Common fixed point of -approximative multivalued mapping in partially ordered metric space
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1 Filomat 7:7 (013), DOI 1098/FIL A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Common fixed point of -approximative multivalued mapping in partially ordered metric space Mujahid Abbas a, Ali Erduran b a Department of Mathematics, Lahore University of Management Sciences, 5479-Lahore, PAKISTAN b Kirikkale UniversityFaculty of Arts and Sciences Department of Mathematics,71450 Yahsihan / Kirikkale Abstract In this paper, we introduce g-approximative multivalued mappings Based on this definition, we gave some new definitions Further, common fixed point results for g-approximative multivalued mappings satisfying generalized contractive conditions are obtained in the setup of ordered metric spaces Our results generalize Theorems 6-9 given in ([1]) 1 Introduction and preliminaries Contractive conditions play an important role in proving the existence of fixed points of single as well as multivalued mappings One of the simplest and most useful results in the fixed point theory is the Banach-Caccioppoli contraction mapping principle [] This principle has been generalized in different directions in different spaces by mathematicians over the years In 1968 Kannan [3] proved a fixed point theorem for a map satisfying a contractive condition that did not require continuity at each point (see, eg, [4] for a listing and comparison of many of these definitions) The concept of weak contractions in Hilbert spaces was defined by Alber and Guerre-Delabriere [5] in 1997 Weak inequalities of the above type have been used to establish fixed point results in a number of subsequent works, some of which are noted in [6] The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler [7] After this, fixed point theory has been developed further and applied to many disciplines to solve functional equations Banach contraction principle has been extended in different directions either by using generalized contractions for multivalued mappings and hybrid pairs of single and multivalued mappings, or by using more general spaces Dhage [8, 9] established hybrid fixed point theorems and obtained some applications of presented results Hong and Shen [10] proved common fixed point results for generalized contractive multivalued operators in complete metric space Also the monotone iterative technique is associated with several nonlinear problem [11] This technique is also employed to prove the existence of fixed points for multivalued monotone operators (see, for example [1]) In [1], the problem of existence and approximation of coupled fixed points for mixed monotone multivalued operators were studied in ordered Banach spaces under the assumption that operators satisfy the condensing condition and upper demicontinuity Hong introduced the concepts of approximative values, comparable approximative values upper and lower comparable approximative values in [1] These definition are very useful tool for proving existence 010 Mathematics Subject Classification Primary 05C38, 15A15, Secondary 05A15, 15A18 Keywords -approximative multivalued mapping, ordered metric space Received: 1 March 01; Accepted: 1 December 01 Communicated by Vladimir Rakočević addresses: mujahid@lumsedupk (Mujahid Abbas), alierduran1@yahoocom (Ali Erduran)
2 M Abbas, A Erduran / Filomat 7:7 (013), of fixed point of multivalued operator in ordered metric space Motivated by the work of [1], for a self map on a ordered metric space, we introduce -approximative multivalued mappings and obtain coincidence and common fixed point results for a hybrid pair of multivalued and single valued mappings Concepts of -comparable approximative, -upper comparable approximative and -lower comparable approximative multivalued mappings are introduced Employing these definitions, common fixed point results for generalized contractive multivalued mappings in the framework of ordered metric spaces are obtained Consequently, Theorem 6-9 in ([1]) are generalized Let (X, d) be a metric space For x X and A X, we denote d(x, A) = infd(x, A) : y A The class of all nonempty bounded and closed subsets of X is denoted by CB(X) Let H be the Hausdorff metric induced by the metric d on X, that is, H(A, B) = max for every A, B CB(X) sup x A d(x, B), sup y B d(y, A), Definition 11 Let X be a nonempty set Then (X,, d) is called an ordered metric space iff: (i) d is a metric on X and (ii) is a partial order on X Definition 1 Let X be an ordered metric space A mapping : X X is said to be (i) weakly L idempotent if x x for x in X (ii) weakly R idempotent if x x for x in X For example, a mapping : [0, 1] [0, 1] given by (x) = x is weakly R idempotent Definition 13 An ordered metric space is said to have a subsequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) x n in X such that x n x, there exists a subsequence x nk of x n with x nk x (x x nk ) respectively Definition 14 An ordered metric space is said to have a sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) x n in X such that x n x implies that x n x (x x n ) respectively Let X be any nonempty set endowed with a partial order and let : X X be a given mapping We define the set X X by = (x, y) X X : x y Note that for each x X, one has (x, x) Example 15 Let X = 0, 1, be endowed with usual order and be a self map on X defined as 0 = 0, 1 = and = 1 Then the subset of X X is = (0, 0), (0, 1), (0, ), (1, 1), (, 1), (, ) Definition 16 Let X be a metric space and : X X A subset Y of X is said to be -approximative for some x in X if Y (X) and the set ( (x)) = y Y : d( (x), y) = d(y, (x)) Y is nonempty Definition 17 Let X be a partially ordered set A mapping F : X X (collection of all nonempty subsets of X ) is said to be: (i) -approximative multivalued mapping (in short - AV multivalued mapping), if Fx is -approximative for each x X, That is, ( (x)) is nonempty for each x in X Fx
3 M Abbas, A Erduran / Filomat 7:7 (013), (ii) -CAV multivalued mapping ( - comparable approximative multivalued mapping) if F is -approximative and for each z X, there exists (y) ( (z)) such that y is comparable to z F(z) (iii) -UCAV( - upper comparable approximative multivalued mapping) if F is -approximative and for each z X, there exists (y) ( (z)) such that (z) (y) F(z) (iv) -LCAV( - lower comparable approximative multivalued mapping) if F is -approximative and for each z X, there exists (y) ( (z)) such that (y) (z) F(z) If F is a single-valued, then -UCAV ( -LCAV) means that Fx x (Fx x) for x X Definition 18 Let : X X and T : X CB(X) A point x in X is said to be: (i) fixed point of if (x) = x : (ii) fixed point of T if x T(x) (iii) coincidence point of a pair (, T) if x Tx : (iv) common fixed point of a pair (, T) if x = x Tx F( ), C(, T) and F(, T) denote set of all fixed points of, set of all coincidence points of the pair (, T) and the set of all common fixed points of the pair (, T), respectively Definition 19 Let f : X X, T : X CB(X), and f Tx CB(X) The pair ( f, T) is called (1)commuting if T f x = f Tx for all x X () weakly compatible [13] if they commute at their coincidence points, that is, f Tx = T f x whenever x C( f, T); (3) (IT) commuting at x X if f Tx T f x Definition 110 f x T f x Let T : X CB(X) The map f : X X, is said to be T weakly commuting at x X if Definition 111 The map f : X X is said to coincidently idempotent with respect to T : X CB(X) if f (x) = f (x) for x in C( f, T) The point x is called point of coincident idempotency Now we present an example of hybrid pair f, T for which f is T weakly commuting at some x C( f, T) Example 11 Let X = [0, ) with usual metric Define f : X X, T : X CB(X) by 0, 0 x < 1 f x = x + 1, 1 x < and Tx = x, 0 x < 1 [1, x + ], 1 x < It can be easily verified that f is T weakly commuting at x = 0 C( f, T) Example 113 Let X = R with usual metric Define f : X X, T : X CB(X) by 1, x 0 f x = x, 0 < x and x, x 1 Tx = [x, 1], 1 < x 1 [1, x], 1 < x < Here C( f, T) = 1 and f is coincidently idempotent with respect to T Let α (0, + ] Ϝ denotes the class of mappings f : [0, α) R which satisfy the following conditions: (i) f (0) = 0 and f (t) > 0 for each t (0, α),
4 M Abbas, A Erduran / Filomat 7:7 (013), (ii) f is continuous, (iii) f is nondecreasing on [0, α) A mapping f is said to be sublinear if f (t 1 + t ) f (t 1 ) + f (t ), whenever t 1, t, t 1 + t (0, α) We define Ϝ s = f : [0, α) R : f is sublinear and f Ϝ Ψ denotes the family of mappings ψ : [0, α) [0, + ) which satisfy the following conditions: (a) ψ(t) < t for each t (0, α), (b) ψ is nondecreasing and right upper semi-continuous, (c) For each t (0, α), lim ψ n (t) = 0 By means for the functions f and ψ given in Ϝ and Ψ respectively, a generalized contractive condition was defined in [9] Let Φ denotes the class of mappings ψ : [0, α) [0, + ) for which ψ(t) < t and n=1 ψn (t) < for each t in (0, α) Definition 114 For two subset A, B of X, we say A 1 B if for each x X, there exists y Y such that x y and A B if each x A, y B implies that x y A multivalued mapping F : X X is said to be -nondecreasing ( -nonincreasing) if x y implies that Fx 1 Fy (Fy 1 Fx) for all x, y X F is said to be - monotone if F is -nondecreasing or -nonincreasing Moreover in what follows (X, ) will be a partially ordered set such that there exists a complete metric d on X Let D = supd(x, y) : x, y X Set α = d if d = and α > d if d < Common fixed point theorems In this section we obtain common fixed point theorems Theorem 1 Suppose that be a nondecreasing self map on X and F : X X is -UCAV and the following holds f (H(Fx, Fy)) ψ ( f (M (x, y)) ) for any (x, y), where f Ϝ s and ψ Φ and M (x, y) = max d( x, y), d( x, Fx), d( y, Fy), d( x, Fy) + d( y, Fx) If X has a limit comparison property and (X) is closed, then F and have a coincidence point x in X Moreover F and have common fixed point if one of the following conditions holds: (i) Pair (F, ) is IT commuting at some x C(F, ) and lim n x = u, for some u X and is continuous at u (ii) Pair (F, ) is IT commuting at some x C(F, ) and x = x (iii) is F weakly commuting at some C(F, ) and is coincidently idempotent with respect to T (iv) is continuous at x for some x C(F, ) for some u X; lim n u = x (v) (C(, F)) is singleton subset of C(, F) Proof Let x 0 X If x 0 Fx 0, then the result is proved If not, then we proceed as follows: As F is -UCAV, Fx 0 (X), F(x 0 ) ( (x 0)) is nonempty so there exists x 1 Fx 0 with x 1 x 0 such that d( x 1, x 0 ) = d(fx 0, x 0 ) for some x 1 X and x 1 x 0 Similarly, there exists x Fx 1 with x 1 x such that (1)
5 M Abbas, A Erduran / Filomat 7:7 (013), d( x 1, x ) = d(fx 1, x 1 ) for some x X, and x x 1 We continue to construct a sequence x n for which either x n 1 Fx n 1 or there exists x n Fx n 1 with x n x n 1 and x n x n 1 such that d( x n, x n 1 ) = d(fx n 1, x n 1 ), for n = 1,, () for some x n in X On the other hand, d(fx n 1, x n 1 ) = sup x Fx n d(x, Fx n 1 ) H(Fx n 1, Fx n ), (3) implies that d( x n, x n 1 ) H(Fx n 1, Fx n ), for n =, 3, (4) Since f is nondecreasing, then we have in which f (d( x n, x n 1 )) f (H(Fx n 1, Fx n )) ψ( f (M (x n 1, x n ))), M (x n 1, x n ) = max d( x n 1, x n ), d(fx n 1, x n 1 ), d(fx n, x n ), d(fx n, x n 1 ) + d(fx n 1, x n ) = max d( x n 1, x n ), d( x n, x n 1 ), d( x n 1, x n ), d( x n 1, x n 1 ) + d( x n, x n ) = max d( x n 1, x n ), d( x n, x n 1 ) If d( x n 1, x n ) > d( x n 1, x n ), then we have f (d( x n, x n 1 )) f (H(Fx n 1, Fx n )) ψ( f (M (x n 1, x n ))) ψ( f (maxd( x n 1, x n ), d( x n 1, x n ))) ψ( f (d( x n, x n 1 ))) < f (d( x n, x n 1 )), a contraction So we have d( x n 1, x n ) d( x n 1, x n ) This yields f (d( x n, x n 1 )) ψ( f (d( x n 1, x n ))) Repeating this process, we have f (d( x n, x n 1 )) ψ( f (d( x n 1, x n ))) For m, n N, n > m, we obtain ψ ( f (d( x n, x n 3 ))) ψ n 1 ( f (d x 0, x 1 )) n 1 d( x n, x m ) d( x i, x i+1 ) i=m
6 M Abbas, A Erduran / Filomat 7:7 (013), This implies f (d( x n, x m )) f (d( x n, x n 1 ) + + d( x m+1, x m )) f (d( x n, x n 1 )) + + f (d( x m+1, x m )) ψ n 1 ( f (d( x 0, x 1 ))) + + ψ m ( f (d( x 0, x 1 ))) n 1 ψ i ( f (d( x 0, x 1 ))) i=m On taking limit as n, m and using ψ n (t) <, it follows that (x n ) is Cauchy sequence in X Since n=1 X is complete and (X) is closed so we have lim x n = x for some x in X Now we prove that d(fx, x) = 0 Suppose that this is not true, then d(fx, x) > 0 For large enough n, we claim that the following equation holds M (x, x n+1 ) = max d( x, x n+1 ), d(fx, x), d(fx n+1, x n+1 ), d(fx, x n+1) + d(fx n+1, x) = d(fx, x) Indeed, since lim x n = x and lim d(fx n+1, x n+1 ) = 0, it follows that 1 lim [d(fx, x n+1) + d(fx n+1, x)] 1 [ lim d(fx, x) + d( x, xn+1 ) + d(fx n+1, x n+1 ) + d( x n+1, x) ] = 1 d(fx, x) So there exists n 0 N such that M (x, x n+1 ) = d(fx, x) for every n > n 0 Note that f (d(fx, x n+ )) f (H(Fx, Fx n+1 )) ψ( f (M (x, x n+1 ))), which on taking limit as n gives f (d(fx, x)) ψ( f (d(fx, x))) < f (d(fx, x)), a contradiction So d(fx, x) = 0, we have x Fx Suppose now that (i) holds Then lim n x = u where u X Since is continuous at u, so we have that u is fixed points of By given assumption, n x C(F, n 1 ) for all n 1 and n x F( n 1 x) Now we prove that d(fu, u) = 0 Suppose that this is not true, then d(fu, u) > 0 Using (1), since f is nondecreasing and sublinear, we obtain, f (d( u, Fu)) f (d( u, n x)) + f (d( n x, Fu)) f (d( u, n x)) + f (H(F( n 1 x), F(u))) f (d( u, n x)) + ψ( f (M ( n 1 x, u)) (5) Where M ( n 1 x, u) = max d( n x, u), d(f n 1 x, n x), d(fu, u), d(fu, n x) + d(f n 1, u) = max d( n x, u), d( n x, n x), d(fu, u), d(fu, n x) + d( n x, u) On taking limit as n, we have M ( n 1 x, u) = d(fu, u)
7 which further implies M Abbas, A Erduran / Filomat 7:7 (013), f (d( u, Fu)) f (d( u, n x)) + ψ( f (d(fu, u))) On taking limit as n, < f (d( u, n x)) + f (d(fu, u)) f (d( u, Fu)) < f (d(fu, u)) (4) a contradiction, so d( u, Fu) = 0 and hence u Fu Consequently u = u Fu Hence u is a common fixed point of F and Suppose now that (ii) holds As x C(F, ), so x Fx F x Now x = x F x implies that that x is a common fixed point of F and Suppose now that (iii) holds The result is obvious Suppose that (iv ) holds As x C(, F) and for some u X, lim n u = x By the continuity of at x, we get x = x Fx Hence x is common fixed point of F and Finally, suppose that (v) holds Let (C(F, )) = x Then x = x = Fx Hence x is common fixed point of F and Similarly, we have following theorem Theorem Suppose that be a nondecreasing self map on X and F : X X is -LCAV and the following holds f (H(Fx, Fy)) ψ ( f (M (x, y)) ) for any (x, y), where f Ϝ s and ψ Φ and M (x, y) = max d( x, y), d( x, Fx), d( y, Fy), d( x, Fy) + d( y, Fx) If X has sequential limit comparison property and (X) is closed, then F and have a coincidence point x in X Moreover F and have common fixed point if any one of conditions (i)-(v) holds as in Theorem 1 Example 3 Let X = 0 [1, ) with usual metric Define : X X, F : X X by 0, x = 0 x = x + 1, 1 x < and Fx = x, x = 0 [1, x + ], 1 x < We can see that function of F and are satisfy condition of Theorem It is clear that F is -UCAV, also (X) is closed and X has a property of limit comparison we can see easly that is F- weakly commuting at x = 0 Besides, is concidently idempotent with respect to F at x = 0 In this case, These functions satisfy condition of (iii) in Theorem 1 Also we can define f (t) = t, ψ(t) = t, then f Ϝ s and ψ Ψ If x = y = 0, we have x = y = 0 and Fx = x, Fy = y f (H(Fx, Fy)) = H(Fx, Fy) = max sup d(z, y), supd(x, t) z x t y = max sup inf d(z, t), sup inf d(p, k) = 0 = max z 0 t 0 = ψ( f (M (x, y))) p 0 k 0 d( x, y), d( x, Fx), d( y, Fy), d( x, Fy) + d( y, Fx)
8 M Abbas, A Erduran / Filomat 7:7 (013), if x = 0, y [1, ), we have x = 0, y = y + 1 and Fx = x, Fy = [1, y + ] f (H(Fx, Fy)) = H(Fx, Fy) = max sup d(z, [1, y + ]), sup d(0, t) z 0 t [1,y+] = 1 also since x < y then we have M (x, y) = max y + 1, x, 1, y x = y + 1 So we satisfy contractive condition Finally, If x, y [1, ), we have x = x + 1, y = y + 1 and Fx = [1, x + ], Fy = [1, y + ] and we can see easly that the contractive condition is satified Hence, satisfy all condition of Theorem 1 It is clear that 0 = x = x Fx that is, x = 0 is common fixed point of F and Corollary 4 Suppose that be a nondecreasing self map on X and F : X X and : X X are self mappings which satisfy f (d(fx, Fy)) ψ( f (M (x, y))) for any (x, y), where f Ϝ s, ψ Φ and M (x, y) = max d( x, y), d(fx, x), d(fy, y), d(fx, y) + d(fy, x) Then F, have a unique coincidence point x X Moreover F and have unique common fixed point if any one of conditions (i)-(v) holds as in Theorem 1 Proof Theorem 1 ensures the existence of coincidence point To prove the uniqueness, let y be another coincidence point of F and If x y, then d( x, y) > 0 Thus, d(fx, y) + d(fy, x) M (x, y) = max d( x, y), d(fx, x), d(fy, y), = d( x, y) This yields f (d( x, y)) = f (d(fx, Fy)) ψ( f (M (x, y))) = ψ( f (d( x, y))) < f (d( x, y)), a contradiction, therefore d( x, y) = 0 The results follows Theorem 5 holds Suppose that be a nondecreasing self map on X and F : X X is -AV and the following f (H(Fx, Fy)) ψ ( f (M (x, y)) ) for any (x, y), where f Ϝ s and ψ Φ and M (x, y) = max d( x, y), d( x, Fx), d( y, Fy), d( x, Fy) + d( y, Fx) If (X) is closed and there exists x 0 X such that x 0 Fx0, then F and have a coincidence point x X Further, an iterative sequence x n with x n Fx n 1 converges to x, where x C(F, ) Moreover F and have common fixed point if any one of conditions (i)-(v) holds as in Theorem 1
9 M Abbas, A Erduran / Filomat 7:7 (013), Proof If x 0 Fx 0, then the proof is finished Otherwise, for any x Fx 0 one has x x 0 As F has -approximative mutlivalued map, for x 1 X, there exists x 1 Fx 0 with x 1 x 0 and d( x 0, x 1 ) = d(fx 0, x 0 ) Similarly, for x X, there exists x Fx 1 with x x 1 and d( x 1, x ) = d(fx 1, x 1 ) We continue the process of constructing a sequence x n such that for x n X, one obtaines x n Fx n 1 with x n x n 1 such that d( x n 1, x n ) = d(fx n 1, x n 1 ) n = 1,, On the other hand, we have So, d(fx n 1, x n 1 ) = sup x Fx n d(x, Fx n 1 ) H(Fx n, Fx n 1 ), d( x n 1, x n ) H(Fx n, Fx n 1 ) for n =, 3, The rest of this proof is the same as that of Theorem 1 Theorem 6 holds Suppose that be a nondecreasing self map on X, F : X X is -CAV and the following f (H(Fx, Fy)) ψ ( f (M (x, y)) ) for any (x, y), where f Ϝ s and ψ Φ and M (x, y) = max d( x, y), d( x, Fx), d( y, Fy), d( x, Fy) + d( y, Fx) If X has a subsequential limit comparison property and (X) is closed, then F and have coincidence point Moreover F and have common fixed point if any one of conditions (i)-(v) holds as in Theorem 1 Proof Following similar arguments to those given in Theorem, and F is -CAV, we obtain a sequence x n whose consecutive terms are comparable, satisfy () and (4) and following hold: x n+1 Fx n, lim x n = x Since X has subsequential limit comparison property so x n has subsequence x nk whose every term is comparable to x Now we prove x Fx Obviously, d( x nk +, Fx) d( x nk +, x nk +1) + d( x nk +1, Fx) d( x nk +, x nk +1) + sup t Fx nk d(t, Fx) d( x nk +, x nk +1) + H(Fx nk, Fx) for k = 0, 1,, For ε > 0, there exists k 0 such that f (d( x nk +, x nk +1)) < ε for all k > k 0 As lim f (d( x n k +, x nk +1)) = 0 k
10 Since x nk is comparable to x for each k, therefore M Abbas, A Erduran / Filomat 7:7 (013), f (d( x nk +, Fx)) f (d( x nk +, x nk +1) + H(Fx nk, Fx)) f (d( x nk +, x nk +1)) + f (H(Fx nk, Fx)) ψ( f (M (x nk, x))) + ε < f (M (x nk, x))) + ε Note that f is continuous and lim k d( x nk, Fx) = d( x, Fx), we obtain by letting k, f (d( x, Fx)) < f (d( x, Fx)) + ε This implies that d( x, Fx) = 0, so we have x Fx By the similar arguments in Theorem, we can show the existence of a common fixed point References [1] S H Hong, Fixed points of multivalued operators in ordered metric spaces with applications, Nonlinear Anal, 7, (010) [] S Banach, Surles operations dans les ensembles abstraits et leurs application aux equations integraes, Fund Math 3, (in French) (19) [3] R Kannan, Some results on fixed points, Bull Calcutta Math soc 60, (1968) [4] B E Rhoades, A comparison of various definitions of contractive mappings, Trans Amer Math Soc 6, (1977) [5] Y I Alber, S G Delabriere, Principle of weakly contractive maps in Hilbert sapces, new results in operator theory, I Gohberg, Yu Lyubich (Eds), Advances and Appl, vol 98, Birkhauser Verlag, Basel, pp 7- (1997) [6] J Harjani, K Sadarangani, Fixed point theoremsfor weakly contractive mappings in partially ordered sets, Nonlinear Anal 71, (009) [7] S B Nadler, Multi-valued contraction mappings, Pacific J Math 0(), (1969) [8] B C Dhage, Hybrid fixed point theory for strictly monotone increasing multivalued mappings with applications, Comput Math Appl 53, (007) [9] B C Dhage, A general multivalued hybrid fixed point theorem and perturbed differential inclusions, Nonlinear Anal TMA 64, (006) [10] M Shen, S H Hong, Common fixed points for generalized contractive multivalued operators in complete metric spaces, Applied Math Letters, (009) [11] G S Ladde, V Lakshmikantham, A S Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, New York, (1985) [1] S S Chang, Y H Ma, Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of function equations arising in dynamic programming, J Math Anal Appl 160, (1991) [13] G Jungck and B E Rhoades, Fixed points for set valued functions without continuity, Indian J Pure Appl Math, 9 (1998), 7-38 [14] R P Agarwal, M A Ei-Gebeily, D O Regan, Generalized contractions in partially ordered metric spaces, Appl Anal 87, (008) [15] A Aliouchhe, Common fixed point theorems of Gregus type for weakly compatible mappings satisfying generalized contractive condition, J Math Anal Appl 341, (008) [16] R Caccioppoli, Un teorema generale sull esistenza di elementi uniti in una transformazione funzionale, Rend Accad dei Lincei 11, (in Italia) (1930) [17] S H Hong, Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications, J Math Anal Appl 8, (003) [18] N B Huy, N H Khanh, Note of fixed point for multivalued increasing operators, J Math Anal Appl 50, (000) [19] D Klim, D Wardowski, Fixed point theorems for set-valued contraction in complete metric space, J Math Anal Appl 334, (007) [0] V Lakshminkantham, L Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal 70, (009) [1] J J Nieto, R Rodriguez-Lopez, Applications of contractive- like mapping princibles to fuzzy equations, Rev Mat Complut 19, (006) [] D O Regan, A Petruşel, Fixed point theorems for generalized contractions in ordered metric space, J Math Anal Appl 341, (008) [3] B E Rhoades, Some theorems on weakly contractive mps, Nonlinear Anal 47, (001)
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