long ngoc volume 2007 article id 20295 17 pages pdf

... ∈ U 2.3 Definition 2.3 see A function f is rd-continuous provided it is continuous at each rightdense point in T and has a left-sided limit at each left-dense point in T The set of rdcontinuous ... boundary value problem on time scales,” Computers & Mathematics with Applications, vol 58, no 9, pp 173 6 174 1, 2009 16 Y Yang and F Meng, “Positive solutions of the singular semipositone boundary value ... and t ∈ Tκ , then the delta derivative of f at the point t is defined to be the number f Δ t provided it exists with the property that for each > 0, there is a neighborhood U of t such that f σ...

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... p-Laplacian dynamic equaitons on time scales,” Advance in Difference Equations, vol 2009, Article ID 262857, 12 pages, 2009 E R Kaufmann and Y N Raffoul, “Positive solutions for a nonlinear functional ... Fx > a, It is obvious that P α, a / ∅ On the other hand, α x a≤x≤ T a, T −η ∀x ∈ ∂P α, a x η 2 .17 a and 2.7 imply for t ∈ 0, η 2.18 Thus, T α Fx T − s φq − B0 a r f x r ,x μ r s η φq η s η − ... form ⎧ ⎨ψ t , t ∈ −r, , ⎩x t , i x t t ∈ 0, T , i 2.25 1, Now, we give theorems, which may be considered as the corollaries of Theorems 2.2 and 2.3 Let f0 lim x→0 f x, ψ s , xp−1 f∞ lim f x, ψ s...

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... found in 17 Definition 2.1 Let X be a Banach space and Φ ∈ C1 X, R1 We say that Φ satisfies the P S condition at the level c ∈ R1 P S c condition for short if any sequence {un } ⊂ X along with ... of Cbms Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986 17 J Mawhin and M Willem, Critical Point Theory and Hamiltonian Systems, vol 74 of Applied ... 3.16 C3 u p , C2 γp Hence, there exists r2 > such that for u ∈ X2 , u ≤ r2 , Φ u ≤ 0, Φ u < 0, 3 .17 for u ∈ X2 , < u ≤ r2 Lastly, the conclusion follows from choosing r Step Φ is coercive on X,...

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... 0, × SN−1 , SN−1 and ζn 0, z G Fn z ζn 1, z z provided n ≥ n0 This completes the proof ∀z ∈ SN−1 , ∀z ∈ SN−1 , 4.42 Boundary Value Problems 17 Theorem 4.10 Assume that q is a positive continuous ... 1997 19 B Gidas, W M Ni, and L Nirenberg, “Symmetry and related properties via the maximum principle,” Communications in Mathematical Physics, vol 68, no 3, pp 209–243, 1979 20 B Gidas, W M Ni, ... exterior domain, where D is a C1,1 bounded domain in ÊN By Lions 2, , Struwe 16 , and Lien et al 17 , we have the following decomposition lemmas Lemma 3.1 Palais-Smale Decomposition Lemma for J...

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... Γ −a n α a α α 1 α a Γn α Γ n−a Γ α Γ −a n α Γα − Γ −a 3.5 −→ as n −→ ∞ This argument is valid provided a is not a positive integer If a is a positive integer, then α xk for ≤ k ≤ a − Then, un ... to have a two-sided inverse α fnk with fa,a However, if we define a new matrix F 1, D has finite norm and is and which agrees with Hμ α elsewhere, then F does possess a unique two-sided inverse Morever, ... Hausdorff matrix,” Proceedings of the National Institute of Sciences of India Part A, vol 27, pp 175 177 , 1961 12 H T Davis, The Summation of Series, The Principia Press of Trinity University, San...

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... Bk t fractional part of t Example 4.3 The error functionals LM , LT , and LS for the midpoint rule, trapezoidal quadrature and Simpson’s rule are, respectively, LM f f t dt − f LT f f t dt − LS ... USA, 2nd edition, 1990 B Bollob´ s, Linear Analysis: An Introductory Course, Cambridge University Press, Cambridge, UK, 2nd a edition, 1999 S Smale, “On the efficiency of algorithms of analysis,” ... Applications Using 2.1 , Smale was able to compute the average error for right rectangle rule, the trapezoidal rule, and Simpson’s rule see 6, Theorem D Later on, Wang and Han in extended and unified results...

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... means and related c c c results,” journal of Inequalities and Applications, vol 2010, Article ID 289730, 16 pages, 2010 M Anwar, J Jakˇ eti´ , J Peˇ ari´ , and A Ur Rehman, “Exponential convexity, ... have k i ul um Υslm ≥ 0, k 1, , q 2.11 l,m This means that the matrix Υi sl valid q sm /2 l,m is positive semidefinite, that is, 2.9 is i b It is assumed that the function s → Υs s ∈ a, b is ... ξi ∈ a, b such that Υi x, p, f Υi x, p, g f ξi , g ξi i 1, , 15, 3.2 provided that the denominators are nonzero The idea of the proofs of Theorems 3.1 and 3.2 is the same as the proofs of Theorems...

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... subordination,” International Journal of Mathematics and Mathematical Sciences, vol 2010, Article ID 275935, 12 pages, 2010 18 H M Srivastava and S S Eker, “Some applications of a subordination theorem ... Similarly, in 16 , some new subclasses of meromorphic functions in the punctured unit disk was considered In 17 , similar results were proved by using the p-valent functions that satisfy the following ... Σp is said to be meromorphically starlike of order α if it is satisfying the following condition: Re − zf z f z > α, z ∈ U∗ , 1.5 for some α α < Similarly recall a function f ∈ Σp is said to be...

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... ranks and each rank is quantized: unnoticeable (−2), not evident (−1), slightly evident (0), evident (1) and very evident (2) Different individuals have different perceptions and visual sensitivities ... information hiding can be divided into two categories: spatial domain information hiding and transform domain (such as the DCT transform domain, the wavelet transform domain, etc.) information hiding ... theoretical guide for hiding information capacity estimation of the real objective images and is not very much contributive to the accomplishment of the project The estimation method proposed for hiding...

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... and p be an E-distance on X Then i X said to be S-complete if for every p-Cauchy sequence {xn } there exists x ∈ X with 0, limn → ∞ p xn , x ii X is said to be p-Cauchy complete if for every ... guarantees the uniqueness of limit of a sequence in uniform spaces An element of uniformity ϑ is said to be symmetrical if V V −1 { y, x : x, y ∈ V } Since each V ∈ ϑ contains a symmetrical W ∈ ϑ ... lemmas in the sequel Definition 2.1 see Let X, ϑ be a uniform space A function p : X × X → R is said to be an E-distance if p1 for any V ∈ ϑ, there exists δ > 0, such that p z, x ≤ δ and p z, y...

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... nonexpansive semigroup in Banach spaces,” Fixed Point Theory and Applications, vol 2011, Article ID 620284, 17 pages, 2011 H K Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: ... on X Then ψ x I x − μF x is a contraction on X for every μ ∈ 0, η/βκ2 Proof If J satisfies , considering the inequality x y ≤ x 2 y, J x y , for all x, y ∈ X, we have ψx − ψy ≤ I − μF x − I − μF ... β κ , then σ ψ x I x −μ F I −T x is a contraction on X With the new imposed conditions and considering the above lemmas, the following corrections should be done in : in Theorem 3.2 and Theorem...

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... Let Y be a metric space, and let IdY : Y → Y be a map given by formula IdY y for each y ∈ Y y Definition 2.15 see A map r : X → Y of a space X onto a space Y is said to be an mr-map if there is ... is an admissible map ϕ : Y X such that r ◦ ϕ IdY Definition 2.16 see 3, A metric space X is called an absolute multiretract notation: X ∈ AMR provided there exists a locally convex space E and ... an mr-map r : E → X from E onto X Definition 2 .17 see 3, A metric space X is called an absolute neighborhood multiretract notation: X ∈ ANMR provided that there exists an open subset U of some...

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... nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol 2010, Article ID 907275, 16 pages, 2010 ... Fx − Fy , for some λ ∈ 0, In , Saeidi and Naseri established a strong convergence theorem for a semigroup of nonexpansive mappings, as follows Theorem Saeidi and Naseri Let {T t : t ∈ S} be ... condition A ≤ v The constant γ , used in Theorem 2, should be chosen in 0, − − δ /λ /α References S Saeidi and S Naseri, “Iterative methods for semigroups of nonexpansive mappings and variational inequalities,”...

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... A Kirk, Topics in Metric Fixed Point Theory, vol 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990 10 H Zhou, “Convergence theorems of fixed points ... spaces,” Journal of Mathematical Analysis and Applications, vol 329, no 1, pp 336–346, 2007 Y Yamada, “The hybrid steepest-descent method for variational inequalities problems over the intesectionof ... Point Theory and Applications Since p, p ∈ C, by replacing p by − t p in the last inequality, dividing by t and taking t → in the just obtained inequality, we obtain F p , p − p ≥ ∀p ∈ C 2.22 The...

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... function F t1 , t2 , t3 , t4 where k > belonging to Ψ does not belongs to Φ as F u, u, 1, < for all u > 0, whereas 15t1 − 13t2 5t3 − 7t4 belonging to Φ does not belongs to Ψ implicit function φ t1 , ... Bn , T q have a point of coincidence each Further, A, B, S and T have a unique common fixed point provided that the pairs A, S and B, T commute pairwise Remark 3 .17 Results similar to Corollary ... of X Then, the pair A, S as well as B, T have a point of coincidence each Further, A, B, S and T have a unique common fixed point provided that both the pairs A, S and B, T are weakly compatible...

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... equation f x y f x f y In this paper, by using the idea of G˘ vruta , we prove the stability of the Jensen a ¸ functional equation and the Pexiderized Cauchy functional equation: f x y g x h y ... satisfying f x −T x Y ≤ψ x 2 .17 for all x ∈ X Moreover, if lim lim max k→∞ n→∞ |2| j ψ 2j x, : k < j ≤ n k 2.18 for all x ∈ X, then T is a unique additive mapping satisfying 2 .17 Generalized Hyers-Ulam ... Now, we show that T1 T2 T3 Replacing x and y by 2n x and in 3.5 , respectively, and dividing both sides of 3.5 by |2|n , we get 2n f x x − 2n g n n 2 Y ≤ |2|n Φ x ,0 2n 3.23 for all x ∈ X By...

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... equation and a Jensen-quadratic equation,” Abstract and Applied Analysis, vol 2007, Article ID 4 5179 , 10 pages, 2007 49 J B Diaz and B Margolis, “A fixed point theorem of the alternative, for ... stability of C∗ -ternary quadratic mappings,” Abstract and Applied Analysis, vol 2007, Article ID 23282, pages, 2007 47 J.-H Bae and W.-G Park, “A functional equation having monomials as solutions,” ... 2009, Article ID 437931, pages, 2009 33 M Eshaghi Gordji and M B Savadkouhi, “Approximation of generalized homomorphisms in quasiBanach algebras,” Analele Stiintifice ale Universitatii Ovidius...

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... extension of Gregus fixed point theorem,” Fixed Point Theory and Applications, vol 2007, Article ID 78628, pages, 2007 ... iv F λx ≤ F x for all λ ∈ K with |λ| ≤ 1, v if λn → and λn ∈ K, then F λn x → In 2007, Olaleru and Akewe considered the existence of fixed point of T when T is defined on a closed convex subset ... above mentioned theorem is not correct Counterexample Example 2.1 Let X R endowed with the Euclidean metric and C X Let T : C → C defined by T x x Let < a < and e > such that a 2e Then for all...

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... u Au u u V V →∞ ∞ lim 3 .17 Remark 3.5 Coercivity of A implies that for any f ∈ V , there is N > such that f u < Au u , 3.18 provided that u V ≥ N In fact, the limit 3 .17 says that, for any M ... both sides yields Ω ∂x1 x1 α φ x dx Ω α φ x dx Ω x1 α α φ x dx ≤ kβ−1 Ω β α φ x sgn φ x ∂x1 φ x dx A.5 The boundary condition makes the left-hand side zero: virtue of Holder’s inequality, identity ... Gauss-Green theorem, the left-hand side becomes − Ω ∇ · J ∇u φ dμ Ω J ∇u · ∇φ dμ n Ω j α ∂xj u ∂xj u ∂xj φ dμ 3.5 10 Fixed Point Theory and Applications We will consider the operator A defined by α...

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... 2007, Article ID 5 0175 , 15 pages, 2007 66 C Park, “Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach,” Fixed Point Theory and Applications, vol 2008, Article ... functional equation in quasi-Banach spaces,” Abstract and Applied Analysis, vol 2009, Article ID 4174 73, 14 pages, 2009 17 K Ravi, J M Rassias, M Arunkumar, and R Kodandan, “Stability of a generalized ... article 114, pp 1–29, 2009 18 M E Gordji and H Khodaei, “On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,” Abstract and Applied Analysis, vol 2009, Article ID...

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