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Abstract: In this paper, characterizations of the existence of the directional derivative and second-order parabolic directional derivative of a locally Lipschitzian function are established. These characterizations involve the adjacent cone and second-order adjacent set of the graph of the function. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to derive two transformation formulae which imply relations between basic and bilateral basic hypergeometric series. Some special cases of them are also discussed. [Copyright &y& Elsevier]

Abstract: Let and be analytic in the unit disk. The Hadamard product of f and g is defined by . This paper gives some characterizations of functions in spaces in terms of the Hadamard products. [Copyright &y& Elsevier]

Abstract: In this paper, we study the stability of a system of wave equations which are weakly coupled and partially damped. Using a frequency domain approach based on the growth of the resolvent on the imaginary axis, we establish the polynomial energy decay rate for smooth initial data. We show that the behavior of the system is sensitive to the arithmetic property of the ratio of the wave propagation speeds of the two equations. [Copyright &y& Elsevier]

Abstract: In this paper, we give a characteristic of abstract convexity structures on topological spaces with selection property. We show that if a convexity structure defined on a topological space has the weak selection property then satisfies -condition. Moreover, in a compact convex subset of a topological space with convexity structure, the weak selection property implies the fixed point property. [Copyright &y& Elsevier]

Abstract: In this paper, the connection between -analogue of special functions and representations of certain two parameter quantum algebras has been developed. The realizations of the representations of these quantum algebras are in terms of -derivative operators. The matrix elements of these representations are computed in terms of the generalized -hypergeometric series. This leads to the developments in the theory of -special functions. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in . Upper and lower bounds are derived for the decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument. [Copyright &y& Elsevier]

Abstract: In this paper we prove, among other things, that the space of all holomorphic functions on an open subset U of a complex metrizable space E, endowed with the Nachbin ported topology, is metrizable only if E has finite dimension. This answers a question by Mujica in [J. Mujica, Gérmenes holomorfos y funciones holomorfas en espacios de Fréchet, Publicaciones del Departamento de Teoría de Funciones, Universidad de Santiago, Spain, 1978]. [Copyright &y& Elsevier]

Abstract: In this paper, we present some fixed-point theorems for families of weakly non-expansive maps under some relatively weaker and more general conditions. Our results generalize and improve several results due to Jungck [G. Jungck, Fixed points via a generalized local commutativity, Int. J. Math. Math. Sci. 25 (8) (2001) 497–507], Jachymski [J. Jachymski, A generalization of the theorem by Rhoades and Watson for contractive type mappings, Math. Japon. 38 (6) (1993) 1095–1102], Guo [C. Guo, An extension of fixed point theorem of Krasnoselski, Chinese J. Math. (P.O.C.) 21 (1) (1993) 13–20], Rhoades [B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–290], and others. [Copyright &y& Elsevier]

Abstract: By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden–Fowler delay dynamic equations on a time scale ; here γ is a quotient of odd positive integers with real-valued positive rd-continuous functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales. Our results in this paper not only extend the results given in [R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second-order delay dynamic equations, Can. Appl. Math. Q. 13 (1) (2005) 1–18] but also unify the oscillation of the second-order Emden–Fowler delay differential equation and the second-order Emden–Fowler delay difference equation. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the global estimates for divergence form elliptic equations in the Orlicz space , where Ω is a bounded Reifenberg domain in . Our result generalizes the estimates. [Copyright &y& Elsevier]

Abstract: A family of algorithms for approximate solution of the bound-constrained minimization problem was introduced in [K.A. Ariyawansa, W.L. Tabor, A class of collinear scaling algorithms for bound-constrained optimization: Derivation and computational results, Technical Report 2003-1, Department of Mathematics, Washington State University, Pullman, WA, 2003, submitted for publication. Available at http://www.math.wsu.edu/math/TRS/2003-1.pdf]. These algorithms employ the standard barrier method, with the inner iteration based on trust region methods. Local models are conic functions rather than the usual quadratic functions, and are required to match first and second derivatives of the barrier function at the current iterate. The various members of the family are distinguished by the choice of a vector-valued parameter, which is the zero vector in the degenerate case that quadratic local models are used. This paper presents a convergence analysis of the family of algorithms presented in [K.A. Ariyawansa, W.L. Tabor, A class of collinear scaling algorithms for bound-constrained optimization: Derivation and computational results, Technical Report 2003-1, Department of Mathematics, Washington State University, Pullman, WA, 2003, submitted for publication. Available at http://www.math.wsu.edu/math/TRS/2003-1.pdf]. Specifically, convergence properties similar to those of barrier methods using quadratic local models are established. [Copyright &y& Elsevier]

Abstract: This paper introduces a vectorial form of equilibrium version of Ekeland-type variational principle. Some equivalent results to our variational principle are given. As applications, we derive the existence of solutions of a vector equilibrium problem in the setting of complete quasi-metric spaces with a W-distance. Caristi–Kirk fixed point theorem for multivalued maps is also established in a more general setting. [Copyright &y& Elsevier]

Abstract: By making use of inclusion theorem, we show in this paper the existence of solutions with a single semicycle for a general second-order rational difference equation. As a corollary, our results positively confirm Conjectures 4.8.3 and 5.4.6 in [M.R.S. Kulenovic, G. Ladas, Dynamics of Second-Order Rational Difference Equations, with Open Problems and Conjectures, Chapman and Hall/CRC, 2002]. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a food-limited population model with impulsive effect. Several explicit sufficient conditions are established for oscillation and nonoscillation of solutions of the equations. [Copyright &y& Elsevier]

Abstract: The fixed point theory of set-valued contractions which was initiated by Nadler [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–488] was developed in different directions by many authors, in particular, by [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972) 26–42; N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188; Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112]. In the present paper, the concept of contraction for set-valued maps in metric spaces is introduced and the conditions guaranteeing the existence of a fixed point for such a contraction are established. One of our results essentially generalizes the Nadler and Feng–Liu theorems and is different from the Mizoguchi–Takahashi result. The second result is different from the Reich and Mizoguchi–Takahashi results. The method used in the proofs of our results is inspired by Mizoguchi–Takahashi and Feng–Liu''s ideas. Comparisons and examples are given. [Copyright &y& Elsevier]

Abstract: We introduce a symmetric dual pair for a class of nondifferentiable multi-objective fractional variational problems. Weak, strong, converse and self duality relations are established under certain invexity assumptions. The paper includes extensions of previous symmetric duality results for multi-objective fractional variational problems obtained by Kim, Lee and Schaible [D.S. Kim, W.J. Lee, S. Schaible, Symmetric duality for invex multiobjective fractional variational problems, J. Math. Anal. Appl. 289 (2004) 505–521] and symmetric duality results for the static case obtained by Yang, Wang and Deng [X.M. Yang, S.Y. Wang, X.T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl. 274 (2002) 279–295] to the dynamic case. [Copyright &y& Elsevier]

Abstract: In this paper we study the hyponormality and subnormality of 2-variable weighted shifts using the Schur product techniques in matrices. As applications, we generalize the result in [R. Curto, J. Yoon, Jointly hyponormal pairs of subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006) 5135–5159, Theorem 5.2] and give a non-trivial, large class satisfying the Curto–Muhly–Xia conjecture [R. Curto, P. Muhly, J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35 (1988) 1–22] for 2-variable weighted shifts. Further, we give a complete characterization of hyponormality and subnormality in the class of flat, contractive, 2-variable weighted shifts with the condition that the norm of the 0th horizontal 1-variable weighted shift of T is a given constant. [Copyright &y& Elsevier]

Abstract: In this paper, we study the weak convergence of “Leray-type” solutions of a magneto-hydro-dynamic systems. The proofs are based on weak compactness arguments and linear algebra methods. [Copyright &y& Elsevier]

Abstract: Consider the higher order nonlinear partial difference equation of neutral type where , , and . In this paper, we first establish the discrete Arzela–Ascoli''s theorem. Next, we obtain some sufficient conditions for the existence of bounded and unbounded nonoscillatory solution of Eq. (∗). [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the relation between the characteristic function of two nonconstant meromorphic functions with three weighted sharing values, which improves a result given by H.X. Yi and Y.H. Li. From this we establish a theorem which improves a result given by P. Li and C.C. Yang. [Copyright &y& Elsevier]

Abstract: In this paper, assume that h is nonnegative and , we prove that if is sufficiently small, then there are at least three positive solutions of Eq. (1) in an exterior cylinder domain. [Copyright &y& Elsevier]

Abstract: In this paper we discuss log-convex solutions of the second order to the functional equation with initial condition given by We prove that if g satisfies an appropriate asymptotic condition, then (E) admits at most one solution f, which is eventually log-convex of the second order. Moreover, f can be defined explicitly in terms of g. If, in addition, g is eventually log-concave of the second order, then (E) has exactly one eventually log-convex of the second order solution. Our results complement similar ones established by R. Webster [R. Webster, Log-convex solutions to the functional equation : Γ-type functions, J. Math. Anal. Appl. 209 (1997) 605–623] and generalize results obtained by L. Lupaş [L. Lupaş, The C-function of E.W. Barnes, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 1 (1990) 5–11]. [Copyright &y& Elsevier]

Abstract: In this paper we will investigate the stability of the generalized (pexiderized) sine functional equation [Copyright &y& Elsevier]

Abstract: Let X be a metric space with metric d, denote the family of all nonempty compact subsets of X and, given , let be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval . The main result of the paper is the following selection theorem: If , , and , then there exists a single-valued function of bounded variation such that for all , , and . We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1–82]. In contrast to this, a multifunction F satisfying for some constant and all with (Lipschitz continuity with respect to ) admits a Lipschitz selection with a Lipschitz constant not exceeding C if and may have only discontinuous selections of bounded variation if . The same situation holds for continuous selections of when it is excess continuous in the sense that as for all and as for all simultaneously. [Copyright &y& Elsevier]

Abstract: We are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so-called degenerated Ginzburg–Landau equation takes the role of the classical Ginzburg–Landau equation as the amplitude equation of the system. This is the case when the relevant nonlinear terms vanish at the bifurcation point. Here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated Ginzburg–Landau equation. In this paper we restrict ourselves to a Swift–Hohenberg–Kuramoto–Shivashinsky equation as a model for such a pattern forming system. [Copyright &y& Elsevier]

Abstract: This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws. We show that the integrability conditions derived from each of the methods are the one and the same, suggesting that there exists a deep connection between these techniques for partial difference equations. [Copyright &y& Elsevier]

Abstract: In this paper, we provide oscillation properties of every solution of the neutral differential equation with positive and negative coefficients where , , , . [Copyright &y& Elsevier]

Abstract: In this paper, we study the normality of a family of meromorphic functions and obtain some normality results for meromorphic functions, which improve and generalize the related results of Gu, Bergweiler and Lin. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of periodic solutions of the second order differential equations . Using continuation lemma, we obtain the existence of periodic solutions provided that () is sublinear when x tends to positive infinity and satisfies a new condition where M, d are two positive constants. [Copyright &y& Elsevier]

Abstract: The sequence space was introduced and studied by Maddox [I.J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2) 18 (1967) 345–355]. In the present paper, the sequence spaces of non-absolute type which is derived by the generalized weighted mean is defined and proved that the spaces and are linearly isomorphic. Besides this, the β- and γ-duals of the space are computed and the basis of that space is constructed. Further, it is established that the sequence space has AD property and given f-dual of the space . Finally, the matrix mappings from the sequence spaces to the sequence space μ and from the sequence space μ to the sequence space are characterized. [Copyright &y& Elsevier]

Abstract: This paper deals with the global attraction and global weak attraction of the origin for the generalized Liénard equation. The problem proposed by Jiang is solved by a negative answer and some new criteria for the globally asymptotically stability are obtained by the ways different from the usual Filippov''s conditions. [Copyright &y& Elsevier]

Abstract: In this paper we establish the exact blow-up rate of the large solutions of a porous media logistic equation. We consider the carrying capacity function with a general decay rate at the boundary instead of the usual cases when it can be approximated by a distant function. Obtaining the accurate blow-up rate allows us to establish the uniqueness result. Our result covers all previous results on the ball domain and can be further adapted in a more general domain. [Copyright &y& Elsevier]

Abstract: In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459–472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278–2295]. Finally, we give two examples to illustrate the obtained results. [Copyright &y& Elsevier]

Abstract: In the present paper a nonlinear size-structured population dynamical model with size and density dependent vital rate functions is considered. The linearization about stationary solutions is analyzed by semigroup and spectral methods. In particular, the spectrally determined growth property of the linearized semigroup is derived from its long-term regularity. These analytical results make it possible to derive linear stability and instability results under biologically meaningful conditions on the vital rates. The principal stability criteria are given in terms of a modified net reproduction rate. [Copyright &y& Elsevier]

Abstract: In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in , and , we obtain that the elliptic system has three nontrivial solutions , and . It is remarkable that the third nontrivial solution is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of sign-changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory. The main theorems are some new three-solution theorems which are different from the famous Amann''s and Leggett-Williams'' three-solution theorems as well as the results in [F. Li, G. Han, Generalization for Amann''s and Leggett–Williams'' three-solution theorems and applications, J. Math. Anal. Appl. 298 (2004) 638–654]. These three solutions are all nonzero. One of them is positive, another is negative, and the third one is a sign-changing solution. Furthermore, the theoretical results are successfully applied to both integral and differential equations. [Copyright &y& Elsevier]

Abstract: The class of σ-regular matrices was defined and characterized by Schaefer [P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972) 104–110] and further studied by Mursaleen [Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford 34 (1983) 77–86], Ahmad and Mursaleen [Z.U. Ahmad, Mursaleen, An application of Banach limits, Proc. Amer. Math. Soc. 103 (1988) 244–246]. In this paper we characterize four-dimensional σ-multiplicative matrices, and establish a core theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions for the oscillation of the linear non-autonomous systems of difference equations with continuous arguments where , , , , ; τ, γ and , , are non-negative constants. Furthermore, assume , where , , , are continuous, bounded and do not change signs on . [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of stationary solutions to the Vlasov–Poisson–Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as . In fact, the stationary Vlasov–Poisson–Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being for some , it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the Vlasov–Poisson–Fokker–Planck system, J. Math. Anal. Appl. 202 (1996) 1058–1075] where the decay rate is assumed. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to obtain general input–output conditions for uniform exponential expansiveness of variational difference equations in terms of the complete admissibility of pairs of sequence spaces. We introduce a large class of Banach sequence spaces and we deduce the connections between the complete admissibility of the pair with and the uniform exponential expansiveness of a system of variational difference equations. We apply our results at the study of the uniform exponential expansiveness of linear skew-product flows. [Copyright &y& Elsevier]

Abstract: Carlitz has introduced an interesting q-analogue of Frobenius–Euler numbers in [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000; L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332–350]. He has indicated a corresponding Stadudt–Clausen theorem and also some interesting congruence properties of the q-Euler numbers. A recent author''s study of more general q-Euler and Genocchi numbers can be found in previous publication [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139–141]. In this paper we give a new construction of q-Euler numbers, which are different from Carlitz''s q-extension and author''s q-extension in previous publication (see [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139–141]). By using our q-extension of Euler numbers, we can also consider a new q-extension of Genocchi numbers and obtain some interesting relations between q-extension of Euler numbers and q-extension of Genocchi numbers. [Copyright &y& Elsevier]

Abstract: Some new generalizations of the Hilbert integral inequality by introducing real functions and . The results of this paper reduce to those of the corresponding inequalities proved by Gao [Mingzhe Gao, On Hilbert''s integral inequality, Math. Appl. 11 (3) (1998) 32–35]. Some applications are considered. [Copyright &y& Elsevier]

Abstract: The classical criterion of asymptotic stability of the zero solution of equations is that there exists a function , for some , such that for some . In this paper we prove that if is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , , where , uniformly for . [Copyright &y& Elsevier]

Abstract: In this paper, we study the thermoelastic system with second sound in one space variable. The mathematical model is a hyperbolic partial differential system with damping term which brings about some parabolic properties. The pointwise estimates of the solution are obtained and from which the generalized Huygens'' principle is shown. Our approach is based on detailed analysis on the Green function of the system. [Copyright &y& Elsevier]

Abstract: This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist? [Copyright &y& Elsevier]

Abstract: In this work we present an extension to arbitrary unital Banach algebras of a result due to Phillips [R.S. Phillips, Spectral theory of semigroups of linear operators, Trans. Amer. Math. Soc. 71 (1951) 393–415] (Theorem 1.1) which provides sufficient conditions assuring the uniform continuity of strongly continuous semigroups of linear operators. It implies that, when dealing with the algebra of bounded operators on a Banach space, the conditions of Phillips''s theorem are also necessary. Moreover, it enables us to derive necessary and sufficient conditions in terms of essential spectra which guarantee the uniform continuity of strongly continuous semigroups. We close the paper by discussing the uniform continuity of strongly continuous groups acting on Banach spaces with separable duals such that, for each , the essential spectrum of is a finite set. [Copyright &y& Elsevier]

Abstract: Given a function f in the class , Chandra [P. Chandra, Trigonometric approximation of functions in -norm, J. Math. Anal. Appl. 275 (2002) 13–26] approximated such an f by using trigonometric polynomials, which are the nth terms of either certain weighted mean or Nörlund mean transforms of the Fourier series representation for f. He showed that the degree of its approximation is . In this paper we obtain the same degree of approximation for a more general class of lower triangular matrices, and deduce some of the results of [P. Chandra, Trigonometric approximation of functions in -norm, J. Math. Anal. Appl. 275 (2002) 13–26] as corollaries. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the problem of framing 2-variable 2-parameter Bessel functions (2V2PBF) into the context of the representation of the 3-dimensional Lie algebra . Further, we derive generating relations and identities involving 2V2PBF. Furthermore, we deduce some new and known relations involving various other forms of generalized Bessel functions (GBF) and Bessel functions (BF) as applications of these results. [Copyright &y& Elsevier]

Abstract: In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826–847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is proved using a fixed point argument based on Schauder''s theorem. Uniqueness is proved using a non-standard technique based on a weak formulation of the problem. [Copyright &y& Elsevier]