Showing papers in "Abstract and Applied Analysis in 2007"
TL;DR: In this paper, the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem (D0) was studied. But the existence of positive solution was not investigated.
Abstract: We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem: D0
90 citations
TL;DR: In this article, the authors define a holomorphic self-map of a unit polydisc of ℂn, and define the Bloch space of all holomorphic functions with domain Dn.
Abstract: Let Dn be the unit polydisc of ℂn, ϕ(z)=(ϕ1(z),…,ϕn(z)) be a holomorphic self-map of Dn, and ψ(z) a holomorphic function on Dn. Let H(Dn) denote the space of all holomorphic functions with domain Dn, H∞(Dn) the space of all bounded holomorphic functions on Dn, and B(Dn) the Bloch space, that is, B(Dn)={f∈H(Dn)|‖f‖B=|f(0)|
82 citations
TL;DR: In this article, the convolution of two functions defined on a time scale is introduced, and the delay in turn is used to introduce convolution for two functions on the time scale.
Abstract: The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider the q-difference equations case.
64 citations
TL;DR: Using the Hyers-Ulam-Rassias stability method of functional equations, this paper investigated homomorphisms in C*-algebras, Lie C *-algeses, and JC *-Algebrases associated with the following Apollonius-type additive functional equation f(z−x) and derived derivations on C * -algebras, Lie c *-bregions and JC* -algesbras associated with f(n−x).
Abstract: Using the Hyers-Ulam-Rassias stability method of functional equations, we investigate homomorphisms in C*-algebras, Lie C*-algebras, and JC*-algebras, and derivations on C*-algebras, Lie C*-algebras, and JC*-algebras associated with the following Apollonius-type additive functional equation f(z−x)
59 citations
TL;DR: In this paper, the generalized stability of C *ternary quadratic mappings in C*ternary rings for the quadrastic functional equation f(x) was proved.
Abstract: We prove the generalized stability of C*-ternary quadratic mappings in C*-ternary rings for the quadratic functional equation f(x
53 citations
TL;DR: This article solved the nonhomogeneous Legendre's differential equation and applied this result to obtaining a partial solution to the Hyers-Ulam stability problem for the non-homogeneous version of the Legendre equation.
Abstract: We solve the nonhomogeneous Legendre's differential equation and apply this result to obtaining a partial solution to the Hyers-Ulam stability problem for the Legendre's equation
34 citations
TL;DR: In this article, sufficient and necessary conditions for analytic functions on the unit ball B with Hadamard gaps were given for f(z)=∑k=1∞Pnk(z) (the homogeneous polynomial expansion of f) satisfying nk.
Abstract: We give some sufficient and necessary conditions for an analytic function f on the unit ball B with Hadamard gaps, that is, for f(z)=∑k=1∞Pnk(z) (the homogeneous polynomial expansion of f) satisfying nk
29 citations
TL;DR: In this article, a fixed point theorem for nonlinear contraction in the modular space is proved and a fixed-point theorem for asymptotic contraction in this space is studied.
Abstract: A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.
26 citations
TL;DR: A Tauberian theorem is proved to recover moderate oscillation of a real sequence out of Abel limitability of the sequence.
Abstract: We prove a Tauberian theorem to recover moderate oscillation of a real sequence u=(un) out of Abel limitability of the sequence (Vn(1)(Δu)) and some additional condition on the general control modulo of oscillatory behavior of integer order of u=(un)
22 citations
TL;DR: In this paper, it was shown that a Cerami sequence can produce a critical point even when a Palais-Smale sequence does not have convergent subsequences (i.e., if G satisfies the PS condition).
Abstract: The concept of linking was developed to produce Palais-Smale (PS) sequences G ( u k ) → a , G ' ( u k ) → 0 for C 1 functionals G that separate linking sets. These sequences produce critical points if
they have convergent subsequences (i.e., if G satisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfying G ( u k ) → a , ( 1 + | | u k | | ) G ' ( u k ) → 0 as well. A Cerami sequence can produce a critical point even when a PS sequence does not. In this situation, it is no longer necessary to show that
G satisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.
22 citations
TL;DR: It is shown that the recently introduced L1TV functional can be used to explicitly compute the flat norm for codimension one boundaries and theflat norm decomposition, and a method for denoising nonboundary or highercodimension sets is obtained.
Abstract: We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for codimension one boundaries. Furthermore, using L1TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of L1TV functional, we obtain a method for denoising nonboundary or higher codimension sets. The flat norm decomposition of differences can made to depend on scale using the flat norm with scale which we define in direct analogy to the L1TV functional. We illustrate the results and implications with examples and figures.
TL;DR: In this paper, the nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied and several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valued Lp− spaces are given.
Abstract: The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valued Lp− spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.
TL;DR: It is shown that Karush-Kuhn-Tucker pseudoinvexity is a necessary and suffcient condition for a vector Karush -Tucker solution to be a weakly efficient solution.
Abstract: We introduce some concepts of generalized invexity for the continuous-time multiobjective programming problems, namely, the concepts of Karush-Kuhn-Tucker invexity and Karush-Kuhn-Tucker pseudoinvexity. Using the concept of Karush-Kuhn-Tucker invexity, we study the relationship of the multiobjective problems with some related scalar problems. Further, we show that Karush-Kuhn-Tucker pseudoinvexity is a necessary and suffcient condition for a vector Karush-Kuhn-Tucker solution to be a weakly efficient solution.
TL;DR: In this article, the Jensen-quadratic functional equation was solved for the cubic functional equation 3 [ g (x + y ) + g ( x − y )+ 6 g( x ) ] and the Jensen functional equation for the Jensen quadratic equation f(x, y, w ) = f( x, z ) + f ( x, w )+ f ( y, z + f( y, y + 6 g (y, w ), w ) + w ).
Abstract: We obtain the general solutions of the cubic functional equation 3 [ g ( x + y ) + g ( x − y ) + 6 g ( x ) ] = 2 g ( 2 x + y ) + 2 g ( 2 x − y ) + g ( − x − y ) + g ( − x + y ) + 6 g ( − x ) and the Jensen-quadratic functional equation f ( ( x + y ) / 2 , z + w ) + f ( ( x + y ) / 2 , z − w ) = f ( x , z ) + f ( x , w ) + f ( y , z ) + f ( y , w ) .
TL;DR: In this paper, upper and lower bounds for Mathieu's series are established, which refine to a certain extent a sharp double inequality obtained by Alzer-Brenner-Ruehr in 1998.
Abstract: Two upper and lower bounds for Mathieu's series are established, which refine to a certain extent a sharp double inequality obtained by Alzer-Brenner-Ruehr in 1998. Moreover, the very closer lower and upper bounds for ζ(3) are deduced.
TL;DR: In this article, the generalized Hyers-Ulam stability of Cauchy-Jensen additive mappings was investigated and it was shown that if a mapping satisfies the functional inequalities with perturbation which satisfies certain conditions, then there exists an additive mapping near the mapping.
Abstract: We investigate the generalized Hyers-Ulam stability of the functional inequalities associated with Cauchy-Jensen additive mappings. As a result, we obtain that if a mapping satisfies the functional inequalities with perturbation which satisfies certain conditions, then there exists a Cauchy-Jensen additive mapping near the mapping.
TL;DR: In this article, the authors presented several characterizations of a local α-times integrated C-semigroup by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem.
Abstract: This paper presents several characterizations of a local α-times integrated C-semigroup {T(t);0≤tlτ} by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem. We also discuss properties concerning the nondegeneracy of T(⋅), the injectivity of C, the closability of subgenerators, the commutativity of T(⋅), and extension of solutions of the associated abstract Cauchy problem.
TL;DR: In this paper, isomorphisms between C *-algebras, Lie C*-algeses, and JC*-Algeses are investigated, and derivations on C * algebra, Lie c*-e c e c * e c e and JC * e e e are associated with the Cauchy-Jensen functional equation 2f((x).
Abstract: We investigate isomorphisms between C*-algebras, Lie C*-algebras, and JC*-algebras, and derivations on C*-algebras, Lie C*-algebras, and JC*-algebras associated with the Cauchy–Jensen functional equation 2f((x
TL;DR: In this article, the authors consider the set of sequences of positive real numbers and show that some subclasses of these sequences have nice selection and game theoretic properties, and they show that these subclasses have certain nice selection properties.
Abstract: We consider the set 𝕊 of sequences of positive real numbers and show that some subclasses of 𝕊 have certain nice selection and game theoretic properties.
TL;DR: In this paper, the authors studied meromorphic functions that share a small function, and proved the following result: if f and g are two transcendental functions in the complex plane and n is a positive integer, then either f(z) or g(z), for a constant satisfying tn
Abstract: We will study meromorphic functions that share a small function, and prove the following result: let f(z) and g(z) be two transcendental meromorphic functions in the complex plane and let n≥11 be a positive integer. Assume that a(z)(≢0) is a common small function with respect to f(z) and g(z). If fnf′ and gng′ share a(z) CM, then either fn(z)f′(z)gn(z)g′(z)≡a2(z), or f(z)≡tg(z) for a constant satisfying tn
TL;DR: In this article, the existence and uniqueness of the strong solution for a multitime evolution equation with nonlocal initial conditions is proved based on a priori estimates and on the density of the range of the operator generated by the considered problem.
Abstract: The existence and uniqueness of the strong solution for a multitime evolution equation with nonlocal initial conditions are proved. The proof is essentially based on a priori estimates and on the density of the range of the operator generated by the considered problem.
TL;DR: In this article, it was shown that for every point from the boundary of a maximal parallelizable region, there exists exactly one orbit contained in this region which is a subset of the first prolongational limit set of the point.
Abstract: We are interested in the first prolongational limit set of the boundary of parallelizable regions of a given flow of the plane which has no fixed points. We prove that for every point from the boundary of a maximal parallelizable region, there exists exactly one orbit contained in this region which is a subset of the first prolongational limit set of the point. Using these uniquely determined orbits, we study the structure of maximal parallelizable regions.
TL;DR: In this paper, the noncommutative neutrix product f ∘ g of f and g is defined to be the neutrix limit of the sequence { f g n }, provided the limit h exists.
Abstract: Let f
and g be distributions and let g n = ( g * δ n ) ( x ) , where δ n ( x ) is a certain sequence converging to the Dirac-delta function δ ( x ) .
The noncommutative neutrix product f ∘ g of f and g is defined to be the neutrix limit of the sequence { f g n } , provided the limit h exists in the sense that N‐ lim n → ∞ 〈 f ( x ) g n ( x ) , φ ( x ) 〉 = 〈 h ( x ) , φ ( x ) 〉 , for all test functions in 𝒟 . In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products x + r ln x + ∘ x − − r − 1 ln x −
and x − − r − 1 ln x − ∘ x + r ln x + are proved to exist and are evaluated for r = 1 , 2 , … . It is consequently seen that these two products are in fact equal.
Abstract: The differential equation u ' ( t ) + A u ( t ) = f ( t ) ( − ∞ t ∞ ) in a general Banach space E with the strongly positive operator A is ill-posed in the Banach space C ( E ) = C ( ℝ , E ) with norm ‖ ϕ ‖ C ( E ) = sup − ∞ t ∞ ‖ ϕ ( t ) ‖ E . In the present paper, the well-posedness of this equation in the Holder space C α ( E ) = C α ( ℝ , E ) with norm ‖ ϕ ‖ C α ( E ) = sup − ∞ t ∞ ‖ ϕ ( t ) ‖ E + sup − ∞ t t + s ∞ (‖ ϕ ( t + s ) − ϕ ( t ) ‖ E / s α ), 0 α 1 , is established. The almost coercivity inequality for solutions of the Rothe
difference scheme in C ( ℝ τ , E ) spaces is proved. The well-posedness of this difference scheme in C α ( ℝ τ , E ) spaces is obtained.
TL;DR: Park and Bae as discussed by the authors improved their results and obtained better results for a Cauchy-Jensen¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯functional equation, and established new theorems for the======generalized Hyers-Ulam stability of a CÀ™¢¿¿¯¯equation.
Abstract: In 2006, W. G. Park and J. H. Bae investigated the Hyers-Ulam
stability of a Cauchy-Jensen functional equation. In this paper, we
improve their results and obtain better results for a Cauchy-Jensen
functional equation. Also, we establish new theorems for the
generalized Hyers-Ulam stability of a Cauchy-Jensen functional
equation.
TL;DR: In this article, the Navier-Stokes equations are perturbed with a maximal monotone operator, in a bounded domain, in 2D and 3D, using the theory of nonlinear semigroups.
Abstract: We study Navier-Stokes equations perturbed
with a maximal monotone operator, in a bounded domain, in 2D and
3D. Using the theory of nonlinear semigroups, we prove existence results
for strong and weak solutions. Examples are also provided.
TL;DR: In this paper, the properties of the weighted Hadamard-type singular integrals are revealed by means of an expression with a kind of integral operators, and applications of strongly singular integral equations are discussed and illustrated.
Abstract: By means of an expression with a kind of integral operators, some
properties of the weighted Hadamard-type singular integrals are revealed. As applications, the
solution for certain strongly singular integral equations is discussed and illustrated.
TL;DR: In this article, the distance between any point and the solution set for variational inequalities is measured by using the strong monotonicity of the perturbed fixed-point map and the normal map associated with variational inequality.
Abstract: By using the strong monotonicity of the perturbed fixed-point map and the normal map associated with cocoercive variational inequalities, we establish two new global bounds measuring the distance between any point and the solution set for cocoercive variational inequalities.
TL;DR: In this article, the authors studied the polyharmonic nonlinear elliptic boundary value problem on the unit ball B in ℝ n ( n ≥ 2 ) ( − △ ) m u + g ( ⋅, u ) = 0, in B (in the sense of distributions) and gave some existence results.
Abstract: Here we study the polyharmonic nonlinear elliptic
boundary value problem on the unit ball B in ℝ n ( n ≥ 2 ) ( − △ ) m u + g ( ⋅ , u ) = 0 , in B (in the sense of distributions)
lim x → ξ ∈ ∂ B ( u ( x ) / ( 1 − | x | 2 ) m − 1 ) = 0 ( ξ ) . Under appropriate conditions related to a Kato class on the nonlinearity
g ( x , t ) , we give some existence results. Our approach is based on estimates for the polyharmonic Green function
on B with zero Dirichlet boundary conditions, including a 3G-theorem,
which leeds to some useful properties on functions belonging to the Kato class.
TL;DR: In this paper, the concept of bounded and unbounded Fredholm operators on Hilbert C*-modules over an arbitrary C *-algebra is discussed and the Atkinson theorem is generalized for bounded and bounded Feredholm operators over C * -algebras of compact operators.
Abstract: The concept of unbounded Fredholm operators on Hilbert C*-modules over an arbitrary C*-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C*-modules over C*-algebras of compact operators In the framework of Hilbert C*-modules over C*-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same