COINCIDENCES AND FIXED POINTS OF RECIPROCALLY CONTINUOUS AND COMPATIBLE HYBRID MAPS

It is proved that a pair of reciprocally continuous and nonvacuously compatible single-valued and multivalued maps on a metric space possesses a coincidence. Besides addressing two historical problems in fixed point theory, this result is applied to obtain new general coincidence and fixed point theorems for single-valued and multivalued maps on metric spaces under tight minimal conditions.


Introduction.
The concept of compatible maps has proven useful for generalizing results in the context of metric fixed point theory for continuous single-valued and multivalued maps (cf.[1,2,4,11,15,16,20,21,22,23,30,31]).Recently, reciprocal continuity for a pair of (discontinuous) single-valued maps has been introduced in [23] and promoted as a means to comprehensive results.
First, we introduce reciprocal continuity for a hybrid pair of single-valued and multivalued maps, and emulate the joint merits of reciprocal continuity and compatibility of a hybrid pair in the setting of metric spaces.We give a general principle (Theorem 2.8) stating that nonvacuously compatible and reciprocally continuous hybrid pair on a metric space has a coincidence.This seems to be of vital interest in view of a historically significant and negatively settled problem that a pair of continuous and commuting self-maps on the closed interval [0, 1] has a common fixed point (see [3,13,17]) and that continuous and commuting maps on a complete metric space need not have a coincidence even (see Remark 2.7(v)).This principle presents another view of a significant result of Jungck [17,Theorem 3.6] on a metric space as well.We apply Theorem 2.8 to obtain a coincidence and fixed point theorem for a hybrid quadruple of maps on a metric space satisfying a very general contractive type condition which includes several general conditions studied by Beg and Azam [1], Ćirić [6], Das and Naik [10], Jungck [16], Kaneko [19,20], Rhoades et al. [27], Singh et al. [29], Tan and Minh [33], and others.One of our results presents another view of recent resolutions to a still open fixed point problem of Simon Reich (see [4,5,8,9,25,26]).Our final result on a compact metric space extends and generalizes fixed point theorems from [7,17,32,33].
In this paper, consistent with [22, page 620], (X, d) denotes a metric space, id the identity map on X, CL(X) (resp., CB(X)) the nonempty closed (resp., closed and bounded) subsets of X and H for the Hausdorff (resp., generalized Hausdorff) metric on CB(X) (resp., CL(X)).Further, d(A, B) denotes the ordinary distance between nonempty subsets A and B of X while d(x, B) stands for d(A, B) when A = {x}.The set of natural numbers is denoted by N.

Reciprocal continuity
Definition 2.1.The maps T : X → CL(X) and f : X → X are reciprocally continuous on X (resp., at t ∈ X) if and only if f T x ∈ CL(X) for each x ∈ X (resp., We may use r.c. for "reciprocal continuity" or "reciprocal continuous" as the situation demands.For self-maps f ,g : X → X, this definition due to Pant [23] reads: f and g are r.c.if and only if lim n gf x n = gt and lim n f gx n = f t whenever {x n } ⊂ X is such that lim n gx n = lim n f x n = t ∈ X.Clearly, any continuous pair is reciprocally continuous but, as the following examples show, the converse is not true. ( These maps are discontinuous at x = 1/2.However, they are r.c.(take a decreasing sequence {x n } converging to 0).
Then T and f are r.c. at x = 0 (take x n = 0, n ∈ N).Notice that there is a discontinuity at their common fixed point (x = 0).
For continuity of multivalued maps at their fixed and common fixed points, refer to [12].The following definition is due to Kaneko and Sessa [20] and Beg and Azam [1] when T : X → CB(X).
Definition 2.4.The maps T : X → CL(X) and f : X → X are compatible if and only if f T x ∈ CL(X) for each x ∈ X and lim n H(T f x n ,f T Evidently commuting maps T , f (i.e., when [19,29]), weakly commuting T , f are compatible, and compatible T , f are weakly compatible (i.e., when f T x = T f x whenever f x ∈ T x, [18]) but the reverse implication is not true.For an excellent discussion on the role of weak compatibility in fixed point considerations, refer to Jungck and Rhoades [18].For self-maps f ,g : X → X, Definition 2.4 due to Jungck [15,16] reads: f and g are compatible if and only if lim n d(gf x n ,f gx n ) = 0 wherever {x n } is a sequence in X such that lim n f x n = lim n gx n = t ∈ X.Notice that the maps g, f of Example 2.2 are not compatible (take {x n } as in Example 2.2).So r.c.need not imply compatibility.Taking X as in Example 2.2 and defining f x = 3x 2 /(x 2 + 8) and gx = x, if x < 2, gx = 2 if x ≥ 2, one may conclude that f , g are r.c.but not compatible (take x n = 4, n ∈ N).Hence we assert that: the r.c. and compatibility are independent concepts.
Following Itoh and Takahashi [14], T : X → CL(X) and f : X → X are IT-commuting (commuting in the sense of Itoh-Takahashi [14] Further, T and f are IT-commuting on X if they are IT-commuting at each point v ∈ X.We remark that the IT-commutativity of a hybrid pair (T , f ) at a point v is more general than its compatibility (cf.[31, Example 1]) and weak compatibility at the point v.
Example 2.5.Let X = [0, ∞) be endowed with the usual metric and We see that T and f are compatible for x < 2 but not for x ≥ 2 (e.g., take x n = 2 + n, n ∈ N).Further, T and f are not r.c.(e.g., take Example 2.6.Let X = [2, ∞) with the usual metric and T x = {1 + x} and f x = 2x + 1.We see that there does not exist a sequence {x n } ⊂ X such that {f x n } and {T x n } both converge to the same element in X.Thus requirements of compatibility are vacuously satisfied .
Remark 2.7.(i) If a compatible pair (T , f ) is such that T t = f M, then it is evident from Definitions 2.1 and 2.4 that the continuity of one of T or f is sufficient to ensure the r.c. of the pair (T , f ).
(ii) If T and f are r.c. and nonvacuously compatible, then T t = f M. See the proof of Theorem 2.8.
(iii) The r.c. at a point t ∈ X may be verified by considering all sequences {x n } ⊂ X such that lim n f x n = t ∈ M = lim n T x n .If there does not exist such a sequence then the definition holds vacuously, and the maps are r.c.(see Example 2.6).This observation applies to the compatibility of T and f as well.Hence nonvacuous compatibility of T and f implies the existence of at least a sequence {x n } in X such that {f x n } and {T x n } both converge as per requirements of Definition 2.4.
(iv) If the pair (T , f ) is compatible at a point v ∈ X and v is a coincidence point of T and f , that is, [20, page 260]).Indeed, commutativity, weak commutativity, and compatibility of T and f are equivalent at a coincidence point v of T and f .(v) A pair of continuous and commuting selfmaps of a complete metric space need not have a coincidence; for example, gx = 1 + x and f x = x, x ∈ [0, ∞).(Notice the vacuous compatibility of g and f .) The following is the main result of this section.In all that follows, C(T , f ) stands for the collection of coincidence points of T and f , that is, C(T , f ) = {v : f v ∈ T v}.Theorem 2.8.Let (X, d) be a metric space and T : X → CL(X) and f : X → X.If T and f are reciprocally continuous and nonvacuously compatible on X then C(T , f ) is nonempty.Further, T and f have a common fixed point f t, provided f f t = f t for some t ∈ C(T , f ).
Proof.Since T and f are nonvacuously compatible, there exits a sequence {x n } in X such that {f x n } and {T x n } converge, respectively, to t ∈ X and M ∈ CL(X) such that t ∈ M and lim n H(T f x n ,f T x n ) = 0. This, in view of the r.c. of T and f , yields This completes the proof.
For a better appreciation of Theorem 2.8 and the relative roles of r.c. and compatibility, consider the following result of Mizoguchi and Takahashi [21, Theorem 3].Theorem 2.9.Let K be a closed convex subset of a uniformly convex Banach space.Let T : K → CB(X) and f : K → K be such that T x is convex for each x ∈ K, H(T x, T y) ≤ q x − y , x, y ∈ K, 0 ≤ q < 1, and f x − f p ≤ x − p , x ∈ K, p ∈ F(f ), where F(f ) denotes the set of fixed points of f .If T and f are IT-commuting on K, then T and f have a common fixed point, that is, there exists z ∈ F(f ) with f z ∈ T z.
Further, Theorem 2.8 applies to discontinuous maps and a common fixed point may be a point of discontinuity as well (see Example 2.3).Notice that, besides several stronger conditions on the space, T of Theorem 2.9 is a multivalued contraction and f is nonexpansive about fixed points.

Results on metric spaces and Reich's problem.
First, we give a very general coincidence and fixed point theorem under very tight conditions.Let ψ denote the family of maps φ from the set R + of nonnegative reals to itself such that φ(t) < t for all t > 0. Theorem 3.1.Let (X, d) be a metric space and S, T : X → CL(X) and f ,g : X → X such that (1) S(X) ⊂ g(X) and the pair (S, f ) is reciprocally continuous and nonvacuously compatible.

M(x, y) = max d(f x,gy),d(f x,Sy),d(gy,T y),d(f x,T y),d(gy,Sx) , (3.1)
then C(S, f ) and C(T , g) are nonempty.Further, (Ia) S and f have a common fixed point f t, provided f f t = f t for some t ∈ C(S, t); (Ib) T and g have a common fixed point gu, provided ggu = gu and T , g are IT-commuting at u ∈ C(T , g); (Ic) S, T , f , and g have a common fixed point, provided (Ia) and (Ib) both are true.
Proof.By Theorem 2.8, (1) implies that C(S, f ) is nonempty, that is, f t ∈ St for some t ∈ X.Since S(X) ⊂ g(X), there is a point u ∈ X such that f t = gu ∈ St.So by (2), So gu ∈ T u and C(T , g) is nonempty.
(Ia) and (Ib) may be shown following the last part of the proof of Theorem 2.8.Now (Ic) is immediate.
Reich [25,26] posed the following question: let (X, d) be a complete metric space and T : X → CB(X) such that H(T x, T y) ≤ k(d(x, y))d(x, y) for all distinct x, y ∈ X, where k : (0 : ∞) → (0, 1) with lim r →t + sup k(r ) < 1 for each t > 0.Then, does T have a fixed point?Mizoguchi and Takahashi [21] have shown that T has a fixed point when lim r →t + k(r ) < 1 for each t ≥ 0. Chang [4] has generalized this result, and Theorem 3.1 presents an extension of Chang's main result [4, Theorem 1].However, Reich's problem remains open and needs further resolution.
The following example shows that the nonvacuous compatibility of one of the pairs (S, f ) or (T , g) is essential even if f = g = id.
If there exists q ∈ (0, 1) such that Proof.Since S(X) ⊂ g(X) and T (X) ⊂ f (X), we construct the sequences {x n }, {y n } ⊂ X as in [11,31] such that, for each n ∈ N, Then as in [11], (ii) The power of Corollary 3.3 is appreciated by observing that the main result of [11,Theorem 2] is obtained under condition (4) when all the maps S, T , f , g are continuous and the pairs (S, f ) and (T , g) are compatible.Moreover, Corollary 3.3 generalizes several other interesting results from [1,16,19,20] and the references therein.
(iii) If f = id under the condition (4) of Corollary 3.3, then the pair (S, f ) is automatically compatible and r.c.Thus if f = g = id then Corollary 3.3 states that S, T : X → CL(X), satisfying condition (4), have a common fixed point in complete X.
Its proof may also be completed using Corollary 3.6 and following the proof of Theorem 4.1.Jungck [16, Theorem 3.2] is Corollary 3.6(IIc) when the pair (T , g) is also compatible on X.

Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities.This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics.It is possible to make a division in two main categories: astronomy and astrodynamics.By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth.Many important topics of research nowadays are related to those subjects.By astrodynamics, we mean topics related to spaceflight dynamics.
It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects.Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts.Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines.The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research.All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
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First
Round of ReviewsOctober 1, 2009 (2)n−2 ,y 2n−1 , d y 2n ,y 2n+1 ≤ q −1/2 d y 2n−1 ,y 2n .Thus, for the sequence {x n } in X, we have {Sx 2n } and {gx 2n−1 } converging, respectively, to M and t ∈ M. Therefore the compatibility of the pair (S, g) is nonvacuous.Hence the proof is immediate from Theorem 3.1 by observing that (4) implies(2).It is evident from the proof of Theorem 3.1 that f t ∈ St and f t = gu ∈ T u, that is, (S, t) and (T , g) may have different coincidence points with f t = gu.See Example 3.5 in support of this observation, which applies to Corollary 3.3 as well.