COINCIDENCE THEOREMS FOR NONLINEAR HYBRID CONTRACTIONS

In this paper, we give some common fixed point theorems for single-valued mappings and multi-valued mappings satisfying a rational inequality Our theorems generalize some results ofB Fisher, M L. Diviccaro et al. and V Popa


INTRODUCTION
Let (X, d) be a metric space and let f and g be mappings from X into itself. In [1], S Sessa defined f and g to be weakly commuting if d(gfx, f g:r,) <_ d(gx, fx) for all x in X It can be seen that two commuting mappings are weakly commuting, but the converse is false as shown in the Example of [2] Recently, G. Jungck [3 extended the concept of weak commutativity in the following way DEFINITION 1.1. Let f and g be mappings from a metric space (X, d) into itself. The mappings f and g are said to be compatible if lim f gx, g f x, 0 whenever (x } is a sequence in X such that lim fx, lim gx, z for some z in X It is obvious that two weakly commuting mappings are compatible, but the converse is not true Some examples for this fact can be found in [3]. Recently, H. Kaneko [4] and S. L. Singh et al. [5] extended the concepts of weak commutativity and compatibility [6] for single-valued mappings to the setting of single-valued and multi-valued mappings, respectively Let (X, d) be a metric space and let CB(X) denote the family of all nonempty closed and bounded subsets of X. Let  It is well-known that (CB(X), H) is a metric space, and if a metric space (X, d) is complete, then (CH(X), H) is also complete. Let 6(A,B) sup{d(z,V) z E A and V B} for all A, B c= CB(X). If A consists of a single point a, then we write r(A,B) 6(a,B) Ifr(A,B) 0, thenA B {a} [7] LEMIIA 1.1 [8]. Let A, B CB(X) and k > 1. Then for each a A, there exists a point b E B such that d(a, b) <_ kH(A, B).
Let (X, d) be a metric space and let f" X X and S" X CB(X) be single-valued and multi- Therefore, f and T are compatible, but f and T are not weakly commuting at x 2 We need the following lemmas for our main theorems, which is due to G Jungck [2] LEMMA 1.2. Let f and g be mappings from a metric space (X, d) into itself If f and g are compatible and fz gz for some z X, then fz z f z Lf z. LEMMA 1.3. Let f and # be mappings from a metric space (X, d) into itself If f and g are compatible and fz,.,, #z, z for some z a X, then we have the following (1) lim #fz, fz if f is continuous at z, (2) fz 9fz and fz #z if f and # are continuous at z

COINCIDENCE THEOREMS FOR NONLINEAR HYBRID CONTRACTIONS
In this section, we give some coincidence point theorems for nonlinear hybrid contractions, e., contractive conditions involving single-valued and multi-valued mappings In the following Theorem 2 1, S(X) and T(X) mean S(X) 12 :ex Sx and T(X) 12 .ex Tx, respectively THEOREM 2.1. Let (X,d) be a complete metric space. Let f, g-X-X be continuous mappings and S, T X CB(X) be H-continuous multi-valued mappings such that T(X) C f(X) and S(X) c g(X), (2.1) the pairs f, S and g, T are compatible mappings, (2 2) cd(fx, Sx)dr'(gy, Ty) + bd(fx, Ty)dr'(gy, Sx) Hr,(Sx, Ty) < 6(fx, Sx) + 6(gy, Ty) for all x, y E X for which 6(fz, Sx) + 6(gy, Ty) O, where p > 1, b > 0 and 1 < c < 2 Then there exists a point z E X such that fx .Sza nd gz Tz, i.e, z is a coincidence point of f, S and of g, T PROOF. Choose a real number k such that 1 < k < (-) and let x0 be an arbitrary point in X Since Sxo c g(X), there exists a point x E X such that gxl Sxo and so there exists a point y such that d(gxl,y) < kH(Sxo, TXl), which is possibly by Lemma Since TXl c f(X), there exists a point z2 X such that y fx2 and  9 and T. Similarly, 6(fz2,+2,Sx2,.,+2)+6(gx2,.,+1,Tx2,+a)=0 for some n 6 N implies that z,,+a is a coincidence point of 9 and T and z2,+2 is a coincidence point of f and S. Now, suppose that 6(fx2,,Sx2,) + 6(gx2,.,+I,Tx2,.,.1) 0 for n 6  Repeating the above argument, since 0 < kr'c-1 < 1, it follows that {gxa,fx2,gx3,.fx4,..., gx2,-l, gx2,.,,gx2,.,+l,-'-} is a Cauchy sequence in X. Since (X, d) is a complete metric space, let lira gX2n+ lim fX2n Z. Now, we will prove that fz Sz, that is, z is a coincidence point of f and S. For every n E N, we have d(fgz,.,+l, Sz) <_ d(.fgx2,,+, Sfx2,.,) + H(S.fx,, Sz). ( 2 7) Thus, from (2 5) for all x, y X for which 5(x, Sx) + ,5(y, Ty) O, where p _> 1, b > 0 and 1 < c < 2. Then S and T have a common fixed point in X, that is, z Sz and z Tz Assuming that f g and S T on X in Theorem 2.1, we have the following COROLLARY 2.3. Let (X, d) be a complete metric space and let f" X X be a continuous single-valued mapping and S X CB(X) be an H-continuous multi-valued mapping such that S(X) C I(X), (2 9) f and S are continuous mappings, for all x, y X for which 6(fx, Sx) + 6(fy, Sy) O, where p > 1, b > 0 and 1 < c < 2 Then there exists a point z E X such that fz Sz, i.e., z is a coincidence point of f and S.

FIXED POINT THEOREMS FOR SINGLE-VALUED MAPPINGS
In this section, using Theorem 2.1, we can obtain some fixed point theore,ms for single-valued mappings in a metric space If S and T are single-valued mappings from a metric space (X, d) into itself in Theorem 2 1, we have the following THEOREM 3.1. Let (X, d) be a complete metric space Let f, g, S and T be continuous mappings from X into itself such that S(X) C g(X) and T(X) C f(X), ( Then f, g, S and T have a unique common fixed point z in X Further, z is the unique common fixed point of f, S and of g, T PROOF. The existence of the point w with fw Sw and gw Tw follows from Theorem 2 From (ii) of (3.3), since d(fw, Sw)+d(gw, Tw)= 0, it follows that d(Sw, Tw)= 0 and so Sw fw w Tw. By Lemma 2, since f and S are compatible mappings and fw Sw, we have S fw SSw fSw f fw, (3 4) which implies that d(fSw, SSw) + d(gw, Tw) 0 and, using the condition (ii) of (3.3), we have S fw SSw Tw gw fw (3 5) and so fw z is a fixed point of S. Further, (3 4) and (3 5) implies that Sz f Sw SSw f z z.
Similarly, since g and T are compatible mappings, we have Tz gz z. Using (ii) of ( 3 3

Call for Papers
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