Strong convergence theorems of Cesàrotype means for nonexpansive mapping in CAT(0) space


 Ami Harrell
 1 years ago
 Views:
Transcription
1 Tang et al. Fixed Point Theory and Applications :00 DOI 0.86/s y R E S E A R C H Open Access Strong convergence theorems of Cesàrotype means for nonexpansive mapping in CAT0 space Jinfang Tang, Shihsen Chang 2* and Jian Dong * Correspondence: 2 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 65022, China Full list of author information is available at the end of the article Abstract The iterative algorithms with Cesàrotype means for a nonexpansive mapping are proposed in CAT0 spaces. Under suitable conditions, some strong convergence theorems for the sequence generated by the algorithms to a fixed point of the nonexpansive mapping are proved. We also proved that this fixed point is also a unique solution to some kind of variational inequality. The results presented in this paper extend and improve the corresponding results of some others. Keywords: Cesàrotype means method; CAT0 space; fixed point Introduction In 975, Baillon [] first proved the following nonlinear ergodic theorem. Theorem. Suppose that C is a nonempty closed convex subset of a Hilbert space E and T : C C is a nonexpansive mapping such that FT. Then, x C, thecesàromeans T n x = n + T i x. weakly converges to a fixed point of T. In 979, Bruck [2] generalized the Baillon Cesàro s means theorem from Hilbert space to uniformly convex Banach space with Fréchet differentiable norms. In 2007, Song and Chen [3] defined the following viscosity iteration {x n } of the Cesàro means for nonexpansive mapping T: x n+ = α n f x n + α n n + T i x,.2 and they proved that the sequence {x n } converged strongly to some point in FT ina uniformly convex Banach space with weakly sequentially continuous duality mapping. In 2009, Yao et al. [4] in a Banach space introduced the following process {x n }: x n+ = α n u + β n x n + γ n Tx n, n Tang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Tang et al. Fixed Point Theory and Applications :00 Page 2 of 3 They proved that the sequence {x n } converged strongly to a fixed point of T under suitable control conditions of parameters. In 204, Zhu and Chen [5] proposed the following iterations with Cesàro s means for nonexpansive mappings: x n+ = α n u + β n x n + γ n n + and viscosity iteration: T i x n, n 0,.4 x n+ = α n f x n +β n x n + γ n n + T i x n, n 0..5 They proved the sequences {x n } both converged strongly to a fixed point of FT inthe framework of a uniformly smooth Banach space. As is well known, an extension of a linear version usually in Banach spaces or Hilbert spaces of this wellknown result to a metric space has its own importance. The above iterative methods..5 involve general convex combinations. If we want to extend these results from Hilbert spaces or Banach spaces to metric spaces, we need some convex structure in a metric space to investigate their convergence on a nonlinear domain. On the other hand, recently the theory and applications of CAT0 space have been studied extensively by many authors. Recall that a metric space X, d is called a CAT0 space, if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT0 space. Other examples of CAT0 spaces include prehilbert spaces see [6], Rtrees see [7], Euclidean buildings see [8], the complex Hilbert ball with a hyperbolic metric see [9], and many others. A complete CAT0 space is often called a Hadamard space. A subset K of a CAT0 space X is convex if, for any x, y K, wehave[x, y] K, where[x, y] is the uniquely geodesic joining x and y. For a thorough discussion of CAT0 spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [6]. Fixed point theory in CAT0 spaces has been first studied by Kirk see [0, ]. He showed that every nonexpansive singlevalued mapping defined on a bounded closed convex subset of a complete CAT0 space always has a fixed point. Motivated and inspired by the research going on in this direction, it is naturally to put forward the following. Open question Can we extend the above Cesàro nonlinear ergodic theorems for nonexpansive mapping in [ 5] from a Hilbert space or Banach space to a CAT0 space? The purpose of this paper is to give an affirmative answer to this question. For solving this problem we first propose the following new iterations with Cesàro s means for nonexpansive mappings in the setting of CAT0 spaces: x n+ = α n u β n x n γ n n + T i x n,.6
3 Tang et al. Fixed Point Theory and Applications :00 Page 3 of 3 and the viscosity iteration: x n+ = α n f x n β n x n γ n n + T i x n,.7 and then study the strong convergence of the iterative sequences.6 and.7. Under suitable conditions, some strong converge theorems to a fixed point of the nonexpansive mapping are proved. We also prove that this fixed point is a unique solution of some kind of variational inequality in CAT0 spaces. The results presented in this paper are new; they extend and improve corresponding previous results. Next we give some examples of iteration.6 or.7 with Cesàro s means for nonexpansive mappings to illustrate the generation of our new iterations. Example If X is a real Hilbert or Banach space, and if α n =0,β n =0, n, then the iteration.6 with Cesàro s means for nonexpansive mappings is reduced as the same iteration of Baillon []orbruck[2]. Example 2 If X is a real Banach space and n =, then the iteration.6 withcesàro s means for nonexpansive mappings is reduced to the same iteration of Yao et al. [4]. If n 2, then the iteration.6canbewritten x n+ = α n u + β n x n + γ n n + T i x n, which is a generalization of results due to Yao et al. [4]. Example 3 The iterations.6 and.7 are a generalization of the iterations of Zhu and Chen [5] from Banach space to CAT0 spaces. Example 4 Let E = R with the usual metric, T be a nonexpansive mapping defined by Tx = sin x,andf be a contractive mapping defined by f x= x 2.Letα n = 2n, β n = 4n 3 4n,and γ n =, n. Then the iteration.6 and.7 with Cesàro s means for nonexpansive 4n mappings Tx = sin x are reduced to the following iterations, respectively: x n+ = 4n 3 u + 2n 4n x n + xn + Tx n + T 2 x n + + T n x n, 4nn +.8 x n+ = 2n 2 x 4n 3 n + 4n x n + xn + Tx n + T 2 x n + + T n x n. 4nn +.9 Example 5 If X, d is a CAT0 space, n =andβ n = 0, then the iteration.7 with Cesàro s means for nonexpansive mappings is reduced to the following iteration, with Cesàro s means for nonexpansive mappings: x n+ = α n f x n γ n 2 x n γ n 2 Tx n,.0 which is a generalization of the iterations in Wangkeeree and Preechasilp [2]fromaBa nach space to CAT0 spaces.
4 Tang et al. Fixed Point Theory and Applications :00 Page 4 of 3 2 Preliminaries and lemmas In this paper, we write tx ty for the unique point z in the geodesic segment joining from x to y such that dx, z =tdx, y, dy, z = tdx, y. 2. Lemma 2. [3] AgeodesicspaceX, d is a CAT0 space, ifandonlyiftheinequality d 2 tx ty, z td 2 x, z+td 2 y, z t td 2 x, y 2.2 is satisfied for all x, y, z Xandt [0, ]. In particular, if x, y, z are points in a CAT0 space and t [0, ], then d tx ty, z tdx, z+tdy, z. Lemma 2.2 [6] Let X, d be a CAT0 space, p, q, r, s X, and λ [0, ]. Then d λp λq, λr λs λdp, r+ λdq, s. 2.3 By induction, we write n m= λ λ m x m := λ n x λ 2 x 2 λ n x n λ n x n. 2.4 λ n λ n λ n Lemma 2.3 [4] Let X, d be a CAT0 space, then for any sequence {λ i } n i= in [0,] satisfying n i= λ i =and for any {x i } n i= X, the following conclusions hold: d λ i x i, x i= λ i dx i, x, x X; 2.5 i= and d 2 i= λ i x i, x λ i d 2 x i, x λ λ 2 d 2 x, x 2, x X. i= Lemma 2.4 Let {x i } and {y i } be any sequences of a CAT0 space X, then for any sequence {λ i } k i= in [0, ] satisfying k i= λ i =and for any {x i } k i= X, the following inequality holds: k d λ i x i, i= k λ i y i i= k λ i dx i, y i. 2.6 i= Proof It is obvious that 2.6holdsfork =2.Supposethat2.6 holds for some k 3. Next we prove that 2.6isalsotruefork +.From2.3and2.4wehave
5 Tang et al. Fixed Point Theory and Applications :00 Page 5 of 3 d k+ k+ λ i x i, λ i y i i= i= k k λ i λ i = d λ k+ x i λ k+ x k+, λ k+ y i λ k+ y k+ λ k+ λ k+ λ k+ d λ k+ i= k k i= k+ = λ i dx i, y i. i= i= λ i λ k+ x i, i= i= k λ i y i + λ k+ dx k+, y k+ λ k+ λ i λ k+ dx i, y i +λ k+ dx k+, y k+ This implies that 2.6holds. Berg and Nikolaev [5] introduced the concept of quasilinearization as follows. Let us denote a pair a, b X X by ab and call it a vector. Then quasilinearization is defined as a map, :X X X X R defined by ab, cd = 2 d 2 a, d+d 2 b, c d 2 a, c d 2 b, d a, b, c, d X. 2.7 It is easy to seen that ab, cd = cd, ab, ab, cd = ba, cd, and ax, cd + xb, cd = ab, cd for all a, b, c, d X.WesaythatX satisfies the CauchySchwarz inequality if ab, cd da, bdc, d 2.8 for all a, b, c, d X. Itiswellknown[5] that a geodesically connected metric space is a CAT0 space if and only if it satisfies the CauchySchwarz inequality. Let C be a nonempty closed convex subset of a complete CAT0 space X. Themetric projection P C : X C is defined by u = P C x du, x=inf { dy, x:y C }, x X. Lemma 2.5 [6] Let C be a nonempty convex subset of a complete CAT0 space X, x X, and u C. Then u = P C x if and only if yu, ux 0, y C. Lemma 2.6 [2] LetCbeaclosedconvexsubsetofacompleteCAT0 space X, and let T : C C be a nonexpansive mapping with FT. Let f be a contraction on C with coefficient α <.For each t [0, ], let {x t } be the net defined by x t = tf x t ttx t. 2.9
6 Tang et al. Fixed Point Theory and Applications :00 Page 6 of 3 Then lim t 0 x t = x, some point FT which is the unique solution of the following variational inequality: xf x, x x 0, x FT. 2.0 Lemma 2.7 [2] Let X be a complete CAT0 space. Then, for any u, x, y X, the following inequality holds: d 2 x, u d 2 y, u+2 xy, xu. Lemma 2.8 [2] Let X be a complete CAT0 space. For any t [0, ] and u, v X, let u t = tu tv. Then, for any x, y X, i ut x, ut y t ux, ut y + t vx, ut y; ii ut x, uy t ux, uy + t vx, uy, and ut x, vy t ux, vy + t vx, vy. Lemma 2.9 [7] Let {a n } be a sequence of nonnegative real numbers satisfying the property a n+ α n a n + α n β n, n 0, where {α n } 0, and {β n } R such that i n=0 α n = ; ii lim sup n β n 0 or n=0 α nβ n <. Then {a n } converges to zero as n. 3 Approximative iterative algorithms Throughout this section, we assume that C is a nonempty and closed convex subset of a complete CAT0 space X,andT : C C is a nonexpansive mappings with FT.Let f be a contraction on C with coefficient k 0,. Suppose {α n }, {β n },and{γ n } are three real sequences in 0, satisfying i α n + β n + γ n =, for all n 0; ii lim n α n =0and n=0 α n = ; iii lim n γ n =0. In the following, we first present two important results. The first one is to prove the mapping S := n n+ T i : C Cisnonexpansive: In fact, for any x, y C, from Lemma 2.4 we get dsx, Sy=d dx, y, n + T i x, n n + d T i x, T i y n + T i y i.e., S is nonexpansive. The second one is to prove the following mapping T t,n : C C is contractive: T t,n x = α nt f x tγ n n + T i x.
7 Tang et al. Fixed Point Theory and Applications :00 Page 7 of 3 In fact, the mapping T t,n can be written as T t,n x = λ n f x λ n n + T i x, x C, where λ n = α nt γ n +tβ n. Therefore this kind of mappings has just a similar form to 2.9. Hence for any x, y C, from Lemma 2.4,wehave d T t,n x, T t,n y = d λ n f x λ n n n + T i x, λ n f y λ n λ n d f x, f y + λ n d λ n kdx, y+ λ n n + d T i x, T i y λ n kdx, y+ λ n dx, y = λ n k dx, y, n + T i x, n n n + T i y n + T i y i.e., T t,n is a contractive mapping. Hence, it has a unique fixed point denote by z t,n, i.e., z t,n = λ n f z t,n λ n n + T i z t,n. From Lemma 2.6,foreachgivenn, we have lim z t,n = x n F t 0 n + T i, 3. and it is the unique solution of the following variational inequality: x n f x n, xxn 0, x F Next we prove that for each n FT=F n + T i. 3.2 n + T i. 3.3 If fact, if x FT, it is obvious that x F n n+ T i, for each n. This implies that FT F n n+ T i foreachn. By using induction, now we prove that F n + T i FT foreachn. 3.4
8 Tang et al. Fixed Point Theory and Applications :00 Page 8 of 3 Indeed, if n =,andx F 2 T i, i.e., x = 2 x 2 Tx,thenwehave 2 x 2 x = 2 x Tx Since the geodesic segment joining any two points in CAT0 space is unique, it follows from 3.5thatx = Tx, i.e., x FT. If 3.4istrueforsomen 2, now we prove that 3.4isalsotrueforn +. Indeed, if x F n+ n+2 T i, i.e., n+ x = n +2 T i x. 3.6 It is easy to see that 3.6canberewrittenas n + n +2 x n +2 x = n + { x n +2 n + Tx n + T n } x n + n +2 T n+ x. 3.7 By the same reasoning showing that the geodesic segment joining any two points in CAT0 space is unique, from 3.7wehave n + n +2 x = n + { x n +2 n + Tx n + T n } x n + This implies that and n +2 x = n +2 T n+ x. x = x n + Tx n + T n x n + and x = T n+ x. 3.8 By the assumption of induction, from 3.8wehavex = Tx.Theconclusion3.4isproved. Therefore the conclusion 3.3isalsoproved. By using 3.3, 3.canbewrittenas lim z t,n = x n FT, 3.9 t 0 and x n is the unique solution of the following variational inequality: x n f x n, xxn 0, x FT. 3.0 Next we prove that for any positive integers m, n, x n = x m where x n is the limit in 3.9. Therefore 3.9canbewrittenas lim t 0 z t,n = x n = x FT, 3. where x is some point, and it is the unique solution of the following variational inequality: xf x, x x 0, x FT. 3.2
9 Tang et al. Fixed Point Theory and Applications :00 Page 9 of 3 In fact, it follows from 3.0that x n f x n, xxn 0, x m f x m, yxm 0, x FT, y FT. Taking x = x m in the first inequality and y = x n in the second inequality and then adding up the resultant two inequalities, we have 0 x n f x n, x m x n x m f x m, x m x n x n f x m, x m x n + f x m f x n, x m x n x m x n, x m x n x n f x m, x m x n = f x m f x n, x m x n xm x n, x m x n d f x m, f x n dx m, x n d 2 x m, x n kd 2 x m, x n d 2 x m, x n =k d 2 x m, x n. Since k 0,, this implies that dx m, x n =0,i.e., x m = x n, m, n. The conclusion is proved. We are now in a position to propose the iterative algorithms with Cesàro s means for a nonexpansive mapping in a complete CAT0 space X, and prove that the sequence generated by the algorithms converges strongly to a fixed point of the nonexpansive mapping which is also a unique solution of some kind of variational inequality. Theorem 3. Let C be a closed convex subset of a complete CAT0 space X, and T : C C be a nonexpansive mapping with FT. Let f be a contraction on C with coefficient k 0, 2. Suppose x 0 C and the sequence {x n } is given by x n+ = α n f x n β n x n γ n n + T i x n, 3.3 where T 0 = I, {α n }, {β n }, and {γ n } are three real sequences in 0, satisfying conditions i iii. Then {x n } converges strongly to x suchthat x = P FT f x, which is equivalent to the following variational inequality: xf x, x x 0, x FT. 3.4 Proof We first show that the sequence {x n } is bounded. For any p FT, we have dx n+, p=d α n f x n β n x n γ n α n d f x n, p + β n dx n, p+γ n d n + T i x n, p n + T i x n, p α n d f xn, f p + d f p, p + β n dx n, p+γ n n + d T i x n, T i p
10 Tang et al. Fixed Point Theory and Applications :00 Page 0 of 3 By induction, we have α n kdx n, p+α n d f p, p + β n dx n, p+γ n dx n, p = α n k dx n, p+α n k { max dx n, p, k d f p, p }. { dx n, p max dx 0, p, k d f p, p } k d f p, p for all n 0. Hence {x n } is bounded, and so are {f x n } and {T i x n }. Let {z t,n } be a sequence in C such that z t,n = α nt f z t,n tγ n n + T i z t,n. It follows from 3.that{z t,n } convergesstronglytoafixedpoint x FT, which is also a unique solution of the variational inequality 3.4. Now we claim that lim sup f x x, xn+ x n It follows from Lemma 2.8 and Lemma 2.4 that d 2 z t,n, x n+ = z t,n x n+, z t,n x n+ α nt f z t,n x n+, z t,n x n+ + tγ n n = α nt [ f z t,n f x, z t,n x n+ + f x x, z t,n x n+ + xzt,n, z t,n x n+ + z t,n x n+, z t,n x n+ ] + tγ n [ n n n + T i z t,n ] n + n + T i x n+ x n+, z t,n x n+ n + T i x n+, z t,n x n+ α nt [ kdzt,n, xdz t,n, x n+ + f x x, z t,n x n+ n + T i z t,n x n+, z t,n x n+ + d x, z t,n dz t,n, x n+ +d 2 z t,n, x n+ ] [ + tγ n n d n + T i z t,n, n + T i x n+ dz t,n, x n+
11 Tang et al. Fixed Point Theory and Applications :00 Page of 3 where + d ] n + T i x n+, x n+ dz t,n, x n+ α nt [ kdzt,n, xdz t,n, x n+ + f x x, z t,n x n+ + d x, z t,n dz t,n, x n+ +d 2 z t,n, x n+ ] [ + tγ n d 2 z t,n, x n+ + d ] T i x n+, x n+ dzt,n, x n+ n + α nt [ Mkdzt,n, x+ f x x, z t,n x n+ + Md x, zt,n ] + d 2 z t,n, x n+ + tγ nmn, M sup { dz t,n, x n+, n 0, 0 < t < }, N sup { d T i x n+, x n+, n 0, i 0 }. This implies that f x x, x n+ z t,n + kmdzt,n, x+ tγ nmn α n t. 3.6 Taking the upper limit as n first, and then taking the upper limit as t 0, we get Since lim sup t 0 lim sup f x x, x n+ z t,n n f x x, xn+ x = f x x, x n+ z t,n + f x x, zt,n x f x x, x n+ z t,n + d f x, x dzt,n, x. Thus, by taking the upper limit as n first, and then taking the upper limit as t 0, it follows from z t,n x and 3.7that Hence lim sup t 0 lim sup f x x, xn+ x 0. n lim sup f x x, xn+ x 0. n Finally, we prove that x n x as n.infact,foranyn 0, let { u n := α n x α n y n, y n := β n α n x n γ n α n n n+ T i x n. 3.8
12 Tang et al. Fixed Point Theory and Applications :00 Page 2 of 3 FromLemma 2.7 andlemma 2.8 we have d 2 x n+, x d 2 u n, x+2 x n+ u n, xn+ x α n 2 d 2 y n, x+2 [ α n f x n u n, xn+ x + α n yn u n, xn+ x ] α n 2 d 2 x n, x+2 [ αn 2 f x n x, xn+ x + α n α n f x n y n, xn+ x + α n α n yn x, xn+ x + α n 2 yn y n, xn+ x ] = α n 2 d 2 x n, x+2 [ αn 2 f x n x, xn+ x + α n α n f x n x, xn+ x ] = α n 2 d 2 x n, x+2α n f x n x, xn+ x = α n 2 d 2 x n, x+2α n f x n f x, xn+ x +2α n f x x, xn+ x α n 2 d 2 x n, x+2α n kdx n, xdx n+, x+2α n f x x, xn+ x α n 2 d 2 x n, x+α n k d 2 x n, x+d 2 x n+, x +2α n f x x, xn+ x. This implies that d 2 x n+, x 2 kα n + αn 2 d 2 x n, x+ α n k = α n2 2k α n α n k Then it follows that d 2 x n, x+ 2α n f x x, xn+ x α n k 2α n f x x, xn+ x. α n k d 2 x n+, x α n d 2 x n, x+α n β n, where α n = α n2 2k α n, β n α n k = 2 f x x, xn+ x. 2 2k α n Applying Lemma 2.9,wecanconcludethatx n x as n. This completes the proof. Letting f x n u for all n N, the following theorem can be obtained from Theorem 3. immediately. Theorem 3.2 Let C be a closed convex subset of a complete CAT0 space X, and let T : C C be a nonexpansive mapping satisfying FT. For given x 0 Carbitrarilyand fixed point u C, the sequence {x n } is given by x n+ = α n u β n x n γ n n + T i x n, 3.9
13 Tang et al. Fixed Point Theory and Applications :00 Page 3 of 3 where T 0 = I, {α n }, {β n }, and {γ n } are three real sequences in 0, satisfying conditions iiii. Then {x n } converges strongly to x suchthat x = P FT u, which is equivalent to the following variational inequality: xu, x x 0, x FT Remark 3.3 Theorem 3. and Theorem 3.2 are two new results. The proofs are simple and different from many others which extend the Cesàro nonlinear ergodic theorems for nonexpansive mappings in Baillon [], Bruck [2], Song and Chen [3], Zhu and Chen [5], Yao et al. [4] from Hilbert space or Banach space to CAT0 spaces. Theorem 3. is also a generalization of the results by Wangkeeree and Preechasilp [2]. Competing interests The authors declare that they have no competing interests. Authors contributions All the authors contributed equally to the writing of the present article. They also read and approved the final paper. Author details Department of Mathematics, Yibin University, Yibin, Sichuan , China. 2 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 65022, China. Acknowledgements The authors would like to express their thanks to the editors and the referees for their helpful suggestions and advises. This work was supported by the Scientific Research Fund of Sichuan Provincial Department of Science and Technology 205JY065 and the Scientific Research Fund of Sichuan Provincial Education Department 4ZA027 and the Scientific Research Project of Yibin University 203YY06. This work was also supported by the National Natural Science Foundation of China Grant No Received: 8 March 205 Accepted: 5 June 205 References. Baillon, JB: Un théorème de type ergodique pour les contractions non linéairs dans un espaces de Hilbert. C. R. Acad. Sci. Paris Sér. AB 28022, Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math , Song, Y, Chen, R: Viscosity approximate methods to Cesàro means for nonexpansive mappings. Appl. Math. Comput. 862, Yao, Y, Liou, YC, Zhou, H: Strong convergence of an iterative method for nonexpansive mappings with new control conditions. Nonlinear Anal., Theory Methods Appl. 706, Zhu, Z, Chen, R: Strong convergence on iterative methods of Cesàro means for nonexpansive mapping in Banach space. Abstr. Appl. Anal. 204,Article ID Bridson, MR, Haefliger, A: Metric Spaces of Nonpositive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 39. Springer, Berlin Kirk, WA: Fixed point theory in CAT0 spaces and Rtrees. Fixed Point Theory Appl , Brown, KS ed.: Buildings. Springer, New York Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Monograph and Textbooks in Pure and Applied Mathematics, vol. 83. Dekker, New York Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis Malaga/Seville, 2002/2003. Colección Abierta, vol. 64, pp University of Seville Secretary of Publications, Seville Kirk, WA: Geodesic geometry and fixed point theory. II. In: International Conference on Fixed Point Theory and Applications, pp Yokohama Publishers, Yokohama Wangkeeree, R, Preechasilp, P: Viscosity approximation methods for nonexpansive semigroups in CAT0 spaces. Fixed Point Theory Appl. 203,ArticleID doi:0.86/ Dhompongsa, S, Panyanak, B: On convergence theorems in CAT0 spaces. Comput. Math. Appl. 560, Tang, JF: Viscosity approximation methods for a family of nonexpansive mappings in CAT0 spaces. Abstr. Appl. Anal. 204, Article ID Berg, ID, Nikolaev, IG: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 33, Dehghan, H, Rooin, J: A characterization of metric projection in CAT0 spaces. In: International Conference on Functional Equation, Geometric Functions and Applications ICFGA 202, 02th May 202, pp Payame Noor University, Tabriz Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 6,
On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces
Bachar and Khamsi Fixed Point Theory and Applications (215) 215:16 DOI 1.1186/s1366315453 R E S E A R C H Open Access On common approximate fixed points of monotone nonexpansive semigroups in Banach
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationQuasi Contraction and Fixed Points
Available online at www.ispacs.com/jnaa Volume 2012, Year 2012 Article ID jnaa00168, 6 Pages doi:10.5899/2012/jnaa00168 Research Article Quasi Contraction and Fixed Points Mehdi Roohi 1, Mohsen Alimohammady
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationSome remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s1366101505080 R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationSumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION
F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Sumit Chandok and T. D. Narang INVARIANT POINTS OF BEST APPROXIMATION AND BEST SIMULTANEOUS APPROXIMATION Abstract. In this paper we generalize and extend
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More information0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup
456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationRelatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations
Wang and Agarwal Advances in Difference Equations (2015) 2015:312 DOI 10.1186/s1366201506500 R E S E A R C H Open Access Relatively dense sets, corrected uniformly almost periodic functions on time
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationQuantum Computation as Geometry
Quantum Computation as Geometry arxiv:quantph/0603161v2 21 Mar 2006 Michael A. Nielsen, Mark R. Dowling, Mile Gu, and Andrew C. Doherty School of Physical Sciences, The University of Queensland, Queensland
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationAn optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationA Simple Proof of Duality Theorem for MongeKantorovich Problem
A Simple Proof of Duality Theorem for MongeKantorovich Problem Toshio Mikami Hokkaido University December 14, 2004 Abstract We give a simple proof of the duality theorem for the Monge Kantorovich problem
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationDegree Hypergroupoids Associated with Hypergraphs
Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationMultiplicative Relaxation with respect to Thompson s Metric
Revista Colombiana de Matemáticas Volumen 48(2042, páginas 227 Multiplicative Relaxation with respect to Thompson s Metric Relajamiento multiplicativo con respecto a la métrica de Thompson Gerd Herzog
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More information10. Proximal point method
L. Vandenberghe EE236C Spring 201314) 10. Proximal point method proximal point method augmented Lagrangian method MoreauYosida smoothing 101 Proximal point method a conceptual algorithm for minimizing
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationTRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh
Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp . TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationComputing divisors and common multiples of quasilinear ordinary differential equations
Computing divisors and common multiples of quasilinear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univlille1.fr
More informationOn using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems
Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University
More informationCONVERGENCE OF SEQUENCES OF ITERATES OF RANDOMVALUED VECTOR FUNCTIONS
C O L L O Q U I U M M A T H E M A T I C U M VOL. 97 2003 NO. 1 CONVERGENCE OF SEQUENCES OF ITERATES OF RANDOMVALUED VECTOR FUNCTIONS BY RAFAŁ KAPICA (Katowice) Abstract. Given a probability space (Ω,
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationMixedÀ¾ Ð½Optimization Problem via Lagrange Multiplier Theory
MixedÀ¾ Ð½Optimization Problem via Lagrange Multiplier Theory Jun WuÝ, Sheng ChenÞand Jian ChuÝ ÝNational Laboratory of Industrial Control Technology Institute of Advanced Process Control Zhejiang University,
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationMATH 425, HOMEWORK 7, SOLUTIONS
MATH 425, HOMEWORK 7, SOLUTIONS Each problem is worth 10 points. Exercise 1. (An alternative derivation of the mean value property in 3D) Suppose that u is a harmonic function on a domain Ω R 3 and suppose
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationSome strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 6 (013), 74 85 Research Article Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationCompletely Positive Cone and its Dual
On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual Peter J.C. Dickinson Luuk Gijben July 3, 2012 Abstract Copositive programming has become a useful tool
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationSome Problems of SecondOrder Rational Difference Equations with Quadratic Terms
International Journal of Difference Equations ISSN 09736069, Volume 9, Number 1, pp. 11 21 (2014) http://campus.mst.edu/ijde Some Problems of SecondOrder Rational Difference Equations with Quadratic
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationarxiv:1502.06681v2 [qfin.mf] 26 Feb 2015
ARBITRAGE, HEDGING AND UTILITY MAXIMIZATION USING SEMISTATIC TRADING STRATEGIES WITH AMERICAN OPTIONS ERHAN BAYRAKTAR AND ZHOU ZHOU arxiv:1502.06681v2 [qfin.mf] 26 Feb 2015 Abstract. We consider a financial
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationNumerical Verification of Optimality Conditions in Optimal Control Problems
Numerical Verification of Optimality Conditions in Optimal Control Problems Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der JuliusMaximiliansUniversität Würzburg vorgelegt von
More informationANALYTICITY OF SETS ASSOCIATED TO LELONG NUMBERS AND THE EXTENSION OF MEROMORPHIC MAPS 1
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 6, November 1973 ANALYTICITY OF SETS ASSOCIATED TO LELONG NUMBERS AND THE EXTENSION OF MEROMORPHIC MAPS 1 BY YUMTONG SIU 2 Communicated
More informationChapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation
Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7 In this section, we discuss linear transformations 89 9 CHAPTER
More informationOn the Algebraic Structures of Soft Sets in Logic
Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 18731881 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationFunctional analysis and its applications
Department of Mathematics, London School of Economics Functional analysis and its applications Amol Sasane ii Introduction Functional analysis plays an important role in the applied sciences as well as
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationDate: April 12, 2001. Contents
2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........
More informationAN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT
Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp 1291260. AN ITERATIVE METHOD FOR THE HERMITIANGENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationTail inequalities for order statistics of logconcave vectors and applications
Tail inequalities for order statistics of logconcave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.TomczakJaegermann Banff, May 2011 Basic
More informationTRIPLE FIXED POINTS IN ORDERED METRIC SPACES
Bulletin of Mathematical Analysis and Applications ISSN: 18211291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON
More informationSOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED MATRIX WITH NEW CONDITIONS
International Journal of Analysis and Applications ISSN 22918639 Volume 5, Number 2 (2014, 191197 http://www.etamaths.com SOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED MATRIX WITH NEW CONDITIONS ABDUL
More information