# Coincidences and fixed points of hybrid contractions.

1. Introduction

Hybrid fixed point theory is a recent development is the ambit of fixed point theorems for contracting single-valued and multivalued maps in metric spaces. Indeed, the study of such maps was initiated during 1980-83 by Bhaskaran and Subrahmanyam (2), Hadzic (10), Kaneko (14), Kulshrestha (18), Kubiak (19), Naimpally et al. (25) and Singh and Kulshrestha (35). For a history of the fundamental work on this line, refer to Singh and Mishra (37), and for more recent work on this line Beg and Azam (1), Jungck and Rhoades (12), Kamran (13), Kaneko (15), Kaneko and Sessa (16), Liu, Wu, and Li (20), Mishra, Singh and Talwar (22), Naidu (24), Pathak et al. (26), Popa (27), Rhoades et al. (28), Shahzad (30), and Singh et al. (31), (33), (34), (36-40). Hybrid fixed point theory has potential applications in functional inclusions, optimization theory, fractal graphics and discrete dynamics for set-valued operators.

The following fundamental coincidence theorem for a pair of multivalued and single-valued maps is essentially due to Singh and Kulshrestha (35) (see also (18) and (37)).

Theorem 1.1 ((35)). Let X be a metric space and (CL(X), H) the Hausdorff metric space induced by d, where CL(X) is the collection of all nonempty closed subsets of X. Let P : X [right arrow] CL(X) and f : X [right arrow] X be such that P(X) [??] f(X) and

H(Px, Py) [less than or equal to] q.max{d(fx, fy), d(fx, Px), d(fy, Py), [d(fx, Py) + d(fy, Px)]/2} (SK) for all x, y [member of] X, where 0 [less than or equal to] q < 1. If f(X) [or P(X)] is a complete subspace of X, then P and f have a coincidence, i.e., there exists a point z [member of] X such that fz [member of] Pz.

We remark that under the conditions of Theorem 1.1, f and P need not have a common fixed point even if f and P are commuting (cf. Def. 2.3) and continuous as the following example shows (see also (25), (33), (37-40)]).

Example 1.1 ((25)). Let X = [0, [infinity]) be endowed with the usual metric, Px = [1 + x, [infinity]) and fx = 2x. Then P(X) [??] f(X) = X. Further H(Px, Py) [less than or equal to] qd(fx, fy), x, y [member of] X, 1/2 [less than or equal to] q < 1. (NSW)

Thus P and f satisfy all the requirements of Theorems 1.1, since (NSW) implies (SK).

Evidently, P and f have a coincidence point z ([greater than or equal to] 1), i.e., fz [member of] Pz for any z [greater than or equal to] 1. Notice that P and f have no common fixed points. Moreover, P is not a multivalued contraction in the sense of Nadler, Jr. (23), since H(Px, Py) = d(x, y), x, y [member of] X. (Recall that Nadler's multivalued contraction is (NSW) with f = the identity map on X, wherein 0 [less than or equal to] q < 1.)

Theorem 1.1 has been generalized and extended on various settings (see, for instance, (1), (20), (24), (27), (28), (34), (36-40)]). In this paper, we obtain a few generalizations and extensions of Theorem 1.1 and other similar results (cf. (15) and (31)). Using these coincidence theorems, we obtain a few fixed point theorems, wherein continuity of maps is not needed, completeness of the space is relaxed to the completeness of a subspace, and the commutativity requirement is tight and minimal.

2. Preliminaries

Consistent with (7) and (32), we use the following notations and definitions.

Definition 2.1 ((7)). Let X be (nonempty) a set and s [greater than or equal to] 1 a given real number. A function

d : X x X [right arrow] [R.sup.+] (nonnegative real numbers) is called a b-metric provided that, for all x, y, z [member of] X,

d(x, y) = 0 iff x = y, (bm-1)

d(x, y) = d(y, x), (bm-2)

d(x, z) [less than or equal to] s[d(x, y) + d(y, z)]. (bm-3)

The pair (X, d) is called a b-metric space.

We remark that a metric space is evidently a b-metric space. However, Czerwik (6), (7) has shown that a b-metric on X need not be a metric on X (see also (8), (9), (32)).

Definition 2.2 ((7)). Let (X, d) be a b-metric space. The Hausdorff b-metric H on CL(X), the collection of all nonempty closed subsets of (X, d) is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In all that follows Y is an arbitrary nonempty set and (X, d) a b-metric space unless otherwise specified. For the following definition in a metric space, one may refer to Itoh and Takahashi (11) and Singh and Mishra (39).

Definition 2.3. Let Y be a nonempty set, f : Y [right arrow] Y and P : Y [right arrow] [2.sup.Y], the collection of all nonempty subsets of Y. Then the hybrid pair (P, f) is (IT)-commuting at x [member of] Y if fPx [??] Pfx for each x [member of] Y.

We cite the following lemmas from Czerwik (7-9) and Singh et al. (31), (32).

Lemma 2.1. For any A, B, C [member of] CL(X),

(i) d(x, B) [less than or equal to] d(x, y) for any y [member of] B,

(ii) d(A, B) [less than or equal to] H(A, B),

(iii) d(x, B) [less than or equal to] H(A, B), x [member of] A

(iv) H(A, C) [less than or equal to] s[H(A, B) + H(B, C)],

(v) d(x, A) [less than or equal to] sd(x, y) + sd(y, A), x, y [member of] X.

Lemma 2.2. Let A and B [member of] CL(X). Then for any x [member of] A and for some 0 < q, k < 1, there exists a y [member of] B such that

[d.sup.2](x, y) [less than or equal to] [q.sup.-k] [H.sup.2](A, B).

For an excellent collection of such results in metric spaces, one may refer to Rus (29).

Lemma 2.2 in a metric space is essentially due to Nadler, Jr. (23) (see also (3) and (5)).

3. Coincidence Theorems

We begin with the following result.

Lemma 3.1. Let (X, d) be a b-metric space and {[y.sub.n]} a sequence in X such that

d([y.sub.n+1], [y.sub.n+2]) [less than or equal to] qd([y.sub.n], [y.sub.n+1]), n = 0, 1, ...,

where 0 [less than or equal to] q < 1. Then the sequence {[y.sub.n]} is Cauchy sequence in X provided that sq < 1.

Proof. For any n,

d([y.sub.n+1], [y.sub.n+2]) [less than or equal to] qd([y.sub.n], [y.sub.n+1]) [less than or equal to] [q.sup.2]d([y.sub.n-1], [y.sub.n]) [less than or equal to] ... [less than or equal to] [q.sup.n+1]d([y.sub.0], [y.sub.1]).

For n < m, by the triangle inequality (cf. Def. 2.1 (bm-3)),

d([y.sub.n], [y.sub.m]) [less than or equal to] sd([y.sub.n], [y.sub.n+1]) + [s.sup.2]d([y.sub.n+1], [y.sub.n+2]) + ... + [s.sup.[m-n-2]][d([y.sub.m-2], [y.sub.m-1]) + d([y.sub.m-1], [y.sub.m])] < s[q.sup.n](1 + sq + [s.sup.2][q.sup.2] + ...)d([y.sub.0], [y.sub.1]) = [s[q.sup.n]/(1 - sq)]d([y.sub.0], [y.sub.1]) [right arrow] 0 as n [right arrow] [infinity],

and {[y.sub.n]} is Cauchy.

Following Liu et al. (21), Singh et al. (33), (35) and Tan et al. (41), we consider the following conditions for f : Y [right arrow] X and P, Q : Y [right arrow] CL(X):

H(Px, Qy) [less than or equal to] q.max {d(fx, fy), d(fx, Px), d(fy, Qy), [d(fx, Qy) + d(fy, Px)]/2}, x, y [member of] X (1)

where q [member of] (0, 1); and

[H.sup.2](Px, Py) [less than or equal to] q.max m(x, y), x, y [member of] X (2)

where q [member of] (0, 1) and

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We remark that (1) with P = Q and Y = X, a metric space is (SK), while the main condition studied in (31) is based on the work of (21) and (41), and is a particular case of (2).

Assume that [beta]: = s[q.sup.1-k][1 + [square root of ((1 + 8[q.sup.-1+k][s.sup.-1))]]/4, where 0 < q, k < 1.

Theorem 3.1. Let Y be an arbitrary nonempty set and (X, d) a b-metric space. Let

P : Y [right arrow] CL(X) and f : Y [right arrow] X be such that P(Y) [??] f(Y) and (2) holds for all x, y [member of] Y. If s[q.sup.1-k] < 1, [beta]s < 1 and one of P(Y) or f(Y) is a complete subspace of X, then fx [member of] Px has a solution, that is P and f have a coincidence. Indeed, for any [x.sub.0] [member of] Y, there exists a sequence {[x.sub.n]} in Y such that

(I) f[x.sub.n+1] [member of] P[x.sub.n], n = 0, 1, 2, ...;

(II) the sequence {f[x.sub.n]} converges to fz for some z [member of] Y, and fz [member of] Pz, that is, P and f have a coincidence at z; and

(III) d(f[x.sub.n], fz) [less than or equal to] [s[[beta].sup.n]/(1 - s[beta])]d/(f[x.sub.0], f[x.sub.1]).

Proof. Pick [x.sub.0] [member of] Y. Let k be a positive number such that k < 1. Following Kulshrestha (18) and Singh and Kulshrestha (35), we construct sequences {[x.sub.n]} [??] Y and {f[x.sub.n]} [??] X in the following manner. Since P(Y) [??] f(Y), we may choose a point [x.sub.1] [member of] Y such that f[x.sub.1] [member of] P[x.sub.0].

If P[x.sub.0] = P[x.sub.1] then [x.sub.1] = z is a coincidence point of P and f, and we are done. So assume that P[x.sub.0] [not equal to] P[x.sub.1].

Now the condition P(Y) [??] f(Y) and Lemma 2.2 allow us to choose a point [x.sub.2] [member of] Y such that [fx.sub.2] [member of] [Px.sub.1] and

[d.sup.2]([fx.sub.1], [fx.sub.2]) [less than or equal to] [q.sup.-k][H.sup.2]([Px.sub.0], [Px.sub.1]).

If [Px.sub.1] = [Px.sub.2], then [x.sub.2] becomes a coincidence point of P and f. If not, continue the process. In general, if [Px.sub.n] [not equal to] [Px.sub.n+1], we choose [fx.sub.n+2] [member of] [Px.sub.n+1] such that

[d.sup.2]([fx.sub.n+1], [fx.sub.n+2]) [less than or equal to] [q.sup.-k][H.sup.2]([Px.sub.n], [Px.sub.n+1]).

Then by (2),

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For the sake of simplicity, we take [y.sub.n] : = [fx.sub.n], [d.sub.n] : = d([y.sub.n], [y.sub.n+1]) and [lambda] : = [q.sup.1-k].

Then the above inequality, after simplification, yields

[d.sub.n+1.sup.2] [less than or equal to] [lambda].max{[d.sub.n.sup.2], [d.sub.n][d.sub.n+1], [d.sub.n][d([y.sub.n], [y.sub.n+2])]/2, [d.sub.n+1][d([y.sub.n], [y.sub.n+2]]/2)},

that is

[d.sub.n+1.sup.2] [less than or equal to] [lambda].max{[d.sub.n.sup.2], [d.sub.n][d.sub.n+1], s([d.sub.n][[d.sub.n], + [d.sub.n+1]]/2), s([d.sub.n+1][[d.sub.n], + [d.sub.n+1]]/2)]}. (3)

We remark that in the construction of sequences {[x.sub.n]} and {[fx.sub.n]}, [x.sub.n] (for each n) is not a coincidence point of P and f. This together with [Px.sub.n] [not equal to] [Px.sub.n+1] means that [fx.sub.n] [not equal to] [fx.sub.n+1]. Indeed, if at any stage [fx.sub.n] = [fx.sub.n+1] then [fx.sub.n] [member of] [Px.sub.n] and {[x.sub.n]} is a coincidence point of P and f. Therefore, according to our construction of the sequences, [d.sub.n] [not equal to] 0. Hence the inequality (3) implies one of the following:

[d.sub.n+1.sup.2] [less than or equal to] [lambda] [d.sub.n.sup.2]

that is

[d.sub.n+1] [less than or equal to] [square root of ([lambda] [d.sub.n])];

[d.sub.n+1.sup.2] [less than or equal to] [lambda][d.sub.n][d.sub.n+1] implies [d.sub.n+1] [less than or equal to] [lambda][d.sub.n];

[d.sub.n+1.sup.2] [less than or equal to] [lambda]s([d.sub.n][[d.sub.n] + [d.sub.n+1]]/2) being a quadratic inequality in [d.sub.n+1] gives

[d.sub.n+1] [less than or equal to] [[lambda]s/4 + [square root of ((([[lambda].sup.2][s.sup.2]/16)]) + [lambda]s/2) ][d.sub.n] = {[lambda]s[1+[square root of ((1+8[([lambda]s).sup.[-1])]]]/4}[d.sub.n];

[d.sub.n+1.sup.2] [less than or equal to] [lambda]s([d.sub.n+1][[d.sub.n] + [d.sub.n+1]]/2) implies [d.sub.n+1] [less than or equal to] [[lambda]s/(2-[lambda]s)][d.sub.n].

These four outcomes together imply

[d.sub.n+1] [less than or equal to] max{[square root of ([lambda], [lambda], [lambda]s)][1+[square root of ((1+8[([lambda]s).sup.-1)]]]/4, [lambda]s/(2-[lambda]s)}[d.sub.n] = [beta][d.sub.n],

where [beta] : = [lambda]s[1+[square root of ((1+8[([lambda]s).sup.-1])]]/4. Notice that 0 < [beta] < 1 and [beta]s < 1. So, by Lemma 3.1, {[fx.sub.n]} is a Cauchy sequence. Now let f(Y) be a complete subspace of X. Then the sequence {[fx.sub.n]} has a limit in f(Y). Call it u. Hence, there exists a point z [member of] Y such that fz = u. Since {[fx.sub.n]} converges to fz,

d([fx.sub.n], [Px.sub.n]) [less than or equal to] d([fx.sub.n], [fx.sub.n+1]) implies that d([fx.sub.n], [Px.sub.n])[right arrow] 0 as n [right arrow] [infinity].

By Lemma 2.1 (iii) and (2),

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Making n [right arrow] [infinity], d(fz, Pz) [less than or equal to] [lambda]d(fz, Pz).

This yields fz [member of] Pz, since Pz is closed and [lambda] < 1. This argument applies to the case when P(Y) is a complete subspace of X, since P(Y) [??] f(Y).

This proves (I) and (II).

For n < m,

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This in the limit (m [right arrow] [infinity]) yields (III).

Now we extend Theorem 3.1 to the setting of a pair of multivalued maps and a single-valued map on Y with values in a b-metric space X.

Theorem 3.2. Let P, Q : Y [right arrow] CL(X) and f : Y [right arrow] X such that P(Y) [union] Q(Y) [??] f(Y) and the following holds for all x, y [member of] Y:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 < q < 1. If one of P(Y), Q(Y) or f(Y) is a complete subspace of X, then fx [member of] Px [intersection] Qx has a solution. Indeed, for any [x.sub.0] [member of] Y, there exists a sequence {[x.sub.n]} in Y such that

(I) [fx.sub.2n+1] [member of] [Px.sub.2n], [fx.sub.2n+2] [member of] [Qx.sub.2n+1], n = 0, 1, ...;

(II) the sequence {[fx.sub.n]} converges to fz for some z [member of] Y, and fz [member of] Pz [intersection] Qz;

(III) d([fx.sub.n], fz) [less than or equal to] [s[[beta].sup.n]/(1-s[beta])]d([fx.sub.0], [fx.sub.1]).

Proof. It may be completed following the proofs of Theorems 3.1 and 3.3.

Assume that 0 < q, k < 1 and [alpha] : = max{[q.sup.1-k], [sq.sup.1-k]/(2 - [sq.sup.1-k])}.

Theorem 3.3. Let Y be an arbitrary nonempty set and (X, d) a b-metric space. Let P, Q : Y [right arrow] CL(X) and f : Y [right arrow] X such that P(Y) [union] Q(Y) [??] f(Y) and the condition (1) for all x, y [member of] Y. If [sq.sup.[1-k]] < 1, as < 1, and one of P(Y), Q(Y) or f(Y) is a complete subspace of X, then fx [member of] Px [intersection] Qx has a solution. Indeed, for any [x.sub.0] [member of] Y, there exists a sequence {[x.sub.n]} in Y such that

(I) [fx.sub.2n+1] [member of] [Px.sub.2n] and [fx.sub.2n+2] [member of] [Qx.sub.2n+1], n = 0, 1 ...;

(II) the sequence {[fx.sub.n]} converges to fz for some z [member of] Y, and fz [member of] Pz [intersection] Qz;

(III) d([fx.sub.n]), fz] [less than or equal to] [s[[alpha].sup.n]/(1 - s[alpha])]d([fx.sub.0], [fx.sub.1]).

Proof. Pick [x.sub.0] [member of] Y. Notice that [q.sup.[-k]] > 1 since 0 < q, k < 1. We construct sequences {[x.sub.n]} in Y and {[fx.sub.n]} in X in the following manner. Since P(Y) [??] f(Y), we can find a point [x.sub.1] [member of] Y such that [fx.sub.1] [member of] [Px.sub.0]. Noting that Q(Y) is also a subspace of f(Y), we, for a suitable point [x.sub.2] [member of] Y, can choose a point [fx.sub.2] [member of] [Qx.sub.1] such that

d([fx.sub.1], [fx.sub.2]) [less than or equal to] [q.sup.[-k]] H([Px.sub.0], [Qx.sub.1]).

We remark that such a choice is possible by Lemma 2.2. In general, we can choose a sequence {[x.sub.n]} in Y such that

[fx.sub.2n+1] [member of] [Px.sub.2n], [fx.sub.2n+2] [member of] [Qx.sub.2n+1], [fx.sub.2n+3] [member of] [Px.sub.2n+2]

and

d([fx.sub.2n+1], [fx.sub.2n+2]) [less than or equal to] [q.sup.-k] H([Px.sub.2n], [Qx.sub.2n+1]), d([fx.sub.2n+2], [fx.sub.2n+3]) [less than or equal to] [q.sup.[-k]] H([Qx.sub.2n+1], [Px.sub.2n+2]).

Taking [y.sub.n]: = [fx.sub.n], [d.sub.n]: = d([y.sub.n], [y.sub.n+1]) and [lambda] : [q.sup.1-k], by (1), [d.sub.2n+1] = d([fx.sub.2n+1], [fx.sub.2n+2]) [less than or equal to] [lambda].max{[d.sub.2n], [d.sub.2n], [d.sub.2n+1], [d([y.sub.2n], [y.sub.2n+2]) + 0]/2} [less than or equal to] [lambda].max{[d.sub.2n],[d.sub.2n+1], s[[d.sub.2n] + [d.sub.2n+1]]/2},

giving [d.sub.2n+2] [less than or equal to] [alpha][d.sub.2n], where [alpha] = max{[lambda], [lambda]s/(2 - [lambda]s)}.

Similarly, by (1),

[d.sub.2n+2] [less than or equal to] [q.sup.-k] H([Px.sub.2n+2], [Qx.sub.2n+1]) [less than or equal to] [lambda].max{[d.sub.2n+1], [d.sub.2n+2], [d.sub.2n+1], [0 + d([y.sub.2n+1], [y.sub.2n+3])]/2}, [less than or equal to] [lambda].max{[d.sub.2n+1], [d.sub.2n+2], s[[d.sub.2n+1] + [d.sub.2n+2]]/2},

giving [d.sub.2n+2] [less than or equal to] [alpha][d.sub.2n+1].

Thus, in general, [d.sub.n+1] [less than or equal to] [alpha][d.sub.n], n = 0, 1,. ...

Note that 0 < [alpha] < 1, and by hypothesis [alpha]s < 1. So, by Lemma 3.1, {[y.sub.n]} is a Cauchy sequence. If we assume that f(Y) is a complete subspace of X, then the sequence {[y.sub.n]} and its subsequences {[y.sub.2n]} and {[y.sub.2n+1]} have a limit in f(Y). Call it u. Then there exists a point z [member of] Y such that fz = u. By (1),

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Making n [right arrow] [infinity], d(fz, Pz) [less than or equal to] qd(fz, Pz).

This gives fz [member of] Pz, since 0 < q < 1 and Pz is closed. Similarly fz [member of] Qz. Thus fz [member of] Pz [intersection] Qz.

The above argument applies to the case when P(Y) or Q(Y) is a complete subspace of X, since P(Y) and Q(Y) are contained in f(Y). This proves (I) and (II). The proof of the last part is analogous to that of Theorem 3.1 (III).

Corollary 3.1. Let P : Y [right arrow] CL(X) and f : Y [right arrow] X such that P(Y) [??] f(Y) and (SK) (cf. Th. 1.1) holds for all x, y [member of] Y. If one of P(Y) or f(Y) is a complete subspace of X, then fx [member of] Px has a solution. Indeed, for any [x.sub.0] [member of] Y, there exists a sequence {[x.sub.n]} in Y such that conclusions (I), (II) of Theorem 3.1 and the conclusion (III) of Theorem 3.3 hold.

Proof. It comes from Theorem 3.3 when P = Q.

We remark that Corollary 3.1 is an extension of Theorem 1.1 to b-metric spaces. Certain results of Czerwik (6), (7) and Singh et al. (32) are particular cases of the above corollary.

4. Fixed Point Theorems

We apply coincidence theorems of the previous section to study solutions of x = fx [member of] Px, x [member of] Px, x = fx [member of] Px [intersection] Qx and x [member of] Px [intersection] Qx, for P, Q : X [right arrow] CL(X) and f : X [right arrow] X.

Theorem 4.1. Let all the hypotheses of Theorem 3.1 be satisfied with Y = X. If f and P are (IT)-commuting just at a coincidence point z (say) of f and P, and if u = fz is fixed point of f, then u is a common fixed point of f and P.

Proof. It comes from Theorem 3.1 that there exist points z, u [member of] X such that

u = fz [member of] Pz.

If u is a fixed point of u = fu and f, P are (IT)-commuting at z then

u = fu = ffz [member of] fPz [??] Pfz = Pu.

This completes the proof.

Theorem 4.2. Let all the hypotheses of Theorem 3.2 be satisfied with Y = X. If f is (IT)-commuting with each of P and Q at their common coincidence point z, and if u = fz is fixed point of f, then f, P and Q have a common fixed point, i.e.,

u = fu [member of] Pu [intersection] Qu

Proof. It comes from Theorem 3.2 that there exist z, u [member of] X such that

u = fz [member of] Pz and u = fz [member of] Qz. Since u = fu, the (IT)-commutativity of f and P implies that

u = fu = ffz [member of] fPz [??] Pfz = Pu. Similarly u = fu [member of] Qu. So u = fu [member of] Pu [intersection] Qu. This completes the proof.

Theorem 4.3. Let all the hypotheses of Theorem 3.3 be satisfied with Y = X. If f is (IT)-commuting with each of P and Q at one of their common coincidences z (say), and if u = fz is a fixed point of f, then f, P and Q have a common fixed point, i.e., u = fu [member of] Pu [intersection] Qu.

Proof. It comes from Theorem 3.3 that there exist points z, u [member of] X such that

u = fz [member of] Pz [intersection] Qz. The rest part of the proof is now evident.

Now we derive some corollaries.

Corollary 4.1. Let (X, d) be a complete b-metric space and P, Q : X [right arrow] CL(X) such that H(Px, Qy) [less than or equal to] q.max{d(x, y), d(x, Px), d(y, Qy), [d(x, Qy) + d(y, Px)]/2} for all x, y [member of] X, where 0 < q, k < 1, [sq.sup.1-k] < 1 with as < 1. Then the functional inclusion x [member of] Px [intersection] Qx has a solution.

Proof. It comes from Theorem 3.3 with Y = X when f = is the identity map on X.

Corollary 4.2. Let (X, d) be a complete b-metric space and P, Q : X [right arrow] CL(X) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 < q, k < 1, [sq.sup.1-k] < 1 with [beta]s < 1. Then x [member of] Px [intersection] Qx has a solution.

Proof. It comes from Theorem 3.2 with Y = X when f = is the identity map on X.

The following result is an extension of the main result of Ciric (3) and Theorem 1.1 with f the identity map on X.

Corollary 4.3. Let (X, d) be a complete b-metric space and P : X [right arrow] CL(X) such that H(Px, Py) [less than or equal to] q.max{d(x, y), d(x, Px), d(y, Py), [d(x, Py) + d(y, Px)]/2} (C-1)

for all x, y [member of] X, where 0 < q, k < 1, [sq.sup.1-k] < 1 with as < 1. Then x [member of] Px has a solution.

Proof. It comes from Corollary 4.1 with P = Q.

Ciric (3) was the first to study the contraction (C-1) in a metric space. Using a similar condition for a pair of multivalued maps in a metric space, Khan (17) obtained some interesting fixed point theorems in metric spaces. We remark that Corollary 4.3 is an improvement in respect of the statement of a main result of Singh et al. [32, Th. 4.1]. Further, the above corollaries improve and extend several fixed point theorems for multivalued maps in metric and b-metric spaces (see, for instance, (1), (5), (6), (7), (17) and (23)).

The following question merits attention: Does the Corollary 4.3 hold when (C-1) is replaced by

H(Px, Py) [less than or equal to] q.max{d(x, y), d(x, Px), d(y, Py), d(x, Py), d(y, Px)}. (C-2)

We remark that (C-2) is the main contraction condition due to Ciric (4) when X is a metric space and P is a single-valued map on X.

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Shyam Lal Singh [dagger]

Gurukula Kangri University, Hardwar Mailing address: 21, Govind Nagar Rishikesh, 249201, India

Stefan Czerwik [double dagger], Krzysztof Krol [section]

Institute of Mathematics Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

and

Abha Singh **

Received June 4, 2008, Accepted June 4, 2008.

* 2000 Mathematics Subject Classification: 54H25; 54C60; 47H10.

[dagger] Email: vedicmri@gmail.com

[double dagger] Email: stefan.czerwik@polsl.pl

[section] Email: krzysztof.krol@polsl.pl

** Email: abha_singh17@rediffmail.com