Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 432130, 16 pages doi:10.1155/2009/432130 Research Article Fixed Point Results for Generaliz ed Contractive Multimaps in Metric Spaces Abdul Latif 1 and Afrah A. N. Abdou 2 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Girls College of Education, King Abdulaziz University, P.O. Box 14884, Jeddah 21434, Saudi Arabia Correspondence should be addressed to Abdul Latif, latifmath@yahoo.com Received 17 May 2009; Accepted 10 August 2009 Recommended by Mohamed A. Khamsi The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, our results either generalize or improve a number of fixed point results including the corresponding recent fixed point results of Ciric 2008, Latif-Albar 2008, Klim-Wardowski 2007, and Feng-Liu 2006. Examples are also given. Copyright q 2009 A. Latif and A. A. N. Abdou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and Preliminaries Let X, d be a metric space, 2 X a collection of nonempty subsets of X, CBX a collection of nonempty closed bounded subsets of X, ClX a collection of nonempty closed subsets of X, KX a collection of nonempty compact subsets of X and H the Hausdorff metric induced by d. Then for any A, B ∈ CBX, H  A, B   max  sup x∈A d  x, B  , sup y∈B d  y, A   , 1.1 where dx, Binf y∈B dx, y. An element x ∈ X is called a fixed point of a multivalued map T : X → 2 X if x ∈ Tx. We denote FixT{x ∈ X : x ∈ Tx}. A sequence {x n } in X is called an orbit of T at x 0 ∈ X if x n ∈ Tx n−1  for all n ≥ 1. A map f : X → R is called lower semicontinuous if for any sequence {x n }⊂X with x n → x ∈ X it implies that fx ≤ lim inf n →∞ fx n . 2 Fixed Point Theory and Applications Using the concept of Hausdorff metric, Nadler 1 established the f ollowing fixed point result for multivalued contraction maps, known as Nadler’s contraction principle which in turn is a generalization of the well-known Banach contraction principle. Theorem 1.1 see 1. Let X, d be a complete metric space and let T : X → CBX be a contraction map. Then FixT /  ∅. Using the concept of the Hausdorff metric, many authors have generalized Nadler’s contraction principle in many directions. But, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorff metric. Recently, Feng and Liu 2 extended Nadler’s fixed point theorem without using the concept of Hausdorff metric. They proved the following result. Theorem 1.2. Let X, d be a complete metric space and let T : X → ClX be a map such that for any fixed constants h, b ∈ 0, 1,h<b,and for each x ∈ X there is y ∈ Tx satisfying the following conditions: bd  x, y  ≤ d  x, T  x  , d  y, T  y  ≤ hd  x, y  . 1.2 Then FixT /  ∅ provided a real-valued function g on X, gxdx, Tx is lower semicontinuous. Recently, Klim and Wardowski 3 generalized Theorem 1.2 and proved the following two results. Theorem 1.3. Let X, d be a complete metric space and let T : X → ClX. Assume that the following conditions hold: i there exist a number b ∈ 0, 1 and a function k : 0, ∞ → 0,b such that for each t ∈ 0, ∞, lim sup r → t  k  r  <b, 1.3 ii for any x ∈ X there is y ∈ Tx satisfying bd  x, y  ≤ d  x, T  x  , d  y, T  y  ≤ k  d  x, y  d  x, y  . 1.4 Then FixT /  ∅ provided a real-valued function g on X, gxdx, Tx is lower semicontinuous. Theorem 1.4. Let X, d be a complete metric space and let T : X → KX. Assume that the following conditions hold: i there exists a function k : 0, ∞ → 0, 1 such that for each t ∈ 0, ∞, lim sup r → t  k  r  < 1, 1.5 Fixed Point Theory and Applications 3 ii for any x ∈ X there is y ∈ Tx satisfying d  x, y   d  x, T  x  , d  y, T  y  ≤ k  d  x, y  d  x, y  . 1.6 Then FixT /  ∅ provided a real-valued function g on X, gxdx, Tx is lower semicontinuous. Note that Theorem 1.3 generalizes Nadler’s contraction principle and Theorem 1.2. Most recently, Ciric 4 obtained some interesting fixed point results which extend and generalize the cited results. Namely, 4, T heorem 5 generalizes 5, Theorem 5, 4, Theorem 6 generalizes  4, Theorems 1.2, 1.3,and3, theorem 7 generalizes Theorem 1.4. In 6, Kada et al. introduced the concept of w-distance on a metric space as follows: A function ω : X × X → 0, ∞ is called w-distance on X if it satisfies the following for each x, y, z ∈ X: w 1  ωx, z ≤ ωx, yωy, z; w 2  a map ωx, · : X → 0, ∞ is lower semicontinuous; that is, if a sequence {y n } in X with y n → y ∈ X, then ωx, y ≤ lim inf n →∞ ωx, y n ; w 3  for any >0, there exists δ>0 such that ωz, x ≤ δ and ωz, y ≤ δ imply dx, y ≤ . Note that, in general for x, y ∈ X, ωx, y /  ωy, x and not either of the implications ωx, y0 ⇔ x  y necessarily hold. Clearly, the metric d is a w-distance on X.LetY, · be a normed space. Then the functions ω 1 ,ω 2 : Y ×Y → 0, ∞ defined by ω 1 x, yy and ω 2 x, yx  y for all x, y ∈ Y are w-distances 6. Many other examples and properties of the w-distance can be found in 6, 7 . The following lemma is crucial for the proofs of our results. Lemma 1.5 see 8. Let K be a closed subset of X and ω be a w-distance on X. Suppose that there exists u ∈ X such that ωu, u0.Thenωu, K0 ⇔ u ∈ K, where ωu, Kinf y∈K ωu, y. Most recently, the authors of this paper generalized Latif and Albar 9, Theorem 1.3 as follows. Theorem 1.6 see 10. Let X, d be a complete metric space with a w-distance ω.LetT : X → ClX be a multivalued map satisfying that for any constant b ∈ 0, 1 and for each x ∈ X there is y ∈ J x b such that ω  y, T  y  ≤ k  ω  x, y  ω  x, y  , 1.7 where J x b  {y ∈ Tx : bωx, y ≤ ωx, Tx} and k is a function from 0, ∞ to 0,b with lim sup r → t  kr <b,for every t ∈ 0, ∞. Suppose that a real-valued function g on X defined by gxωx, Tx is lower semicontinuous. Then there exists v o ∈ X such that gv o 0. Further, if ωv o ,v o 0, then v 0 ∈ FixT. The aim of this paper is to present some more general results on the existence of fixed points for multivalued maps satisfying certain conditions. Our results unify and generalize 4 Fixed Point Theory and Applications the corresponding results of Mizoguchi and Takahashi 5, Klim and Wardowski 3,Latif and Abdou 10, Ciric 4, Feng and Liu 2, Latif and Albar 9 and several others. 2. The Results First we prove a theorem which is a generalization of Ciric 4, Theorem 5 and due to Klim and Wardowski 3, Theorem 1.4. Theorem 2.1. Let X, d be a complete metric space with a w-distance ω. Let T : X → ClX be a multivalued map. Assume that the following conditions hold: i there exists a function ϕ : 0, ∞ → 0, 1 such that for each t ∈ 0, ∞ lim sup r → t  ϕ  r  < 1 2.1 ii for any x ∈ X, there exists y ∈ T x satisfying ω  x, y  ≤  2 − ϕ  ω  x, y  ω  x, T  x  , ω  y, T  y  ≤ ϕ  ω  x, y  ω  x, y  2.2 iii the map f : X → R, defined by fxωx, Tx is lower semicontinuous. Then there exists v 0 ∈ X such that fv 0 0. Further if ωv 0 ,v 0 0, then v 0 ∈ Tv 0 . Proof. let x 0 ∈ X be any initial point. Then there exists x 1 ∈ Tx 0  such that ω  x 0 ,x 1  ≤  2 − ϕ  ω  x 0 ,x 1   ω  x 0 ,T  x 0  , ω  x 1 ,T  x 1  ≤ ϕ  ω  x 0 ,x 1  ω  x 0 ,x 1  . 2.3 From 2.3 we get ω  x 1 ,T  x 1  ≤ ϕ  ω  x 0 ,x 1   2 − ϕ  ω  x 0 ,x 1   ω  x 0 ,T  x 0  . 2.4 Define a function ψ : 0, ∞ → 0, ∞ by ψ  t   ϕ  t   2 − ϕ  t    1 −  1 − ϕt  2 . 2.5 Using the facts that for each t ∈ 0, ∞,ϕt < 1 and lim r → t  sup ϕr < 1, we have ψ  t  < 1 , 2.6 lim sup r → t  ψ  r  < 1 ∀t ∈  0, ∞  2.7 Fixed Point Theory and Applications 5 From 2.4 and 2.5, we have ω  x 1 ,T  x 1  ≤ ψ  ω  x 0 ,x 1  ω  x 0 ,T  x 0  . 2.8 Similarly, for x 1 ∈ X, there exists x 2 ∈ Tx 1  such that ω  x 1 ,x 2  ≤  2 − ϕ  ω  x 1 ,x 2   ω  x 1 ,T  x 1  , ω  x 2 ,T  x 2  ≤ ϕ  ω  x 1 ,x 2  ω  x 1 ,x 2  . 2.9 Thus ω  x 2 ,T  x 2  ≤ ψ  ω  x 1 ,x 2  ω  x 1 ,T  x 1  . 2.10 Continuing this process we can get an orbit {x n } of T in X satisfying the following: ω  x n ,x n1  ≤  2 − ϕ  ω  x n ,x n1   ω  x n ,T  x n  , 2.11 ω  x n1 ,T  x n1  ≤ ψ  ω  x n ,x n1  ω  x n ,T  x n  , 2.12 for each integer n ≥ 0. Since ψt < 1 for each t ∈ 0, ∞ and from 2.12, we have for all n ≥ 0 ω  x n1 ,T  x n1  <ω  x n ,T  x n  . 2.13 Thus the sequence of nonnegative real numbers {ωx n ,Tx n } is decreasing and bounded below, thus convergent. Therefore, there is some δ ≥ 0 such that lim n →∞ ω  x n ,T  x n   δ. 2.14 From 2.11,as ϕt < 1 for all t ≥ 0, we get ω  x n ,T  x n  ≤ ω  x n ,x n1  < 2ω  x n ,T  x n  , 2.15 Thus, we conclude that the sequence of nonnegative reals {ωx n ,x n1 } is bounded. Therefore, there is some θ ≥ 0 such that lim inf n →∞ ω  x n ,x n1   θ. 2.16 Note that ωx n ,x n1  ≥ ωx n ,Tx n  for each n ≥ 0, so we have θ ≥ δ. Now we will show that θ  δ. Suppose that δ  0. Then we get lim n →∞ ω  x n ,x n1   0. 2.17 6 Fixed Point Theory and Applications Now consider δ>0. Suppose to the contrary, that θ>δ.Then θ − δ>0andsofrom2.14 and 2.16 there is a positive integer n 0 such that ω  x n ,T  x n  <δ θ − δ 4 ∀n ≥ n 0 , 2.18 θ − θ − δ 4 <ω  x n ,x n1  ∀n ≥ n 0 . 2.19 Then from 2.19, 2.11 and 2.18,weget θ − θ − δ 4 <ω  x n ,x n1  ≤  2 − ϕ  ω  x n ,x n1   ω  x n ,T  x n  <  2 − ϕ  ω  x n ,x n1    δ  θ − δ 4  . 2.20 Thus for all n ≥ n 0 ,  2 − ϕ  ω  x n ,x n1   > 3θ  δ 3δ  θ , 2.21 that is, 1   1 − ϕ  ω  x n ,x n1   > 1  2  θ − δ  3δ  θ , 2.22 and we get −  1 − ϕ  ωx n ,x n1    2 < −  2θ − δ 3δ  θ  2 . 2.23 Thus for all n ≥ n 0 , ψ  ω  x n ,x n1   1 −  1 − ϕ  ωx n ,x n1    2 < 1 −  2θ − δ 3δ  θ  2 . 2.24 Thus, from 2.12 and 2.24,weget ω  x n1 ,T  x n1  ≤ hω  x n ,T  x n  ∀n ≥ n 0 , 2.25 Fixed Point Theory and Applications 7 where h  1 − 2θ − δ/3δ  θ 2 . Clearly h<1asθ>δ.From 2.18 and 2.25, we have for any k ≥ 1 ω  x n 0 k ,T  x n 0 k  ≤ h k ω  x n 0 ,T  x n 0  . 2.26 Since δ>0andh<1, there is a positive integer k 0 such that h k 0 ωx n 0 ,Tx n 0  <δ.Now, since δ ≤ ωx n ,Tx n  for each n ≥ 0, by 2.26 we have δ ≤ ω  x n 0 k 0 ,T  x n 0 k 0  ≤ h k 0 ω  x n 0 ,T  x n 0  <δ. 2.27 a contradiction. Hence, our assumption θ>δis wrong. Thus δ  θ. Now we will show that θ  0. Since θ  δ ≤ ωx n ,Tx n  ≤ ωx n ,x n1 , then from 2.16 we can read as lim inf n →∞ ω  x n ,x n1   θ, 2.28 so, there exists a subsequence {ωx n k ,x n k 1 } of {ωx n ,x n1 } such that lim k →∞ ω  x n k ,x n k 1   θ  . 2.29 Now from 2.7 we have lim sup ωx n k ,x n k 1  → θ ψ  ω  x n k ,x n k 1  < 1, 2.30 and from 2.12, we have ω  x n k ,T  x n k 1  ≤ ψ  ω  x n k ,x n k 1  ω  x n k ,T  x n k  2.31 Taking the limit as k →∞and using 2.14,weget δ  lim sup k →∞ ω  x n k1 ,T  x n k 1  ≤  lim sup k →∞ ψ  ω  x n k1 ,x n k 1   lim sup k →∞ ω  x n k ,T  x n k    ⎛ ⎝ lim sup ωx n k ,x n k 1  → θ ψ  ω  x n k ,x n k 1  ⎞ ⎠ δ. 2.32 If we suppose that δ>0, then from last inequality, we have lim sup ωx n k ,x n k 1  → θ ψ  ω  x n k ,x n k 1  ≥ 1, 2.33 8 Fixed Point Theory and Applications which contradicts with 2.30.Thusδ  0. Then from 2.14 and 2.15, we have lim n →∞ ω  x n ,T  x n   0, 2.34 and thus lim n →∞ ω  x n ,x n1   0  . 2.35 Now, let α  lim ωx n k ,x n k 1  → 0  sup ψ  ω  x n k ,x n k 1  . 2.36 Then by 2.7, α<1. Let q be such that α<q<1. Then there is some n 0 ∈ N such that ψ  ω  x n ,x n1  <q ∀n ≥ n 0 . 2.37 Thus it follows from 2.12, ω  x n1 ,T  x n1  ≤ qω  x n ,T  x n  ∀n ≥ n 0 . 2.38 By induction we get ω  x n1 ,T  x n1  ≤ q n1−n 0 ω  x n 0 ,T  x n 0  ∀n ≥ n 0 . 2.39 Now, using 2.15 and 2.39, we have ω  x n ,x n1  ≤ 2q n−n 0 ω  x n 0 ,T  x n 0  ∀n ≥ n 0 . 2.40 Now, we show that {x n } is a Cauchy sequence, for all m>n≥ n 0 , we get ω  x n ,x m  ≤ m−1  kn ω  x k ,x k1  ≤ 2 m−1  kn q k−n 0 ω  x n 0 ,T  x n 0  ≤ 2  q n−n 0 1 − q  ω  x n 0 ,T  x n 0  . 2.41 Hence we conclude, as q<1, that {x n } is Cauchy sequence. Due to the completeness of X, there exists some v 0 ∈ X such that lim n →∞ x n  v 0 . Since f is lower semicontinuous and from 2.34, we have 0 ≤ f  v 0  ≤ lim inf n →∞ f  x n   ω  x n ,T  x n   0, 2.42 Fixed Point Theory and Applications 9 and thus, fv 0 ωv 0 ,Tv 0   0. Since ωv 0 ,v 0 0, and Tv 0  is closed, it follows from Lemma 1.5 that v 0 ∈ Tv 0 . We also have the following interesting result by replacing the hypothesis iii of Theorem 2.1 with another natural condition. Theorem 2.2. Suppose that all the hypotheses of Theorem 2.1 except (iii) hold. Assume that inf { ω  x, v   ω  x, T  x  : x ∈ X } > 0, 2.43 for every v ∈ X with v / ∈ Tv. Then FixT /  ∅. Proof. Following the proof of Theorem 2.1, there exists a Cauchy sequence {x n } with x n ∈ Tx n−1 . Due to the completeness of X, there exists v 0 ∈ X such that lim n →∞ x n  v 0 . Since ωx, · is lower semicontinuous and x m → v 0 ∈ X, it follows for all n ≥ n 0 ω  x n ,v 0  ≤ lim m →∞ inf ω  x n , x m  ≤  2q n−n 0 1 − q  ω  x n 0 ,T  x n 0  , ω  x n ,T  x n  ≤ ω  x n ,x n1  ≤ 2q n−n 0 ω  x n 0 ,T  x n 0  . 2.44 Assume that v 0 / ∈ Tv 0 . Then, we have 0 < inf { ω  x, v 0   ω  x, T  x  : x ∈ X } ≤ inf { ω  x n ,v 0   ω  x n ,T  x n  : n ≥ n 0 } ≤ inf  2q n−n 0 1 − q  ω  x n 0 ,T  x n 0   2q n−n 0 ω  x n 0 ,T  x n 0  : n ≥ n 0   2  2 − q   1 − q  q n 0 ω  x n 0 ,T  x n 0  inf  q n : n ≥ n 0   0, 2.45 which is impossible and hence v 0 ∈ FixT. Now, we present an improved version of Ciric 4, Theorem 6 and which also generalizes due to Latif and Abdou 10, Theorem 1.6 and due to Klim and Wardowski 3, Theorem 1.3. Theorem 2.3. Let X, d be a complete metric space with a w-distance ω. Let T : X → ClX,bea multivalued map. Assume that the following condition hold: i there exist functions ϕ : 0, ∞ → 0, 1 and μ : 0, ∞ → b, 1, with b>0,μ nondecreasing such that ϕ  t  <μ  t  , lim sup r → t  ϕ  r  < lim sup r → t  μ  r  , 2.46 10 Fixed Point Theory and Applications ii for any x ∈ X, there exists y ∈ T x satisfying the following conditions: μ  ω  x, y  ω  x, y  ≤ ω  x, T  x  , ω  y, T  y  ≤ ϕ  ω  x, y  ω  x, y  , 2.47 iii the map f : X → R, defined by fxωx, Tx is lower semicontinuous. Then there exists v 0 ∈ X such that fv 0 0. Further if ωv 0 ,v 0 0, then v 0 ∈ Tv 0 . Proof. Let x 0 be an arbitrary, then there exists x 1 ∈ Tx 0  such that μ  ω  x 0 ,x 1  ω  x 0 ,x 1  ≤ ω  x 0 ,T  x 0  , ω  x 1 ,T  x 1  ≤ ϕ  ω  x 0 ,x 1  ω  x 0 ,x 1  . 2.48 From 2.48 we have ω  x 1 ,T  x 1  ≤ ϕ  ω  x 0 ,x 1  μ  ω  x 0 ,x 1  ω  x 0 ,T  x 0  . 2.49 Define a function ψ : 0, ∞ → 0, ∞ by ψ  t   ϕ  t  μ  t  ∀t ∈  0, ∞  . 2.50 Since ϕt <μt, we have ψ  t  < 1, 2.51 lim sup r → t  ψ  r  < 1 ∀t ∈  0, ∞  . 2.52 Thus from 2.49 ω  x 1 ,T  x 1  ≤ ψ  ω  x 0 ,x 1  ω  x 0 ,T  x 0  . 2.53 Similarly, there exists x 2 ∈ Tx 1  such that μ  ω  x 1 ,x 2  ω  x 1 ,x 2  ≤ ω  x 1 ,T  x 1  , ω  x 2 ,T  x 2  ≤ ϕ  ω  x 1 ,x 2  ω  x 1 ,x 2  . 2.54 Then by definition of ψ, we get ω  x 2 ,T  x 2  ≤ ψ  ω  x 1 ,x 2  ω  x 1 ,T  x 1  . 2.55 [...]... Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings,” Journal of Mathematical Analysis and Applications, vol 317, no 1, pp 103–112, 2006 3 D Klim and D Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol 334, no 1, pp 132–139, 2007 4 L Ciric, Fixed point theorems... Theory, Methods & Applications, vol 67, no 1, pp 187–199, 2007 9 A Latif and W A Albar, Fixed point results in complete metric spaces,” Demonstratio Mathematica, vol 41, no 1, pp 145–150, 2008 10 A Latif and A A N Abdou, Fixed points of generalized contractive maps,” Fixed Point Theory and Applications, vol 2009, Article ID 487161, 9 pages, 2009 ... minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol 44, no 2, pp 381–391, 1996 7 W Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000 8 L.-J Lin and W.-S Du, “Some equivalent formulations of the generalized Ekeland’s variational principle and their applications,” Nonlinear Analysis: Theory,... x2 , ⎪ ⎪ 2 ⎨ ⎪ 17 1 ⎪ ⎪ ⎩ , , 96 4 for x ∈ 0, for x 15 32 15 32 ∪ 15 ,1 , 32 3.2 14 Fixed Point Theory and Applications Define now ϕ : 0, ∞ → 0, 1 as follows ⎧8 ⎪ t, ⎪ ⎪5 ⎨ ⎪4 ⎪ ⎪ , ⎩ 5 ϕt for t ∈ 0, 1 for t ∈ ,∞ 2 1 , 2 3.3 Note that f x ⎧ ⎪ 1 x2 , ⎪ ⎪ ⎨2 ⎪ 17 ⎪ ⎪ ⎩ , 96 ω x, T x for x ∈ 0, 15 32 ∪ 15 ,1 , 32 3.4 15 32 for x and f is lower semicontinuous Moreover for each x ∈ 0, 15/32 ∪ 15/32, 1 we... point theorems for multi-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol 348, no 1, pp 499–507, 2008 5 N Mizoguchi and W Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces,” Journal of Mathematical Analysis and Applications, vol 141, no 1, pp 177–188, 1989 6 O Kada, T Suzuki, and W Takahashi, “Nonconvex minimization... decreasing Now let lim sup ψ ω xn , xn n→∞ 1 α 2.62 Thus by 2.52 , α < 1 Then for any q ∈ α, 1 , there exists n0 ∈ N such that ψ ω xn , xn 1 . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 432130, 16 pages doi:10.1155/2009/432130 Research Article Fixed Point Results for Generaliz ed Contractive Multimaps. of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, our results. general results on the existence of fixed points for multivalued maps satisfying certain conditions. Our results unify and generalize 4 Fixed Point Theory and Applications the corresponding results