ai ; 8y 2 Aiþ1 . S S Sp Sp p Define also the composed self-map F p : Xj pi¼1 Ai ! Xj pi¼1 Ai as from the self-map i¼1 Ai 3 y ¼ F x 2 i¼1 Ai Sp Sp p , are defined via the restriction F p jAi :¼ F : Xj i¼1 Ai ! Xj i¼1 Ai whose restrictions to Ai , F : XjAi ! XjAi ; 8i 2 p  by y ¼ F p x 2 Ai for each x 2 Ai ; 8i 2 p . Note that the domain of the self-map F p of X is F pi jApi      F iþ1 jAiþ1  F i jAi ; 8i 2 p Sp p i¼1 Ai while that of F jAi is Ai . The manuscript investigates, under two types of integral-type contractive conditions of the T self-map F of X, the existence of fixed points of such a self-map in pi¼1 Ai , provided that the intersection it is nonempty. S S In that case, the fixed points coincide with those of the self-map F p : Xj pi¼1 Ai ! Xj pi¼1 Ai . It is also investigated the existence T  for the case that pi¼1 Ai ¼ ;. In such a case, the of best proximity points between adjacent and non-adjacent subsets Ai ; 8i 2 p  are also fixed points of the composed self-maps best proximity points at each pair of adjacent subsets Ai ; Aiþ1 ; 8i 2 p  even under weaker contractive integral-type conditions. A key basic result F p : XjAi ! XjAi from each subset Ai to itself; 8i 2 p used in the mathematical proofs is that the distance between any pair of (adjacent or non-adjacent) subsets is identical for nonexpansive contractions, [13,19,20,29]. It is first assumed that the integral-type contractive Condition 1 below holds for an A-distance d : X  X ! R0þ : Condition 1:

Z

dðF npþjþ1 x;F npþjþ1 yÞ

uðtÞdt 6 ð1  ajn Þ

0

Z

dðF npþj x;F npþj yÞ

uðtÞdt þ ajn 0

Z

distðAj ;Ajþ1 Þ

uðtÞdt; 8j 2 p;

ð3:1Þ

0

P1

 where fajn g1 n¼1 are sequences of nonnegative real numbers subject to n¼1 ajn ¼ 1; 8j 2 p, 8n 2 Zþ . The self-map F of X is said to be reasonably nonexpansive [19,20], [29] through this manuscript if

Z

dðF np x;F np yÞ

uðtÞdt 6 K M

0

Z

dðx;yÞ

0

dðF ip x; F ip yÞ 6 K M dðx; yÞ þ K M

uðtÞdt þ K 0M 8x; y 2 X;

8n 2 Z þ

for some nonnegative real constants K M and K M . In particular, F is reasonably nonexpansive if F p is nonexpansive, [17,19,20]. The following result follows from Condition 1: Theorem 3.1. The following properties hold under Condition 1 for any A-distance d : X  X ! R0þ :  satisfying Eq. (3.1) are all nonexpansive and so it is the self-map (i) The restricted self-maps F p jAi ; 8i 2 p S S F : Xj pi¼1 Ai ! Xj pi¼1 Ai . . (ii) 1 > distðAi ; Aiþ1 Þ ¼ g P 0; 8i 2 p R dðF ðnþ1Þp x;F ðnþ1Þp yÞ (iii) lim sup 0 uðtÞdt 6 M < 1; 8ðx; yÞ 2 Ai  Aiþ1 ; 8i 2 p with M ¼ MðgÞ and M : R0þ ! R0þ being monotone n!1

increasing with Mð0Þ ¼ 0. T T , that is i2p Ai –;, then there is a fixed point  x 2 ð i2p Ai Þ \ FixðF p Þ of the self-map F p of X and of (iv) If distðAi ; Aiþ1 Þ ¼ 0; 8i 2 p S T T S its restrictions to i2p Ai and i2p Ai defined through the natural set inclusions i2p Ai # i2p Ai  X. Also,  x 2 FixðFÞ for the Sp Sp self-map F : Xj i¼1 Ai ! Xj i¼1 Ai .

Proof. Consider some A-distance d : X  X ! R0þ . Note that for each

[ V# Ai  i2p

!

[  Ai

! 2 U;

; 9d ¼ dðVÞ > maxdistðAi ; Aiþ1 Þ P 0; 8i 2 p  i2p

 i2p

such that

maxðdðz; xÞ; dðz; yÞÞ 6 d for some z 2

[ Ai  X ) ½ðx; yÞ 2 V ^ ðy; xÞ 2 V () x; y are V  close:  i2p

, then ðx; yÞ 2 V if maxðdðz; xÞ; dðz; yÞÞ 6 d with any d > distðAi ; Aiþ1 Þ P 0 and z 2 Ai [ Aiþ1 ; If, in particular, V # Ai  Aiþ1 ; 8i 2 p , and if V # ðAi [ Aiþ1 Þ  ðAi [ Aiþ1 Þ then ½ðx; yÞ 2 V ^ ðy; xÞ 2 V () x; y are V  close. It is first proven that the self-map 8i 2 p F p of X satisfying Eq. (3.1) is nonexpansive. Proceed by contradiction by assuming that it is expansive. Then, one gets by Qn defining a real sequence fan g1 n¼1 with general term an :¼ 1  j¼1 ½1  ajn  2 ½0; 1:

5414

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426 p Y

½1  ajn 

!Z

Z dðF np x;F np yÞ

p Y p X 

 1  ajn a‘n

Z

Z

uðtÞdt þ min ð1  an Þ 

dðF ðnþ1Þp x;F ðnþ1Þp yÞ

uðtÞdt >

P

distðA‘ ;A‘þ1 Þ

uðtÞdt 0

‘¼1 j¼‘þ1

j2p

0

Z

uðtÞdt þ

0

j¼1

P an

dðF np x;F np yÞ

distðAj ;Ajþ1 Þ

uðtÞdt

0

Z dðF np x;F np yÞ

uðtÞdt

0

0

) min

Z

 j2p

distðAj ;Ajþ1 Þ

uðtÞdt >

0

Z dðF np x;F np yÞ

uðtÞdt ) min distðAj ; Ajþ1 Þ > dðF np x; F np yÞ  dðx1 ; y1 Þ;  j2p

0

Sp

for some x1 ; y1 2 i¼1 Ai which is a contradiction and the self-map F p (and then the self-map F) of X is nonexpansive and prop, Assume 9i; j 2 p  satisfying 1 6 j 6 p  i erty (i) holds. Now, it is proven by contradiction that distðAi ; Aiþ1 Þ ¼ g P 0; 8i 2 p such that distðAi ; Aiþ1 Þ > distðAiþj ; Aiþjþ1 Þ. Then there are best proximity points zi 2 Ai , niþj 2 Aiþj and some zi 2 Ai such that, since p > j and the self-map F of X is nonexpanding, one gets:

    dðzi ; Fzi Þ ¼ distðAi ; Aiþ1 Þ > distðAiþj ; Aiþjþ1 Þ ¼ dðniþj ; Fniþj Þ ¼ d F j zi ; F jþ1 zi P d F pzi ; F pþ1 zi ¼ dð^zi ; F ^zi Þ 6 distðAi ; Aiþ1 Þ; for some ^zi 2 Ai with the last inequality being strict unless ^zi ¼ zi what is a contradiction if ^zi –zi , Now, assume that ^zi ¼ F p zi ¼ zi , then the best proximity point zi ¼ ^zi 2 FixðF p Þ since F p^zi ¼ F 2p zi ¼ F p zi ¼ ^zi ¼ zi and distðAi ; Aiþ1 Þ ¼ 0, i.e.,  and A \ A –;. This is a contradiction to the assumption distðAi ; Aiþ1 Þ > distðAiþj ; Aiþjþ1 Þ. Then, distðAi ; Aiþ1 Þ ¼ g P 0; 8i 2 p Ti p iþ1 S Ai –; if and only if g ¼ 0. Since the self-map F p of X restricted to pi¼1 Ai is nonexpansive then the self-map F of X restricted i¼1 S to pi¼1 Ai is reasonably nonexpansive. It also follows by contradiction that g < 1. Assume that g ¼ 1, Then, the following contradiction follows from Eq. (3.1):

Z dðF ðnþ1Þp x;F ðnþ1Þp yÞ

06

!Z Z p p Y p dðF np x;F np yÞ Y X     1  ajn uðtÞdt þ 1  ajn a‘n

uðtÞdt 6

0

0

j¼1

distðA‘ ;A‘þ1 Þ

uðtÞdt < 0;

0

‘¼1 j¼‘þ1

R np np S , unless 0dðF x;F yÞ uðtÞdt ¼ þ1 (and then dðF np x; F np yÞ ¼ 1; 8x; y 2 pi¼1 Ai ) provided that ajn > 0. Such a for some n 2 Zþ j 2 p P1 ajn always exists since n¼1 ajn ¼ 1; 8j 2 p. Then, 1 > distðAi ; Aiþ1 Þ ¼ g P 0; 8i 2 p and Property (ii) follows. Note that Eq. (3.1) yields directly via recursion:

Z dðF ðnþ1Þp x;F ðnþ1Þp yÞ

p Y 

uðtÞdt 6

0

1  ajn



!Z

dðF np x;F np yÞ

0

j¼1

n1 Y

0

m¼iþ1

j¼1

8n 2 Zþ ; Note that qn :¼

8x 2 A i ;

Qp

j¼1 ½1

q :¼ lim sup m!1

i¼1

p Y p X 

 1  ajn a‘n

p Y 

1  ajm



!!

j¼1

Z

distðA‘ ;A‘þ1 Þ

uðtÞdt

0

‘¼1 j¼‘þ1

!Z p n1 Y n1 dðF p x;F p yÞ Y X   6 1  aj‘ uðtÞdt þ ‘¼1

uðtÞdt þ

p Y p X 



1  aji a‘i

Z

distðA‘ ;A‘þ1 Þ

!

uðtÞdt ;

0

‘¼1 j¼‘þ1

8y 2 Aiþ1 :

ð3:2Þ

 ajn  2 ½0; 1. Define Zþ :¼ Z s [ Z c with Z s :¼ fn 2 Zþ : qn < 1g, Z c :¼ fn 2 Zþ : qn ¼ 1g, and

lY ðZ sm Þ

!1=m

qn

;

n¼1

Z sm :¼ fn 2 Zþ : ½qn < 1 ^ n 6 mg  Z s # Zþ : P Note also that the cardinal (or discrete measure) of Z s is lðZ s Þ ¼ v0 (i.e., infinity numerable), since otherwise, 1 n¼1 ajn < 1 Pm  (contrarily to one of the given hypothesis) and lðZ c Þ 6 v0 . Since n¼1 ajn ¼ 1 and lðZ s Þ ¼ v0 , so that q  2 ð0; 1Þ, for somej 2 p it follows that:

lim

m Y

m!1

qn ¼

Y n2Z s

n¼1

!

qn

Y

!

qn ¼

n2Z c

Y

!

qn ¼ lim q m ¼ 0:

ð3:3Þ

m!1

n2Z s

Then, since the distance between any two adjacent sets Ai ; Aiþ1 is a real constant g, one gets from Eqs. (3.2) and (3.3):

lim sup n!1

Z dðF ðnþ1Þp x;F ðnþ1Þp yÞ

uðtÞdt ¼ lim sup n!1

0

¼

a  1q

n1 X

n1 Y

p Y

i¼1

m¼iþ1

j¼1

Z

0

g



!! ½1  ajm 

p Y p X 

 ½1  aji  a‘i

‘¼1 j¼‘þ1

uðtÞdt < 1; 8x 2 Ai ; 8y 2 Aiþ1 ; 8i 2 p;

Z

g

!

uðtÞdt 6 M :

0

ð3:4Þ

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

 :¼ supj2p;n2Zþ ajn 6 1 since lim where a

n!1

06

n1 X

n1 Y

i¼1

m¼iþ1

p Y ½1  ajm 

!!

j¼1

5415

Qn1 Qp n ‘¼1 ð j¼1 ½1  aj‘ Þ ¼ lim q ¼ 0, and

p Y p X 



½1  aji  a‘i

n!1

Z

!

g

uðtÞdt < 1; 8x 2 Ai ; 8y 2 Aiþ1 :

ð3:5Þ

0

‘¼1 j¼‘þ1

Note that M ¼ MðgÞ is monotone increasing with it argument g and that Mð0Þ ¼ 0. Property (iii) has been proven. If g ¼ 0 S S S then i2p Ai –; so that there is a fixed point  x of F^p1 : Xj i2p Ai ! X which is also a fixed point of its extensions F^p2 : Xj i2p Ai ! X p p S p p T p p T p ^ ^ ^ ^ and F : X ! X since F j i2p Ai ¼ F 2 , F j i2p Ai ¼ F 1 and F 2 j i2p Ai ¼ F 1 . It turns out that  x is also a fixed point of S S F : Xj pi¼1 Ai ! pi¼1 Ai , [6]. Property (iv) has been proven. h Note that the proved boundedness property of the A-distance 1 > distðAi ; Aiþ1 Þ also relies on the fact that this is a distance between best proximity points in adjacent sets. It is well-known that a distance map in a metric space has always an uniform equivalent distance which is finite. The following two concluding results from Eq. (3.4) are direct since Re uðtÞdt > 0; 8e 2 Rþ : 0  that Condition 1 holds and that Corollary 3.2. Assume that g :¼ distðAi ; Aiþ1 Þ > 0; 8i 2 p

9lim

n1 X

n!1

i¼1

n1 Y

p Y

m¼iþ1

j¼1

!! ½1  ajm 

p Y p X 

 ½1  aji  a‘i

Z

!

g

uðtÞdt < 1:

0

‘¼1 j¼‘þ1

g of card S ¼ p such that Then, there is a set S :¼ fxi 2 Ai : i 2 p

 ga dðxi ; xiþ1 Þ ¼ lim dðF x; F yÞ ¼ ;  n!1 1q np

np

8ðx; yÞ 2

[ Ai

!

! [  Ai ;

 i2p

8i 2 p:

 i2p

S  ¼1q  so that S is a set of best proximity points in i2p Ai of the  then the points of the set S satisfy g ¼ dðxi ; xiþ1 Þ; 8i 2 p If a  and satisfies Fxi ¼ xiþ1 and self-maps F and F p of X. Each xi 2 S is a fixed point of FjAi , a best proximity point of F p jAi ; 8i 2 p . xi 2 FixðF p jAi Þ so that F p xi ¼ xi ; 8i 2 p T  then Corollary 3.2 still holds with the set S consisting only of a set of identical Corollary 3.3. If g ¼ 0 so that i2p Ai –;; 8i 2 p T . points  x in i2p Ai such that  x 2 FixðFÞ,  x 2 FixðF p Þ and  x 2 F p jAi ; 8i 2 p Since an E-distance is also an A-distance, the following conclusion is direct from Theorem 3.1 and Corollary 3.1: Corollary 3.4. Theorem 3.1 and Corollary 3.2 also hold if d : X  X ! R0þ is an E-distance. An important relaxation of Condition 1 allows the reformulation of Theorem 3.1 and Corollaries 3.1–3.4 except in the result that  x 2 FixðFÞ when g ¼ 0 as follows: Corollary 3.5. Assume that Condition 1 is reformulated as the p-cyclic contractive Condition 2 below Condition 2:

Z dðF ðnþ1Þp x;F ðnþ1Þp yÞ

uðtÞdt 6 ð1  an Þ

0

Z dðF np x;F np yÞ 0

uðtÞdt þ an min  j2p

Z

distðAj ;Ajþ1 Þ

uðtÞdt;

ð3:6Þ

0

for a new real sequence fan g1 n¼1 under the weaker constraints:

an :¼ 1 

p Y ½1  ajn  2 ½0; 1; j¼1

8n 2 Zþ and

1 X

an ¼ þ1;

ð3:7Þ

n¼1

and that the finite limit of Corollary 3.2 exists. Then, the following properties hold:

(i) Theorem 3.1 and Corollaries 3.1–3.4 still hold, except that in the case that the distance between adjacent sets g is zero  have a nonempty intersection), the property  x 2 FixðFÞ is not guaranteed, since the restricted (i.e., if all subsets Ai , i 2 p . self-maps F : XjAi ! XjAiþ1 can be expansive for some i 2 p g of cardS ¼ p of best proximity points of the self-map F of X such that (ii) If g > 0 then it exists a set S :¼ fxi 2 Ai : i 2 p   xi 2 FixðF p jAi Þ; 8i 2 p and there are Cauchy sequences fxi ðnÞg1 8i 2 p which satisfy n¼1 ; . The points xi ; xiþ1 are ðAi [ Aiþ1 Þ  ðAi [ Aiþ1 Þ – close xi ðn þ 1Þ ¼ F p xi ðnÞ ¼ Fxi1 ðn þ 1Þ and xi ðnÞ ! xi as n ! 1; 8i 2 p a g  via some existing real constant d > 1 for each i 2 p q in Definition 2.2. Also, the pairs of Cauchy sequences 1 1 1 fxi ðnÞgn¼1 , fxiþ1 ðnÞgn¼1 have sub-sequences fxi ðnÞgn¼N , fxiþ1 ðnÞg1 n¼N which are ðAi [ Aiþ1 Þ  ðAi [ Aiþ1 Þ – close via a real a g constant d0 ðeÞ ¼ e þ d > e þ 1  in Definition 2.2 for any given e > 0 and some integer N ¼ Nðd; eÞ. q

5416

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

Outline of Proof. First note that Theorem 3.1 (i)–(iii) is independent of the above modification. Note also that now Q 1  an ¼ nj¼1 ½1  ajn  < 1 on a subset of Zþ of infinite discrete measure so that Eqs. (3.2), (3.3), (3.4), (3.5) still hold except that  x 2 FixðFÞ is not guaranteed when g ¼ 0 [last part of Theorem 3.1 (iv) and Corollary 3.3], since ajn  1 for j belonging to ; 8n 2 Zþ . It still holds that  some proper nonempty subset of p x 2 FixðF p Þ. Property (i) has been proven. Now, note from Cor ollary 3.2 that from Theorem 3.1 there is a set S of p points each being a fixed point of the restricted self-map F p jAi ; 8i 2 p under the pair-wise constraints:

 ga dðxi ; xiþ1 Þ ¼ lim dðF x; F yÞ ¼ ;  n!1 1q np

np

[ Ai

8ðx; yÞ 2

!

! [  Ai ;

 i2p

8i 2 p;

 i2p

which are necessarily in disjoint adjacent sets since the distances between all the sets are a constant g > 0 and FðAi Þ # Aiþ1 ;  ga . Then the A-distance dðx; yÞ of any pair ðx; yÞ 2 Ai  Aiþ1 ; 8i 2 p  converges to a constant distance 1 8i 2 p  . Then, there is a q 1 p . Those seconvergent sequence fxi ðnÞgn¼1 of points in Ai verifying xi ðnÞ ! xi as n ! 1 since xi 2 FixðF jAi Þ for each i 2 p quences are Cauchy sequences since each convergent sequence in a metric space is a Cauchy sequence [8,12,27]. Furthermore, xi ðn þ 1Þ ¼ F p xi ðnÞ ¼ Fxi1 ðn þ 1Þ since xi ðnÞ 2 Ai implies xi 2 A i , xi ðn þ 1Þ ¼ F p xi ðnÞ 2 Ai and  xiþ1 ðnÞ ¼ Fxi ðnÞ 2 Aiþ1 ; 8n 2 Zþ ; 8i 2 p. The remaining parts of Property (ii) concerning closeness according to Definition 2.2 follow the fact that the best proximity points of the self-map F of X are also fixed points of restricted composed maps to a g which Cauchy sequences of points converge and whose distance is 1 q . Property (ii) has been proven. h Since the validity of Theorem 3.1 (iii) is independent of the modification of Condition 1 to the weaker one Condition 2 implying the use of the sequence fan g1 n¼1 (see proof of Corollary 3.5), Condition 2 of Corollary 3.5 may be replaced with that below: Condition 3:

Z dðF ðnþ1Þp x;F ðnþ1Þp yÞ

uðtÞdt 6 ð1  an Þ

0

Z dðF np x;F np yÞ

uðtÞdt þ an min 

Z

j2p

0

g

uðtÞdt:

ð3:8Þ

0

The above discussion may be discussed under any of the following replacements of Conditions 1–3: Condition 4:

Z dðF npþjþ1 x;F npþjþ1 yÞ

uðtÞdt 6 w

0

Z dðF npþj x;F npþj yÞ

uðtÞdt þ

Z

!

g

uðtÞdt ; 8j 2 p:

ð3:9Þ

0

0

Condition 5:

Z dðF npþjþ1 x;F npþjþ1 yÞ

uðtÞdt 6 w1

Z dðF npþj x;F npþj yÞ

0

!

uðtÞdt þ w2

Z



g

uðtÞdt ; 8j 2 p:

ð3:10Þ

0

0

Condition 6:

Z dðF ðnþ1Þp x;F ðnþ1Þp yÞ

Z dðF np x;F np yÞ

uðtÞdt 6 w

0

uðtÞdt þ

Z

!

g

uðtÞdt :

ð3:11Þ

0

0

Condition 7:

Z dðF npþjþ1 x;F npþjþ1 yÞ

uðtÞdt 6 w1

0

Z dðF npþj x;F npþj yÞ

!

uðtÞdt þ w2

Z

g



uðtÞdt :

ð3:12Þ

0

0

where w; w1 ; w2 : R0þ ! R0þ are comparison functions, namely, monotone increasing satisfying lim wn ðtÞ ¼ wn1 ðtÞ ¼ wn2 ðtÞ ¼ 0; 8t 2 R0þ . t!1 Thus, wð0Þ ¼ w1 ð0Þ ¼ w2 ð0Þ ¼ 0 and wðtÞ ¼ w1 ðtÞ ¼ w2 ðtÞ < t; 8t 2 Rþ as a consequence of their above properties to be comparison functions. In addition, w : R0þ ! R0þ satisfies the sub-additive condition wðt1 þ t2 Þ 6 wðt1 Þ þ wðt2 Þ. As a result of the above properties, note that: (a) Conditions 4 and 5 imply:

Z dðF npþjþ1 x;F npþjþ1 yÞ 0

uðtÞdt 6

Z dðF npþj x;F npþj yÞ 0

uðtÞdt þ

Z 0

g

uðtÞdt; 8x; y 2

[ Ai ;

8j 2 p;

ð3:13Þ

 i2p

S  and some x; y 2 i2p Ai if and only if g ¼ dðF npþj x; F npþj yÞ ¼ 0, that is, the distance with the equality standing for some j 2 p between relevant points in the upper-limits of the integral and between all the adjacent sets are zero.

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

(b) Conditions 6 and 7 imply:

Z

dðF ðnþ1Þp x;F ðnþ1Þp yÞ

uðtÞdt 6

0

Z

dðF np x;F np yÞ

uðtÞdt þ

Z

0

g

uðtÞdt; 8x; y 2

0

[ Ai ;

5417

ð3:14Þ

 i2p

S with the equality standing for some x; y 2 Ai if and only if g ¼ dðF np x; F np yÞ ¼ 0.  i2p The following results follow: Corollary 3.6. Theorem 3.1 and Corollaries 3.1–3.4 hold ‘‘mutatis-mutandis’’ under any of the p-cyclic contractive Conditions 6 and 7. Corollary 3.7. Theorem 3.1 and Corollaries 3.1–3.4 hold ‘‘mutatis-mutandis’’ under any of the p-cyclic contractive Condi have a nonempty tions 4 and 5 except that  x 2 FixðFÞ if the distance between adjacent sets g is zero (i.e., all sets Ai , i 2 p intersection). The proofs are direct as that of Theorem 3.1 (see also that of Corollary 3.5) by using the properties Eq. (3.14) for that of Corollary 3.6 and Eq. (3.13) for that of Corollary 3.7. The subsequent additional result proves that the restricted maps F : XjAi ! Aiþ1 and are not asymptotically expansive. Proposition 3.8. If Theorem 3.1 holds under Condition 1 and, in addition, d : X  X ! R0þ is an E-distance then the restricted . maps F : XjAi ! Aiþ1 and F p : XjAi ! Ai are neither expansive not asymptotically expansive; 8i 2 p Proof. Proceed by contradiction. Given any e 2 Rþ and any sufficiently large real constant M 1 > 2M þ e with the constant M being defined in Theorem 3.1 (iii), there exists a sufficiently large N ¼ NðM; eÞ 2 Zþ such that one gets after using the triangle inequality for E-distances:

M1 6

R dðF np x;F np zÞ 0

R d F np x;F np yÞ R d F np y;F np zÞ uðtÞdt 0 ð uðtÞdt þ 0 ð uðtÞdt 6 2M þ e < 1;

 what is a contradiction, and also 8ðx; z; yÞ 2 Ai  A i  Aiþ1 and any i 2 p R dðF np x;F np zÞ R np np R np np M 1 6 lim sup 0 uðtÞdtlim sup 0dðF x;F yÞ uðtÞdt þ lim sup 0dðF y;F zÞ uðtÞdt 6 2M < 1; n!1

n!1

n!1

 what is a contradiction. Then F p : XjAi ! Ai is neither expansive nor asymptotically 8ðx; z; yÞ 2 Ai  A i  Aiþ1 and any i 2 p . By replacing np ! np þ 1, the same contradiction occurs so that F : XjAi ! Aiþ1 are not expansive or expansive; 8i 2 p . h asymptotically expansive; 8i 2 p Note that Proposition 3.8 also hold by replacing Condition 1 by Condition 2 or any of Conditions 3–7. 4. Main combined results of fixed point and best proximity points concerning two self-maps of X S S  :¼ f1; 2; :::; pg. The objective Consider p-cyclic self-maps F; G : Xjð i2p Ai Þ ! Xjð i2p Ai Þ subject to FðAi Þ [ GðAi Þ # Aiþ1 ; 8i 2 p is to first investigate if each of them has a fixed point and then if they have a common fixed point through contraction conditions on Lebesgue integrals and use of comparison functions. The following problems are investigated through this section: . Problem 1 (F and G have a coincidence point): 9x 2 Ai such that Fx ¼ Gx for some i 2 p S  such that Fx ¼ Gy. Problem 2 (F and G take a common value in i2p Ai ): 9ðx; yÞ 2 Ai  Aj for some i; jð6 p  iÞ 2 p T Problem 3 (F and G have a common fixed point): Fx ¼ Gx ¼ x for some x 2 i2p Ai ð–;Þ. Problem 4 (F and G have a common best proximity point which is not a fixed point): 9x 2 FixðF p jAi Þ \ FixðGp jAi Þð–;Þ for T  such that F p x ¼ Gp x ¼ x provided that i2p Ai ¼ ;. some i 2 p Condition 1 can be modified as follows: Condition 8:

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ

uðtÞdt 6 ð1  an Þ

0

Z dðF np x;Gnp yÞ

uðtÞdt þ ajn

Z

distðAi ;Aj Þ

uðtÞdt; 8ðx; yÞ 2 Ai  Aj ; 8i; j 2 p; 8n

0

0

2 Zþ :

ð4:1Þ

: If Ai ¼ Aj then Eq. (4.1) becomes for any x; y 2 A i and some i 2 p

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ 0

uðtÞdt 6 ð1  an Þ

Z dðF np x;Gnp yÞ

uðtÞdt;

ð4:2Þ

0

8n 2 Zþ and some sequence fa n g1 n¼1 subject to an 2 ½0; 1 and one of Conditions 9 and 10 below instead of Condition 8:

P1

n¼1

an ¼ þ1. For the same purposes, one can introduce

5418

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

Condition 9:

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ

Z dðF np x;Gnp yÞ

uðtÞdt 6 w

0

uðtÞdt þ

Z

!

g

uðtÞdt ;

ð4:3Þ

0

0

, 8n 2 Zþ with g ¼ 0 if Ai ¼ Aj where w : R0þ ! R0þ is a comparison function. If x; y 2 Ai for some i 2 p  8ðx; yÞ 2 Ai  Aj ; 8i; j 2 p then Eq. (4.3) becomes:

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ

Z dðF np x;Gnp yÞ

uðtÞdt 6 w

0

!

uðtÞdt ; 8n 2 Zþ :

ð4:4Þ

0

Condition 10:

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ

Z

uðtÞdt 6 w

!

dðF np x;Gnp yÞ

uðtÞdt þ w1

0

0

8ðx; yÞ 2 Ai  Aj ; where w; w

1

8i; j 2 p;

Z

g



uðtÞdt ;

0

8n 2 Zþ with g ¼ 0 if Ai ¼ Aj

ð4:5Þ

 then Eq. (4.5) becomes: : R0þ ! R0þ are comparison functions. If x; y 2 Ai for some i 2 p

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ

Z dðF np x;Gnp yÞ

uðtÞdt 6 w

0

!

uðtÞdt ; 8n 2 Zþ :

ð4:6Þ

0

Following a close proof to that of Theorem 3.1 (iii) under any of Conditions 8–10 under their simpler versions Eqs. (4.2), (4.4) or (4.6), one gets instead of Eq. (3.4) since g ¼ 0:

Z dðF ðnþ1Þp x;Gðnþ1Þp yÞ

lim

n!1

0

; uðtÞdt ¼ 0; 8x; y 2 Ai ; 8i 2 p ) n!1 lim dðF np x; Gnp yÞ ¼ 0; 8x; y 2 A i ; 8i 2 p

ð4:7Þ

. Also, for any given d 2 Rþ , there exists z  Gnp y 2 Ai such that and 9 x 2 Ai such that F p  x ¼ Gp  x ¼ FðF p  xÞ ¼ GðGp  xÞ 2 Aiþ1 ; 8i 2 p nj p nj p nj p dðF x; G yÞ ¼ maxðdðF x; zÞ; 0Þ 6 d and nj 2 SN for some existing SN  Zþ .  Remark 4.1. Conditions 8–10 might be generalized to more general versions via sequences fajn g1 n¼1 ; j 2 p as used in Conditions 1, 3, 4, 5 and 7 of Section 3 instead of fan g1 n¼1 . The subsequent results in this section can be generalized to the above generalization without difficulty although its presentation is more involved and then omitted. The following result follows as an answer to Problem 1 proving that the existence of a coincidence point ensures that of at . least p coincidence points each allocated at one of the Ai ; i 2 p Theorem 4.2. Let d : X  X ! R0þ be an A-distance. If Eq. (4.2) holds (i.e., if Condition 8 holds for any points x, y in the same ) then there is a set of p points CðF; GÞ :¼ f subset Ai for some i 2 p xi 2 Ai :  xiþj ¼ F j  xi ¼ Gj  xi ; 8i; jð6 p  iÞg which are S coincidence points of F and G in i2p Ai . Each of them is also a coincidence point of the restricted self-maps F p jAi and Gp jAi for 1 . Also, for any given x; y 2 Ai and real d 2 Rþ there exist two p-Cauchy sequences fxn ðiÞg1 each i 2 p n¼1 and fyn ðiÞgn¼1 with i ; 8i 2 p  which contain subsequences fxjn ðiÞg and fyj ðiÞg which x1 ðiÞ ¼ x and y1 ðiÞ ¼ y which converge respectively to  xi and y n . are V i -close for any V i  ðAi  Ai Þ \ ðfxi g  fxi gÞ; 8i 2 p The above results remain valid if Condition 8 in the form Eq. (4.2) is replaced with the simplified Conditions 9 or 10 in the forms Eqs. (4.4) or (4.6), respectively. S S Remark 4.3. A coincidence point of F; G : Xjð i2p Ai Þ ! Xjð i2p Ai Þ implies also that they have a common value at the same . Therefore, Theorem 4.2 allows to find some common values of F and G However, F and G can possess common values Ai ; i 2 p S . Some more common values can be found under in i2p Ai which are not coincidence points, i.e., reached at the same Ai ; i 2 p the more general versions of Conditions 8–10, Eqs. (4.1), (4.3) and (4.5), respectively. The following theorem is useful for that purpose. The following result follows as an answer to Problem 2. The proof follows directly from Eq. (3.4) and Corollary 3.2. Theorem 4.4. Let d : X  X ! R0þ be an A-distance. Assume that Eq. (4.1) holds (i.e., if Condition 8 holds for any points S x; y 2 i2p Ai ) and that the following limit exists:

lim

n1 X

n!1

i¼1

n1 Y m¼iþ1

½1  am ai

!Z

g

uðtÞdt < 1:

0

S ^ GÞ :¼ f  2 i2p Ai : F  g CðF; GÞ which are common values of the maps F and G in Then, there is a set of points CðF; x; y x ¼ Gy S  Ai . The above results remain valid if Condition 8 in the form Eq. (4.1) is replaced with the Condition 9 or Condition i2p 10 in the forms Eqs. (4.3) or (4.5), respectively.

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426

5419

Note that the existence of the limit required in Theorem 4.3 is a particular version of the existence of the finite limit of Corollary 3.2, namely,

9lim

n1 X

n!1

i¼1

n1 Y

p Y

m¼iþ1

j¼1

!! ½1  ajm 

p Y p X

ð½1  aji Þa‘i

Z

!

g

uðtÞdt < 1; 0

‘¼1 j¼‘þ1

p since the sequence fan g1 n¼1 has a constant value kept over each cycle of the F self-maps of domains restricted to any subset  (see also Remark 4.1). Necessary conditions for solving Problems 3 and 4 concerning the existence of common fixed Ai , i 2 p points and common best proximity points of the self-maps F and G are easily solved by combining some of the necessary results of Section 3 with the above ones in this section. For instance, it exists a common fixed point of F and G if and only if it exists a fixed point of F (sufficiency-type conditions have been obtained in Section 3) and a coincidence point of F and G (i.e., it is jointly a fixed point of F and a solution to Problem 1) under the further necessary condition that all subsets intersect. Also, it exists a common best proximity point of F (sufficiency – type conditions have been obtained in Section 3) and G if it exists a best proximity of F and a coincidence point of F and G (Problem 2). Those combined conditions are only necessary since the corresponding fixed points/best proximity points are only ensured to exist but not calculated. The following simple result is useful.

T Proposition 4.5. Assume that i2p Ai –;, FixðFÞ – ;, CðF; GÞ – ; and both Conditions 1 (or, alternatively, any of Conditions 2– 7) and 8 (or, alternatively, either Condition 9 or Condition 10) hold. Then x 2 FixðFÞ \ CðF; GÞ ) x 2 FixðGÞ and ^ GÞ ) x 2 FixðGÞ. x 2 FixðFÞ \ CðF; Proof. The first part is direct since

½x 2 FixðFÞ \ CðF; GÞ () x ¼ Fx ¼ Gx ) ½x ¼ Gx () x 2 FixðGÞ: T ^ GÞÞ so that y–x ¼ Fx ¼ Gy, that is, To prove the second part, proceed by contradiction by assuming 9x; yð–xÞ 2 ðð i2p Ai Þ \ CðF; ^ GÞ is not necessarily in FixðGÞ and yð–xÞ 2 CðF; ^ GÞ \ FðGÞ.  x 2 FixðFÞ \ CðF; Take, for instance Condition 8, Eq. (4.2). Then, since d : X  X ! R0þ is an A. distance, one has:

06

Z

dðx;yÞ

uðtÞdt 6 ð1  an Þ

0

Z

dðF p x;Gp xÞ

uðtÞdt ¼ 0 ) ½dðx; yÞ ¼ 0

() x ¼ y;

0

which is a contradiction to x–y. h Related to Proposition 4.5, note that FixðFÞ–; can be tested from some of the results in Section 2 (for instance, from Cor^ GÞ–; can be tested from Theorem 4.4. ollary 3.2), CðF; GÞ–; can be tested from Theorem 4.2 and CðF; 5. Examples This section contains some examples, including numerical simulations, which illustrate the results introduced in the previous sections. Three examples are presented. The first one is concerned with weakly contractive self-maps in intersecting sets giving to a fixed point. The second one considers a contractive self-map in non-intersecting sets which leads to proximity points. Moreover, this example is then extended to the case of two self-maps having common proximity points. Finally, the third example refers to dynamic non-periodic sampling systems. 5.1. Weakly contractive cyclic self-map involving intersecting sets Consider the following three sets:

n o A1 ¼ ðx; 0; 0Þ 2 R3 ; x P 0 ;

n o A2 ¼ ð0; y; 0Þ 2 R3 ; y P 0 ;

whose intersection is non-empty and given by

Fðx; 0; 0Þ ¼

x ð0; x; 0Þ; xþ1

Fð0; y; 0Þ ¼

T

i¼1;2;3 Ai

n o A3 ¼ ð0; 0; zÞ 2 R3 ; z P 0 ;

¼ fð0; 0; 0Þg. The self-map F is defined as:

y ð0; 0; yÞ; yþ1

Fð0; 0; zÞ ¼

z ðz; 0; 0Þ: zþ1

 with p ¼ 3. In addition, F is contractive since for x1 ¼ ðx1 ; 0; 0Þ 2 A1 and F is a cyclic self-map since FðAi Þ # Aiþ1 for all i 2 p x2 ¼ ðx2 ; 0; 0Þ 2 A1 where x2 > x1 without loss of generality we have for the Euclidean distance:

dðx1 ; x2 Þ ¼ x2  x1 P dðFx1 ; Fx2 Þ ¼

1 dðx1 ; x2 Þ; ðx1 þ 1Þðx2 þ 1Þ

i.e., dðFx1 ; Fx2 Þ 6 dðx1 ; x2 Þ. However, given any positive real constant 0 < k < 1 there always exist two values x1 and x2 such 1 that ðx1 þ1Þðx > k, if they are chosen small enough. The latter implies that there is not any constant 0 < k < 1 satisfying 2 þ1Þ

5420

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426 5 4

Z

3 2 1 0 0 2 4 6

1

0

4

3

2

5

Y

X

Fig. 1. Evolution of the iterations to a fixed point at the origin.

8

Z

6

4

2

0 0 2

8 4

6 4

6 8

2 0

Y

X

Fig. 2. Evolution of the iterations to a fixed point at the origin when x0 ¼ ½ 8

0

0 .

6 5

Z

4 3 2 1 0 0

6

2

4 4

2 6

X

0

Y

Fig. 3. Evolution of the iterations to a fixed point at the origin when x0 ¼ ½ 0

6

0 .

dðFx1 ; Fx2 Þ 6 kdðx1 ; x2 Þ for all x1 ¼ ðx1 ; 0; 0Þ 2 A1 and x2 ¼ ðx2 ; 0; 0Þ 2 A1 . Thus, the self-map F only satisfies the more general contraction constraint dðFx1 ; Fx2 Þ 6 dðx1 ; x2 Þ, i.e., F is only weakly contractive. This argument remains true for any initial condition in A2 or A3 , as it can be checked straightforwardly. Despite such a contraction constant k does not exist, we can still

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7 6 5

Z

4 3 2 1 0 0

6 2

4

4

2

6 8

0

Y

X

Fig. 4. Evolution of the iterations to a fixed point at the origin when x0 ¼ ½ 0

0

7 .

apply the results in Theorem 3.1 to conclude the existence of a fixed point. For this, we must first verify the Condition 1 given by Eq. (3.1). Since the sets are intersecting, distðAi ; Aiþ1 Þ ¼ 0 and Condition 1 reads in this case:

Z dðF npþjþ1 x1 ;F npþjþ1 x2 Þ

uðtÞdt 6 ð1  ajn Þ

0

Z dðF npþj x1 ;F npþj x2 Þ

uðtÞdt;

0

which arbitrarily holds for ajn ¼ 0 since dðFx1 ; Fx2 Þ 6 dðx1 ; x2 Þ. Thus, at the light of Theorem 3.1, F has a fixed point in T i¼1;2;3 Ai ¼ fð0; 0; 0Þg. This result is illustrated graphically in Fig. 1. Fig.1 shows the evolution of the sequence of iterates with initial point at x0 ¼ ½ 5 0 0 . The first steps of iterations are marked with arrows to appreciate the evolution. As stated by Theorem 3.1 the iterates converge to the unique fixed point at the origin. Figs. 2–4 show that the fixed point is reached regardless the value of the initial point and the set where it initially belongs. R dðF ðnþ1Þp x;F ðnþ1Þp yÞ Moreover, notice that in this case limn!1 0 uðtÞdt ¼ 0, as Theorem 3.1 (iii) states since for any initial condin tion x0 we have limn!1 F x0 ¼ 0 and thus, limn!1 dðF ðnþ1Þp x; F ðnþ1Þp yÞ ¼ 0. If the self-map F is considered to define a discrete-time dynamic system xkþ1 ¼ Fðxk Þ, the developed fixed-point type methods allow proving that the origin is an asymptotically stable equilibrium point. Thus, the theory is highly applicable to the study of stability of dynamic and control systems. The following example is related to contractive self-maps in non-intersecting sets and proximity points. 5.2. Contractive cyclic self-maps involving non-intersecting sets Consider the following sets in the plane, defined in a complex way:

r 2 ½r0 ; 1Þ;

h p p io h2  ; ; 6 6

15

A4

10

5

Y

n A1 ¼ P ¼ rejh 2 R2 ;

0

A1

A3

d(Ai,A i+1)

-5

A2

-10

-15 -15

-10

-5

0 X

5

10

Fig. 5. Representation of the sets in the plane.

15

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M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 5410–5426 40 30 Proximity points 20

Y

10 0 -10 -20 Proximity points -30 -40 -40

-30

-20

-10

0 X

10

20

30

40

Fig. 6. Existence of proximity points for non-intersecting sets.

n A2 ¼ P ¼ rejh 2 R2 ;

r 2 ½r 0 ; 1Þ;

h p p p pio h2   ;  ; 6 2 6 2

n A3 ¼ P ¼ rejh 2 R2 ;

r 2 ½r 0 ; 1Þ;

h p io p h 2   p;  p ; 6 6

P ¼ rejh 2 R2 ;

r 2 ½r 0 ; 1Þ;

 p 3p p 3p h2   ;  ; 6 2 6 2

A4 ¼

with r 0 > 0. Fig. 5 shows the representation of these sets in the plane for r0 ¼ 5. As it can be appreciated in Fig. 5, the sets are non-intersecting and symmetrically distributed in the plane. Furthermore, the Euclidean distance between them can be calculated as:

  p p  p  dðAi ; Aiþ1 Þ ¼ r 0 ej6  r 0 ejð62Þ  ¼ 0:5176r 0 : The self-map F is defined as the

  F rejh ¼



p=2-radians clockwise rotation of the point along with a movement towards the origin:

1 jðhp2Þ e r : r

The self-map F satisfies the integral-type constraint given by Condition 1, reformulated as Eq. (3.6). Thus, according to Corollary 3.5 there will exist proximity points for F belonging to each set Ai . This situation is depicted in Fig. 6 where the evop lution of the iterates for initial value P ¼ 35ej10 along with the proximity points are shown:

40 30 20

Y

10 0 -10 -20 -30 -40 -40

-30

-20

-10

0 X

10

20

30

40

Fig. 7. Existence of proximity points for non-intersecting sets and different initial condition.

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The values of the proximity points are: p

P1 ¼ 5ej10 2 A1 ;

p p

P2 ¼ 5ejð102Þ 2 A2 ;

p

P3 ¼ 5ejð10pÞ 2 A3 ;

p 3p

P4 ¼ 5ejð10 2 Þ 2 A4 :

Notice that the distance between them is bounded as Corollary 3.5 (ii) states. The value of the proximity points depends on the initial condition. Thus, for a different initial condition a different value for the proximity points is obtained. Fig. 7 shows p the obtained proximity points for initial condition P ¼ 15ej2 : The proximity points are given in this case by: p

P2 ¼ 5ej2 2 A2 ;

P3 ¼ 5ejp 2 A3 ;

p

P4 ¼ 5ej2 2 A4 ;

P 1 ¼ 5 2 A1 ;

and similar properties as before are obtained. 5.2.1. Existence of best proximity common points for two self-maps The Example 5.2 can be generalized to study the existence of common best proximity points for two self-maps. For this, define a self-map as in Section 5.2 but with a drift in the phase to the east boundary of each one of the sets Ai . Thus, the mapping would be defined as:

  F rejh ¼

 1 jðhp2þ0:1dÞ e r ; r

ð5:1Þ

for a point P in the complex plane, P ¼ rejh , where d is the angular distance between the location of the image and the east boundary of the corresponding region. Fig. 8 depicts the evolution of this self-map: 20

best proximity points 15 10

Y

5 0 -5 -10 -15 -20 -15

-10

-5

0

5

10

X

Fig. 8. Evolution and existence of proximity maps for self-map Eq. (5.1).

15

10

Y

5

0

-5

-10

-15 -15

-10

-5

0

5

10

X

Fig. 9. Evolution and existence of proximity maps for self-map Eq. (5.1) under a different initial condition.

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15

10

Y

5

0

-5

-10

-15 -15

-10

-5

0

5

10

15

X

Fig. 10. Combined iteration of both self-maps. The iteration of F is marked with ⁄ while + denotes the iteration of G.

Fig. 8 shows that the self-map given by Eq. (5.1) moves a point towards the origin and towards the right boundary of each region. Thus, this self-map possesses best proximity points located exactly at the corner of each region, as Fig. 8 indicates. Furthermore, these best proximity points are always the same regardless the initial condition of the self-map as the following Fig. 9 for other initial condition shows: Now, we can consider another self-map, G, defined in the same way as Eq. (5.1) except for the fact that in this case the rotation of the point is made counter clockwise:

  F rejh ¼



r

1 jðhþpþ0:1dÞ e 2 ; r

ð5:2Þ

where the argument of Eq. (5.2) now possesses a positive phase p2 instead of the phase of  p2 appearing in Eq. (5.1). Again, d is the angular distance between the location of the image and the east boundary of the corresponding region. In conclusion, we have two self-maps, one moving the image of a point clockwise and another moving it counter clockwise. Fig. 10 shows the combined iteration of both self-maps: Fig. 10 shows that both self-maps possess the same best proximity point regardless the initial condition. They are located at the corner of each region. This fact can be proved from Condition 8 in Section 4 since the potential situation dðF ðnþ1Þpx ; Gðnþ1Þpy Þ > dðF npx ; Gnpy Þ, which would avoid proving the existence of a common proximity point in a classical setting, R distðAi ;Aiþ1 Þ is damped with the presence of the integral term 0 /ðtÞdt in Eq. (4.1) which in this case is positive since the sets are non-intersecting. Therefore, Theorem 4.2 applies and both self-maps possess a common set of best proximity points. Thus, the presented integral-type conditions help proving the existence of such common points. 5.3. Loss functions involving integrals over non-constant integration intervals Note that there is a simple way valid for the interpretation of the various integral-type conditions invoked to prove the existence of the fixed/best proximity points in terms of the ajn parameters. For instance, note that Condition 1 is equivalent to:

  R 0  npþjx ;F npþjy  npþJþ1y F uðtÞdt  d F npþJþ1x ;F d ; ajn R 0 npþj npþj x y ðF ;F Þ  distðAj ; Ajþ1 ÞuðtÞdt d

8j 2 p;

ð5:1Þ

P1  with fajn g1 n¼1 being sequences of nonnegative real numbers subject to n¼1 ajn ¼ 1; 8j 2 p, 8n 2 Zþ . In many situations the . If there exist distances between consecutive sets in a cyclic self-mapping are identical, that is distðAj ; Ajþ1 Þ ¼ D P 0; 8j 2 p . The test function uðtÞ fixed/best proximity points of the map F under any of the given results then ajn ! 1 as n ! 1; 8j 2 p has to be positive but it can be dependent on particular problem at hand. For instance, the integrand of a loss function under the integral symbol of an optimization min–max problem could be used as test function to solve a fixed/best proximity point associated with such a problem. Consider the loss function over a sliding time horizon of variable size for an optimal control problem:

J ðt k  hðt k ÞÞ ¼

Z

tk

t k hðt k Þ

GðxðsÞ; uðxðsÞÞ; sÞds ¼

Z

hðtk Þ

Gðxðs þ tk  hðtk ÞÞ; uðxðs þ t k  hðtk ÞÞÞ; sÞds;

0

P where ft k gk2N is a strictly ordered sequence of positive sampling time instants defined by tk ¼ k1 j¼0 T j where fT j gj2N0 is the sequence of sampling periods (or intervals) in-between each two consecutive sampling instants defined by T k ¼ t kþ1  t k ;

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8k 2 N0 ¼ N [ f0g and hðt k Þ ¼ t k  t k‘k is the variable time-sliding horizon size of the loss function. The integrand function G : X  U  R0þ ! R0þ , with R0þ ¼ fz 2 R : z P 0g, is assumed to be nonnegative and globally convex in the state and control variables, defined by the continuous time-differentiable n-vector function x : Rn jX  R0þ ! Rn , where X is the state space, and u : Rm jU  R0þ ! Rm is the piecewise continuous p-vector control, where U is the control space. In general, f‘j gj2N0 and fT j gj2N0 , may be problem dependent that, is dependent on the states and controls. Consider two disjoint sequences of potential sampling periods

A1 ¼ A ¼

T An ¼

T A 1 þ en ðxðt n Þ; uðt n ÞÞ

and A2 ¼ B ¼ n2N0

T Bn ¼

T B ; 1  qn ðxðt n Þ; uðt n ÞÞ n2N0

ð5:2Þ

being respectively non-decreasing and non-increasing, where tn ¼ tn ðt 0 ; t 1 ; :::; tn1 Þ; n 2 N0 , subject to the following constraints:

(a) T B  T A ¼ DP0 q TB e TA (b) 0 < n1T B n1 6 qn ¼ qn ðxðtn Þ; uðt n ÞÞ < 1, fqn gn2N0 converges to zero as n ! 1 and 0 < en ¼ en ðxðtn Þ; uðt n ÞÞ 6 n1T A n1 so that n n fen gn2N0 is bounded and converges to zero as n ! 1. Note that the above constraints ensure that the sequences fen gn2N0 and fqn gn2N0 converge to zero and the sets of sampling periods fT 2n gn2N0 and fT 2nþ1 gn2N0 converge, respectively, to T A and T B or vice-versa depending on the initial conditions. Now, consider Condition 1, in its equivalent form Eq. (5.1), with

uðsÞ ¼ Gðxðs þ tk  hðtk ÞÞ; uðxðs þ tk  hðtk ÞÞÞ; sÞ; 8s 2 ½0; T k Þ; 8k 2 N0 ; and a 2-cyclic self-mapping F on A [ B with FðT j Þ ¼ T jþ1 2 B if T j 2 A and FðT j Þ ¼ T jþ1 2 A if T j 2 B. Note that such a mapping is not necessarily a strong contraction for any sequences fqn gn2N0 and fen gn2N0 fulfilling the given conditions. For any sequence of sampling instants obtained from the corresponding ones of sampling periods, one gets:

  d F 2n T k ; F 2nþ1 T k ¼ jT kþ2nþ1  T kþ2n j ¼ tkþ2nþ1  t kþ2n ;

8k 2 N 0 :

Then, if the conditions

R tkþ2nþ2 2tkþ2nþ1 tkþ2n

!

R T kþ2nþ1 T kþ2n n X uðtÞdt uðtÞdt ¼ R0T T  þT  > 0; lim an P R tkþ2nþ1 tkþ2n T  þT  aj ¼ 1; kþ2n n!1 B B A A uðtÞdt uðtÞdt j¼0 0 0 0

ð5:3Þ

then the sequence of sampling periods fT 2n gn2N0 and fT 2nþ1 gn2N0 tend to best proximity points T A and T B , or vice-versa, depending on the initial conditions of the iteration, which coincide with a unique fixed point if T A ¼ T B while the sliding horizon size of the loss function converges to a constant value. Note that if Eq. (5.3) hold for a 2-cyclic self-mapping F on A [ B for sets A and B defined in a different way as Eq. (5.2) then the result still holds. Note also that if Eq. (5.3) hold for n P n0 and P limn!1 ð nj¼n0 aj Þ ¼ 1 for some finite n0 2 N0 then the above result still holds. 6. Conclusions This paper has investigated some properties of nonexpansive and contractive p-cyclic self-maps on the union of a finite number of subsets of a uniform metric space under a number of distinct integral-type contractive conditions expressed in closed forms. The existence of fixed points and best proximity points has been studied. The results have been then extended to the study of common best proximity and common fixed points of pairs of cyclic self-maps in uniform metric spaces under a number of closed form integral-type constraints. It is suggested that the main results on fixed points are also applicable to the parallel properties of self-maps on one single subset of the uniform metric space. Examples have been also given in order to illustrate the formal methodology of the main body of the manuscript. Acknowledgments The authors are grateful to the Spanish Ministry of Education for its partial support of this work through Grant DPI201230651. They are also grateful to the Basque Government for its support through Grants IT378-10 and SAIOTEK S-PE09UN12 and to the University of Basque Country by its support through Grant UFI 11/07. References [1] M. De la Sen, Total stability properties based on fixed point theory for a class of hybrid dynamic systems, Fixed Point Theory Appl. 2009, Article ID 826438, 19 pp. http://dx.doi.org/10.1155/2009/826438. [2] M. De la Sen, About robust stability of dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl. 2008, Article ID 480187, 20 pp. http://dx.doi.org/10.1155/2008/480187. [3] M. De la Sen, A. Ibeas, Stability results for switched linear systems with constant discrete delays, Math. Probl. Eng. 2008 (2008), http://dx.doi.org/ 10.1155/2008/543145 (Article ID 543145, 28 pages).

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