Coincidence and common fixed point theorems for Suzuki type hybrid contractions and applications

Coincidence and common fixed point theorems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multivalued maps in a metric space are obtained. In addition, the existence of a common solution for a certain class of functional equations arising in a dynamic programming is also discussed. MSC:47H10, 54H25.


Introduction
Consistent with [] (see also []), Y denotes an arbitrary nonempty set, (X, d) a metric space and CL(X) (resp. CB(X)), the collection of all nonempty closed (resp. closed bounded) subsets of X. The hyperspace (CL(X), H) (resp. (CB(X), H)) is called the generalized Hausdorff (resp. the Hausdorff ) metric space induced by the metric d on X. For the sake of brevity, we choose the following notations, wherein S, T, f , and g are maps to be defined specifically in a particular context, while x and y are elements of some specific domain: Let CB(X) denote the class of all nonempty closed bounded subsets of X. A map T : X → CB(X) is called a Nadler multivalued contraction if there exists k ∈ [, ) such that, for every x, y ∈ X, H(Tx, Ty) ≤ kd(x, y). http://www.fixedpointtheoryandapplications.com/content/2014/1/147 The classical multivalued contraction theorem due to Nadler [] states that Nadler multivalued contraction on a complete metric space X has a fixed point in X, that is, there exists z ∈ X such that z ∈ Tz. For a detailed discussion of this theorem on generalized Hausdorff metric spaces and applications, one may refer to [-], and [].
The following important result involving two pairs of hybrid maps on an arbitrary nonempty set with values in a metric space is due to Singh and Mishra [] (see also []). and T commute at w; (v) S, T, f , and g have a common fixed point provided that (iii) and (iv) both are true.
Theorem . Let X be a complete metric space and T : X → CB(X). Assume there exists r ∈ [, ) such that, for every x, y ∈ X, and others) for maps in metric spaces satisfying Suzuki type contractive conditions is that the contractive conditions are required to be satisfied not for all points of the domain. For example, the condition (.) of Theorem . puts some restrictions on the domain of the map T.
In all that follows, we take a nonincreasing function ϕ from [, ) onto (, ] defined by is a complete subspace of X, then there exists a point z ∈ Y such that fz ∈ Sz ∩ Tz. Further, if Y = X, and fz is a fixed point of f , then fz is common fixed point of S and T provided that f is IT (Itoh-Takahashi)-commuting [] with S and T at z. Now a natural question arises whether Theorem . can further be generalized. In this paper, we answer this question affirmatively under tight minimal conditions. Our main result (Theorem .) also presents an extension of Theorem . for a quadruplet of maps. Some recent results are discussed as special cases. Further, using two corollaries of the main result (Theorem .), we obtain other common fixed point theorems for multivalued and single-valued maps on metric spaces. We also deduce the existence of common solution for a certain class of functional equations arising in dynamic programming. Examples are given to justify applications.

Main results
The following definition is due to Itoh and Takahashi [] (see also []).
Definition . Let T : X → CL(X) and f : X → X. Then the hybrid pair (T, f ) is ITcommuting at z ∈ X if fTz ⊆ Tfz.
Evidently a pair of commuting multivalued map T : X → CL(X) and a single-valued map f : X → X are IT-commuting but the reverse implication is not true [, p.]. However, a pair of single-valued maps f , g : X → X are IT-commuting (also called weakly compatible by Jungck and Rhoades []) at x ∈ X if fgx = gfx when fx = gx. http://www.fixedpointtheoryandapplications.com/content/2014/1/147 We shall need the following lemma, essentially due to Nadler [  Proof Without loss of generality, we may take r >  and f , g non-constant maps.
Let ε >  be such that β = r + ε < . We construct two sequences {x n } and {y n } in Y as follows.
Let x  ∈ Y and y  = gx  ∈ Sx  . By Lemma ., there exists y  = fx  ∈ Tx  such that Similarly, there exists y  = gx  ∈ Sx  such that Continuing in this manner, we find a sequence {y n } in Y such that yielding (.). So, in both cases we obtain (.). In an analogous manner, we show that We conclude from (.) and (.) that, for any n ∈ N , d(y n+ , y n ) ≤ βd(y n , y n- ).
Therefore the sequence {y n } is Cauchy. Assume that the subspace g(Y ) is complete. Notice that the sequence {y n } is contained in g(Y ) and has a limit in g(Y ). Call it u. Let z ∈ f - u.
Then z ∈ Y and fz = u. The subsequence {y n+ } also converges to u. Let z  ∈ g - u. Then  Since ga ∈ Sz implies ga = fz, we have from (.) and (.), Therefore, by the given assumption, This contradicts fz / ∈ Sz. Consequently fz ∈ Sz, and C(S, f ) is nonempty.
In an analogous manner, we can prove in the case  ≤ r <   that C(T, g) is nonempty. This gives fz ∈ Sz, that is, z is a coincidence point of f and S. Analogously, fz ∈ Tz. Thus (I) and (II) are completely proved.
Further, if Y = X, and fz is a fixed point of f , and S and f are IT-commuting at z, then fSz ⊆ Sfz. Therefore, fz ∈ Sz implies ffz ∈ fSz ⊆ Sfz, so fz ∈ Sfz. This proves that u = fz is a common fixed point of f and S. Therefore (.) implies that u is a common fixed point of f and S. This proves (III). Analogously, T and g have a common fixed point gz  . Therefore (.) implies that u is a common fixed point of T and g. This proves (IV). Now (V) is immediate.
Remark . In Theorem ., the hypothesis 'fz is a fixed point of f ' is essential for the existence of a common fixed point of S and f (see [, ] and the following example). Similarly, the hypothesis 'gz  is a fixed point of g' is essential for the existence of a common fixed point of T and g.
, and all other hypotheses of Theorem . with Y = X = R + are satisfied for r =   = ϕ(r). Notice that gz  = Tz  =   , where z  =  -/ . Thus g and T have a coincidence at z  , but gz  =   is not a fixed point of g and hence not a common fixed point of g and T. Note that z =   is a coincidence point of f and S, and We remark that in general the coincidence points z and z  guaranteed by Theorem . or Corollary . may be different. However, if we take f = g in Theorem ., the maps S, T, and f have a common coincidence point. So we have a slightly sharp result. Then there exists an element z ∈ X such that z ∈ Sz.
Proof It follows from Theorem . when Y = X, T = S, and f , g are the identity maps on X.
The following result extends a common fixed point theorem of [, Theorem .]. Proof Set Sx = {Px} and Tx = {Qx} for every x ∈ Y . Then it easily comes from Theorem . that C(P, f ) and C(Q, g) are nonempty. Furthermore, if Y = X and f and g commute, respectively, with P and Q at z, then ffz = fPz = Pfz, ffz = fQz = Qfz, ggz = gPz = Pgz, and ggz = gQz = Qgz.
Also This says that fz is fixed point of f and P. Analogously gz is fixed point of g and Q. The uniqueness of the common fixed point follows easily.
The following result extends and generalizes coincidence and common fixed point theorems of Goebel   Then C(P, f ) and C(P, g) are nonempty. Further, if Y = X and if P commutes with f and g at a common coincidence point, then f , g, and P have a unique common fixed point, that is, there exists a unique point z ∈ X such that fz = gz = Pz = z.
Proof It follows from Corollary . when Q = P.
Corollary . Let (X, d) be a complete metric space and f , g : X → X be onto maps. Assume there exists r ∈ [, ) such that, for every x, y ∈ X,

Then f and g have a unique common fixed point.
Proof It follows from Corollary . when Y = X and P, Q both are the identity maps on X.
Corollary . Let (X, d) be a complete metric space and f : X → X be an onto map. Assume there exists r ∈ [, ) such that, for every x, y ∈ X,
The following example shows that Theorem . is indeed more general than Theorem ..

Example . Consider a metric space
Let S, T, f and g : X → X be such that Here we note that the value of r is /, so by definition, ϕ(r) = /, so ϕ(r) min{d(fx, Sx), d(gy, Ty)} =  >  = d(fx, gy).
Thus S, T, f , and g satisfy the assumption of Theorem . (and also Corollary .).
In the following example, we show that two multivalued maps and two single-valued maps satisfy all the hypotheses of Theorem . to ensure common coincidence points of pairwise maps.
Now we give an application of Corollary .. and this implies (.). Therefore ≤ r · r -λ max r λ d(fx, gy), r λ ρ(fx, Sx), r λ ρ(gy, Ty), where r = r -λ < . Hence by Corollary ., there exist z  , z  ∈ Y such that h  z  = fz  and h  z  = gz  . This implies that z  is a coincidence point of f and S, and z  is a coincidence point of g and T. Then there exists z ∈ Y such that fz ∈ Sz ∩ Tz.
Proof It follows from Theorem . when g = f .

Corollary . []
Let X be a complete metric space and let S, T : X → BN(X). Assume there exists r ∈ [, ) such that, for every x, y ∈ X, Then there exists a unique point z ∈ X such that z ∈ Sz ∩ Tz.
Proof It follows from Theorem . when f and g are the identity maps on X. Then there exists z ∈ Y such that fz ∈ Sz.
Proof It follows from Theorem . when g = f and T = S.
Corollary . Let X be a complete metric space and let S : X → BN(X). Assume there exists r ∈ [, ) such that, for every x, y ∈ X, Then there exists a unique point z ∈ X such that z ∈ Sz.
Proof It follows from Theorem . that S has a fixed point when f = g is the identity map on X and T = S. The uniqueness of the fixed point follows easily. http://www.fixedpointtheoryandapplications.com/content/2014/1/147

Applications
Throughout this section, we assume that U and V are Banach spaces, W ⊆ U and D ⊆ V . Let R denote the field of reals, τ : W × D → W , g, g : W × D → R and G  , G  , F  , F  : Considering W and D as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations: (DP-) G  , G  , F  , F  , g, and g are bounded.
(DP-) Let ϕ(r) be defined as in the previous sections. Assume that there exists r ∈ [, ) such that, for every (x, y) ∈ W × D, h, k ∈ B(W ), and t ∈ W , and A  , A  , J  , and J  are defined as follows: (DP-) For any h, k ∈ B(W ), there exist u, v ∈ B(W ) such that and . Pick x ∈ W , and choose y  , y  ∈ D such that Therefore, the first inequality in (DP-) becomes and this together with (.), (.), and (.) implies Similarly, (.), (.), and (.) imply So, from (.) and (.), we obtain (  .  ) http://www.fixedpointtheoryandapplications.com/content/2014/1/147 As λ >  is arbitrary and (.) is true for any x ∈ W , taking supremum, we find from (.) and (.) that Therefore, Corollary . applies, wherein A  , A  , J  , and J  correspond, respectively, to the maps P, Q, f , and g. So For any h, k ∈ B(W ), and i = , , define p i , q i : W → R by x (x + )(y + ) sin y y +  +  ; Notice that G  , G  , F  , F  , g, and g are bounded. Also We see that Thus and this implies Thus all the assumptions of Theorem . are satisfied. So the system of equations (.a) and (.b) has a unique solution in B(W ).

Corollary . Suppose that the following conditions hold:
(i) G, F  , F  , g, and g are bounded.
(ii) Let ϕ(r) be defined as in the previous sections. Assume that there exists r ∈ [, ) such that, for every (x, y) ∈ W × D, h, k ∈ B(W ), and t ∈ W , and A, J  , and J  are defined as follows: Ah(x) = J  u(x) and Ak(x) = J  v(x), x ∈ W .
(iv) There exist h, k ∈ B(W ) such that and  Proof It follows from Theorem . when F  = F  = F.
As an immediate consequence of Theorem . and Corollary ., we obtain the following.

Corollary . []
Suppose that the following conditions hold: (i) G  , G  , and g are bounded.
(ii) There exists r ∈ [,) such that, for every (x, y) ∈ W × D, h, k ∈ B(W ), and t ∈ W , where A  and A  are defined as follows: Corollary . Suppose that the following conditions hold: (i) G and g are bounded.
Then the functional equation (.a) with G  = G  = G possesses a unique bounded solution in W .
Proof It follows from Corollary . when G  = G  = G.