Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 682465, 12 pages doi:10.1155/2012/682465 Research Article Global Dynamical Systems Involving Generalized f-Projection Operators and Set-Valued Perturbation in Banach Spaces Yun-zhi Zou,1 Xi Li,2 Nan-jing Huang,2 and Chang-yin Sun1 1 2 School of Automation, Southeast University, Jiangsu, Nanjing 210096, China Department of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China Correspondence should be addressed to Nan-jing Huang, nanjinghuang@hotmail.com Received 29 February 2012; Accepted 16 May 2012 Academic Editor: Zhenyu Huang Copyright q 2012 Yun-zhi Zou et al. 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. A new class of generalized dynamical systems involving generalized f -projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value. 1. Introduction It is well known that dynamics system has long time been an interest of many researchers. This is largely due to its extremely wide applications in a huge variety of scientific fields, for instance, mechanics, optimization and control, economics, transportation, equilibrium, and so on. For details, we refer readers to references 1–10 and the references therein. In 1994, Friesz et al. 3 introduced a class of dynamics named global projective dynamics based on projection operators. Recently, Xia and Wang 7 analyzed the global asymptotic stability of the dynamical system proposed by Friesz as follows: dx PK x − ρNx − x, dt 1.1 where N : Rn → Rn is a single-valued function, ρ > 0 is a constant, PK x denotes the projection of the point x on K; here K ⊂ Rn is a nonempty, closed, and convex subset. 2 Journal of Applied Mathematics Later, in 2006, Zou et al. 9 studied a class of global set-valued projected dynamical systems as follows: dxt ∈ PK gxt − ρNxt − gxt , dt for a.a. t ∈ 0, J, 1.2 x0 b, where N : Rn → 2R is a set-valued function, g : Rn → Rn is a single-valued function, ρ > 0 is a constant, PK x denotes the projection of the point x on K, b is a given point in Rn . The concept of generalized f-projection operator was first introduced by Wu and Huang 11 in 2006. They also proved that the generalized f-projection operator is an extension of the projection operator PK in Rn and it owns some nice properties as PK does; see 12, 13. Some applications of generalized f-projection operator are also given in 11–13. Very recently, Li et al. 14 studied the stability of the generalized f-projection operator with an application in Banach spaces. We would like to point out that Cojocaru 15 introduced and studied the projected dynamical systems on infinite Hilbert spaces in 2002. To explore further dynamic systems in infinite dimensional spaces in more general forms has been one of our major motivations and efforts recently, and this paper is a response to those efforts. In this paper, we introduce and study a new class of generalized dynamical systems involving generalized f-projection operators. By using the fixed-point theorem due to Nadler 16, we prove that the equilibrium points set of this class of generalized global dynamical systems is nonempty and closed. We also show that the solutions set of the systems with set-valued perturbation is continuous with respect to the initial value. The results presented in this paper generalize many existing results in recent literatures. n 2. Preliminaries Let X be a Banach space and let K ⊂ X be a closed convex set, let N : X → 2X be a set-valued mapping, and let g : X → X be a single-valued mapping. The normalized duality mapping J from X to X∗ is defined by Jx x∗ ∈ X∗ : x, x∗ x2 x∗ 2 , 2.1 for x ∈ X. For convenience, we list some properties of J· as follows.X is a smooth Banach space, J· is single valued and hemicontinuous; that is, J is continuous from the strong topology of X to the weak ∗ topology of X∗ . Let CX denote the family of all nonempty compact subsets of X and let H·, · denote the Hausdorff metric on CX defined by HA, B max sup inf da, b, sup inf da, b , a∈A b∈B b∈B a∈A ∀A, B ∈ CX. 2.2 Journal of Applied Mathematics 3 In this paper, we consider a new class of generalized set-valued dynamical system, that is, to find those absolutely continuous functions x· from 0, h → X such that dxt f ∈ ΠK gxt − ρNxt − gxt, dt for a.a. t ∈ 0, h, 2.3 x0 b, where b ∈ X, ρ > 0 is a constant and f : K → R ∪ {∞} is proper, convex, and lower semif continuous and ΠK : X → 2K is a generalized f-projection operator denoted by f ΠK x u ∈ K : GJx, u inf GJx, ξ , ξ∈K ∀x ∈ X. 2.4 It is well known that many problems arising in the economics, physical equilibrium analysis, optimization and control, transportation equilibrium, and linear and nonlinear mathematics programming problems can be formulated as projected dynamical systems see, e.g., 1–10, 15, 17 and the references therein. We also would like to point out that problem 2.3 includes the problems considered in Friesz et al. 3, Xia and Wang 7, and Zou et al. 9 as special cases. Therefore, it is important and interesting to study the generalized projected dynamical system 2.3. Definition 2.1. A point x∗ is said to be an equilibrium point of global dynamical system 2.3, if x∗ satisfies the following inclusion: f 0 ∈ ΠK gx − ρNx − gx. 2.5 Definition 2.2. A mapping N : X → X is said to be i α-strongly accretive if there exists some α > 0 such that Nx − N y , J x − y ≥ αx − y2 , ∀x, y ∈ K; 2.6 ii ξ-Lipschitz continuous if there exists a constant ξ ≥ 0 such that Nx − N y ≤ ξ x − y , ∀x, y ∈ K. 2.7 Definition 2.3. A set-valued mapping T : X → X is said to be ξ-Lipschitz continuous if there exists a constant ξ > 0 such that H T x, T y ≤ ξ x − y , ∀x, y ∈ K, 2.8 where H·, · is the Hausdorff metric on CX. Lemma 2.4 see 14. Let X be a real reflexive and strictly convex Banach space with its dual X∗ and let K be a nonempty closed convex subset of X. If f : K → R ∪ {∞} is proper, convex, and 4 Journal of Applied Mathematics f f lower semicontinuous, then ΠK is single valued. Moreover, if X has Kadec-Klee property, then ΠK is continuous. Lemma 2.5 see 18. Let X be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and only if there exists a constant Cq > 0 such that, for all x, y ∈ X, x y q ≤ xq q y, Jq x Cq y q . 2.9 Lemma 2.6 see 19. Let X, d be a complete metric space and let T1 , T2 be two set-valued contractive mappings with same contractive constants θ ∈ 0, 1. Then HFT1 , FT2 ≤ 1 sup HT1 x, T2 x, 1 − θ x∈X 2.10 where FT1 and FT2 are fixed-point sets of T1 and T2 , respectively. Lemma 2.7 see 19. Let X be a real strictly convex, reflexive, and smooth Banach space. For any f f x1 , x2 ∈ X, let x1 ΠK x1 and x2 ΠK x2 . Then x1 − x2 , Jx1 − Jx2 , x1 − x2 ≥ 2M δ 2M 2 2.11 where M x1 2 x2 2 . 2 2.12 We say that X is 2-uniformly convex and 2-uniformly smooth Banach space if there exist k, c > 0 such that δX ≥ k2 , 2.13 ρX t ≤ ct2 , where x y : x y 1, x − y ≥ , δX inf 1 − 2 1 x y x − y − 1 : x 1, y ≤ t . ρX t sup 2 2.14 Based on Lemma 2.7, we can obtain the following lemma. Lemma 2.8. Let X be 2-uniformly convex and 2-uniformly smooth Banach space. Then c f f ΠK x − ΠK y ≤ 64 x − y , k ∀x, y ∈ X. 2.15 Journal of Applied Mathematics 5 Proof. According to Lemma 2.7, we have ⎛ f ⎞ f x − Π y Π f K K f ⎠, Jx − J y , ΠK x − ΠK y ≥ 2M12 δ⎝ 2M1 2.16 where M1 f f ΠK x2 ΠK y2 . 2 2.17 Since δX ≥ k2 , 2.16 yields 2 f f ΠK x − ΠK y ≤ Jx − J y . k 2.18 From the property of J·, we have 2M22 ρX 4 x − y /M2 Jx − J y ≤ x − y ≤ 32c x − y . 2.19 It follows from 2.18 and 2.19 that c f f ΠK x − ΠK y ≤ 64 x − y . k 2.20 This completes the proof. 3. Equilibrium Points Set In this section, we prove that the equilibrium points set of the generalized set-valued dynamical system 2.3 is nonempty and closed. Theorem 3.1. Let X be 2-uniformly convex and 2-uniformly smooth Banach space. Let N : X → CX be μ-Lipschitz continuous and let g : X → X be α-Lipschitz continuous and β-strongly accretive. If c 1 α2 − 2βC2 64 α ρμ < 1, k 3.1 then the equilibrium points set of the generalized set-valued dynamical system 2.3 is nonempty and closed. 6 Journal of Applied Mathematics Proof. Let f T x x − gx ΠK gx − ρNx , ∀x ∈ K. 3.2 f Since N : X → CX and ΠK are continuous, we know that T : X → CX. From Definition 2.1, it is easy to see that x∗ is an equilibrium point of the generalized set-valued dynamical system 2.3 if and only if x∗ is a fixed-point of T in X, that is: f x∗ ∈ T x∗ x∗ − gx∗ ΠK gx∗ − ρNx∗ . 3.3 Thus, the equilibrium points set of 2.3 is the same as the fixed-points set of T . We first prove that FT is nonempty. In fact, for any x, y ∈ X and a1 ∈ T x, there exists u ∈ Nx such that f a1 x − gx ΠK gx − ρu . 3.4 Since u ∈ Nx, and N : X → CX, it follows from Nadler 16 that there exists v ∈ Ny such that u − v ≤ H Nx, N y . 3.5 f a2 y − g y ΠK g y − ρv . 3.6 Let Then a2 ∈ T y. From 3.4 to 3.6, we have f f a1 − a2 x − y − gx − g y ΠK gx − ρu − ΠK g y − ρv f f ≤ x − y − gx − g y ΠK gx − ρu − ΠK g y − ρv . 3.7 Since g is α-Lipschitz continuous and β-strongly accretive, x − y − gx − g y 2 ≤ x − y 2 − 2 gx − g y , J x − y C2 gx − gy 2 2 ≤ 1 α2 − 2βC2 x − y . 3.8 f From Lemma 2.8, where ΠK is Lipchitz continuous, we have c f f ΠK gx − ρu − ΠK g y − ρv ≤ 64 gx − g y ρu − v k c ≤ 64 α x − y ρu − v . k 3.9 Journal of Applied Mathematics 7 From the selection of v and the Lipschitz continuity of N, u − v ≤ H Nx, N y ≤ μ x − y . 3.10 In light of 3.7–3.10, we have a1 − a2 ≤ 1 α2 c − 2βC2 64 α ρμ x − y L x − y , k 3.11 where L 1 α2 − 2βC2 64 c α ρμ . k 3.12 Now 3.11 implies that d a1 , T y inf a1 − a2 ≤ L x − y . a2 ∈T y 3.13 Since a1 ∈ T x is arbitrary, we have sup d a1 , T y ≤ L x − y . a1 ∈T x 3.14 Similarly, we can prove sup dT x, a2 ≤ L x − y . a2 ∈T y 3.15 From 3.14, 3.15, and the definition of the Hausdorff metric H on CX, we have H T x, T y ≤ L x − y , ∀x, y ∈ K. 3.16 Now the assumption of the theorem implies that L < 1 and so T x is a set-valued contractive mapping. By the fixed-point theorem of Nadler 16, there exists x∗ such that x∗ ∈ T x∗ , and thus x∗ is the equilibrium point of 2.3. This means that FT is nonempty. Now we prove that FT is closed. Let {xn } ⊂ FT with xn → x0 n → ∞. Then xn ∈ T xn and 3.16 imply that HT xn , T x0 ≤ Lxn − x0 . 3.17 Thus, dx0 , T x0 ≤ x0 − xn dxn , T xn HT xn , T x0 ≤ 1 Lxn − x0 −→ 0 as n → ∞. It follows that x0 ∈ FT and so FT are closed. This completes the proof. 3.18 8 Journal of Applied Mathematics Remark 3.2. Theorem 3.1 is a generalization of Theorem 1 in Zou et al. 9 from Rn to Banach space X. 4. Sensitivity of the Solutions Set In this section, we study the sensitivity of the solutions set of the generalized dynamical system with set-valued perturbation for 2.3 as follows: dxt f ∈ ΠK gxt − ρNxt − gxt Fxt, dt for a.a.t ∈ 0, h, 4.1 x0 b, where g and b are the same as in 2.3, F : X → 2X is a set-valued mapping, and N : X → X is a single-valued mapping. Let Sb denote the set of all solutions of 4.1 on 0, h with x0 b. Now we prove the following result. Theorem 4.1. Let X be 2-uniformly convex and 2-uniformly smooth Banach space. Let g : X → X be α-Lipschitz continuous, let N : X → X be μ-Lipschitz continuous, and let F : X → CX be a ω-Lipschitz continuous set-valued mapping with compact convex values. If 64 c α ρμ α ω < 1, k c h 64 α ρμ α ω < 1, k 4.2 then Sb is nonempty and continuous. Proof. Let f Mx ΠK gx − ρNx − gx Fx. 4.3 Then M : X → CX is a set-valued mapping with compact convex values since F : X → CX is a set-valued mapping with compact convex values. For any x1 , x2 ∈ X and a1 ∈ Mx1 , there exists u ∈ Fx1 such that f a1 ΠK gx1 − ρNx1 − gx1 u. 4.4 Since u ∈ Fx1 , and F : X → CX, it follows from Nadler 16 that there exists v ∈ Fx2 such that u − v ≤ HFx1 , Fx2 . 4.5 f a2 ΠK gx2 − ρNx2 − gx2 v. 4.6 Let Journal of Applied Mathematics 9 Then a2 ∈ Mx2 . From 4.4 and 4.6, we have f f a1 − a2 ΠK gx1 − ρNx1 − ΠK gx2 − ρNx2 − gx2 − gx2 u − v f f ≤ gx1 − gx2 ΠK gx − ρNx1 − ΠK g y − ρNx2 u − v. 4.7 Since g is α-Lipschitz continuous, gx1 − gx2 ≤ αx2 − x2 . 4.8 f From Lemma 2.8, ΠK is Lipschitz continuous. It follows from the continuity of N and g that c f f ΠK gx1 − ρNx1 −ΠK gx2 −ρNx2 ≤ 64 gx1 −gx2 ρNx1 −Nx2 k c ≤ 64 α ρμ x1 − x2 . k 4.9 From the selection of v and the Lipschitz continuity of F, we know u − v ≤ HFx1 , Fx2 ≤ ωx1 − x2 . 4.10 In light of 4.7–4.10, we have a1 − a2 ≤ α 64 c α ρμ ω x1 − x2 θ x − y , k 4.11 where θ α 64 c α ρμ ω. k 4.12 Now 4.11 implies that da1 , Mx2 inf a1 − a2 ≤ θx1 − x2 . a2 ∈Mx2 4.13 Since a1 ∈ Mx1 is arbitrary, we obtain sup da1 , Mx2 ≤ θx1 − x2 . 4.14 sup dMx1 , a2 ≤ θx1 − x2 . 4.15 a1 ∈Mx1 Similarly, we can prove a2 ∈Mx2 10 Journal of Applied Mathematics From 4.13, to 4.15, and the definition of the Hausdorff metric H on CX, we have HMx1 , Mx2 ≤ θx1 − x2 , ∀x1 , x2 ∈ X. 4.16 Now 4.2 implies that 0 < θ < 1, and so Mx is a set-valued contractive mapping. Let Qx, b y ∈ C0, h, X | yt b t zsds, zs ∈ Mxs , 4.17 0 where C0, h, X f : 0, h −→ X | f is continuous . 4.18 Since M : X → CX is a continuous set-valued mapping with compact convex values, by the Michael’s selection theorem see, e.g., Theorem 16.1 in 20, we know that Qx, b is nonempty for each x and b ∈ X. Moreover, it is easy to see that the set of fixed-points of Qx, b coincides with Sb. It follows from 21 or 8 that Qx, b is compact and convex for each x and b ∈ X. Suppose that bm is the initial value of 4.1; that is, x0 bm m 0, 1, 2, . . . and bm → b0 m → ∞. Since Qx, b0 Qx, bm − bm b0 , 4.19 it is obvious that Qx, bm converges uniformly to Qx, b0 . Next we prove that Qx, bm is a set-valued contractive mapping. For any given x1 , x2 ∈ C0, h, X, since M : X → CX is a continuous set-valued mapping with compact convex values, by the Michael’s selection theorem see, e.g., Theorem 16.1 in 20, we know that Mx1 s has a continuous selection r1 s ∈ Mx1 s. Let c1 t bm t r1 sds. 4.20 0 Then c1 ∈ Qx1 , bm . Since r1 s ∈ Mx1 s is measurable and Mx2 s is a measurable mapping with compact values, we know that there exists a measurable selection r2 s ∈ Mx2 s such that r1 s − r2 s ≤ HMx1 s, Mx2 s. 4.21 Thus, it follows from 4.16 that r1 s − r2 s ≤ HMx1 s, Mx2 s ≤ θx1 − x2 . 4.22 Let c2 t bm t 0 r2 sds. 4.23 Journal of Applied Mathematics 11 Then c2 ∈ Qx2 , bm and c1 − c2 ≤ h r1 s − r2 sds ≤ hHMx1 s, Mx2 s 0 4.24 ≤ hθx1 − x2 . Hence, we have dc1 , Qx2 , bm inf c2 ∈Qx2 ,bm c1 − c2 ≤ hθx1 − x2 . 4.25 Since c1 ∈ Qx1 , bm is arbitrary, we obtain sup c1 ∈Qx1 ,bm dc1 , Qx2 , bm ≤ hθx1 − x2 . 4.26 dQx1 , bm , c2 ≤ hθx1 − x2 . 4.27 Similarly, we can prove that sup c2 ∈Qx2 ,bm From the definition of the Hausdorff metric H on CX, 4.26 and 4.27 imply that HQx1 , bm , Qx2 , bm ≤ hθx1 − x2 , ∀x1 , x2 ∈ X, m 0, 1, 2, . . . . 4.28 Since hθ < 1, it is easy to see that Qx, b has a fixed-point for each given b ∈ X, and so Sb is nonempty for each given b ∈ X. Setting Wm x Qx, bm , m 0, 1, 2, . . . , 4.29 we know that Wm x are contractive mappings with the same contractive constant hθ. By Lemma 2.6 and 4.28, we have HFWm , FW0 ≤ 1 sup HWm x, W0 x −→ 0. 1 − hθ x∈X 4.30 Thus, FWm → FW0 , which implies that Sbm → Sb; that is, the solution of 4.1 is continuous with respect to the initial value of 4.1. This completes the proof. Remark 4.2. Theorem 4.1 is a generalization of Theorem 2 in Zou et al. 9 from Rn to Banach space X. Acknowledgments The authors appreciate greatly the editor and two anonymous referees for their useful comments and suggestions, which have helped to improve an early version of the paper. This 12 Journal of Applied Mathematics work was supported by the National Natural Science Foundation of China 11171237 and the Key Program of NSFC Grant no. 70831005. References 1 J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1984. 2 P. Dupuis and A. Nagurney, “Dynamical systems and variational inequalities,” Annals of Operations Research, vol. 44, no. 1–4, pp. 9–42, 1993, Advances in equilibrium modeling, analysis and computation. 3 T. L. Friesz, D. Bernstein, N. J. Mehta, R. L. Tobin, and S. Ganjalizadeh, “Day-to-day dynamic network disequilibria and idealized traveler information systems,” Operations Research, vol. 42, no. 6, pp. 1120– 1136, 1994. 4 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1992. 5 G. Isac, “Some solvability theorems for nonlinear equations with applications to projected dynamical systems,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 1, pp. 3–13, 2009. 6 Y. S. Xia, “Further results on global convergence and stability of globally projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 122, no. 3, pp. 627–649, 2004. 7 Y. S. Xia and J. Wang, “On the stability of globally projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 106, no. 1, pp. 129–150, 2000. 8 D. Zhang and A. Nagurney, “On the stability of projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 85, no. 1, pp. 97–124, 1995. 9 Y. Z. Zou, K. Ding, and N. J. Huang, “New global set-valued projected dynamical systems,” Impulsive Dynamical Systems and Applications, vol. 4, pp. 233–237, 2006. 10 Y.-z. Zou, N.-j. Huang, and B.-S. Lee, “A new class of generalized global set-valued dynamical systems involving H, η-monotone operators in Hilbert spaces,” Nonlinear Analysis Forum, vol. 12, no. 2, pp. 183–191, 2007. 11 K.-q. Wu and N.-j. Huang, “The generalised f-projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006. 12 K.-q. Wu and N.-j. Huang, “Properties of the generalized f-projection operator and its applications in Banach spaces,” Computers & Mathematics with Applications, vol. 54, no. 3, pp. 399–406, 2007. 13 K.-Q. Wu and N.-J. Huang, “The generalized f-projection operator and set-valued variational inequalities in Banach spaces,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2481–2490, 2009. 14 X. Li, N. J. Huang, and Y. Z. Zou, “On the stability of generalized f-projection operators with an application,” Acta Mathematica Sinica, vol. 54, pp. 1–12, 2011. 15 M. G. Cojocaru, in Projected dynamical systems on Hilbert spaces [Ph.D. thesis], Queen’s University, Kingston, Canada, 2002. 16 S. B. Nadler, Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475– 488, 1969. 17 M.-G. Cojocaru, “Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 3, pp. 793–804, 2006. 18 H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis, vol. 16, no. 12, pp. 1127–1138, 1991. 19 T.-C. Lim, “On fixed point stability for set-valued contractive mappings with applications to generalized differential equations,” Journal of Mathematical Analysis and Applications, vol. 110, no. 2, pp. 436–441, 1985. 20 L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, vol. 495 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1999. 21 Y. Xia and J. Wang, “A general projection neural network for solving monotone variational inequalities and related optimization problems,” IEEE Transactions on Neural Networks, vol. 15, no. 2, pp. 318–328, 2004. Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Algebra Hindawi Publishing Corporation http://www.hindawi.com Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Journal of Hindawi Publishing Corporation http://www.hindawi.com Stochastic Analysis Abstract and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com International Journal of Mathematics Volume 2014 Volume 2014 Discrete Dynamics in Nature and Society Volume 2014 Volume 2014 Journal of Journal of Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Applied Mathematics Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014