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Green's formula and Gauss's formula are two important formulas in vector calculus. In 2008, Tarasov [12] developed the fractional Green and Gauss formulas and also suggested two possible extensions of his fractional vector formulas (see V.E. Tarasov (2008) [12] ). The first possible extension is to prove his fractional integral theorems for a general form of domains and boundaries, such as elementary regions. The second one is to generalize the formulations of fractional integral theorems for α > 1 . The purpose of this article is to follow the above two interesting suggestions and present the higher-order fractional Green and Gauss formulas that are the extensions of fractional integral theorems obtained by Tarasov. In particular, the obtained formulas can be reduced to the classical Green and Gauss formulas when α = 1 and the fractional Green and Gauss formulas in [12] when 0 < α ≤ 1 , respectively. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish the strong convergence of possible solutions to the following nonhomogeneous second order evolution system to an element of , with an exponential rate of convergence when , where A is a general maximal monotone operator in a real Hilbert space H, is a real constant and is a given function. We show also that the curve u is almost nonexpansive, and present some applications of our result. [Copyright &y& Elsevier]

Abstract: We prove the Korn''s inequality for the conformal Killing operator on pseudo-Euclidean space , and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which is a pseudo-Euclidean conformal generalization of Donati''s theorem for Euclidean Killing operator. [Copyright &y& Elsevier]

Abstract: In this paper we prove a new version of Cowling–Price''s theorem for the Dunkl transform on . [Copyright &y& Elsevier]

Abstract: Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant and for each there are cones with normal constant . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results. [Copyright &y& Elsevier]

Abstract: This paper studies the uniqueness problem on entire function that share a finite, nonzero value with their linear differential polynomials and proves some theorems which generalize some results given by Jank, Mues and Volkmann, P. Li, J.P. Wang and H.X. Yi. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems. [Copyright &y& Elsevier]

Abstract: This paper is devoted to studying the growth of solutions of equations of type where , and are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen. [Copyright &y& Elsevier]

Abstract: We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians. [Copyright &y& Elsevier]

Abstract: A ratio-dependent Leslie system with feedback controls is studied. By using a comparison theorem and constructing a suitable Lyapunov function, some sufficient conditions for the existence of a unique almost periodic solution (periodic solution) and the global attractivity of the solutions are obtained. Examples show that the obtained criteria are new, general, and easily verifiable. [Copyright &y& Elsevier]

Abstract: By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a three-stage-structured predator–prey system with time delay is established. [Copyright &y& Elsevier]

Abstract: In this paper, we prove a strong convergence theorem by the hybrid method for a family of nonexpansive mappings which generalizes Nakajo and Takahashi''s theorems [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], simultaneously. Furthermore, we obtain another strong convergence theorem for the family of nonexpansive mappings by a hybrid method which is different from Nakajo and Takahashi. Using this theorem, we get some new results for a single nonexpansive mapping or a family of nonexpansive mappings in a Hilbert space. [Copyright &y& Elsevier]

Abstract: Drewnowski and Paúl proved in [L. Drewnowski, P.J. Paúl, The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000) 485–503] that for any strongly nonatomic submeasure η on the power set of the ideal has the Nikodym property (NP); in particular, this result applies to densities defined by strongly regular matrices A. Grahame Bennett and the authors stated in [G. Bennett, J. Boos, T. Leiger, Sequences of 0''s and 1''s, Studia Math. 149 (2002) 75–99] that the strong null domain of any strongly regular matrix A has the Hahn property (HP). Moreover, Stuart and Abraham [C.E. Stuart, P. Abraham, Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems, J. Math. Anal. Appl. 300 (2) (2004) 351–361] pointed out that the said results are in some sense dual and that the last one follows from the first one by considering the density (defined by A) as submeasure on and the ideal as well by identifying with the set χ of sequences of 0''s and 1''s. In this paper we aim at a better understanding of the intimated duality and at a characterization of those members of special classes of matrices A such that has NP (equivalently, has HP). [Copyright &y& Elsevier]

Abstract: -convexity was introduced in [W. Briec, C. Horvath, -convexity, Optimization 53 (2004) 103–127]. Separation and Hahn–Banach like theorems can be found in [G. Adilov, A.M. Rubinov, -convex sets and functions, Numer. Funct. Anal. Optim. 27 (2006) 237–257] and [W. Briec, C.D. Horvath, A. Rubinov, Separation in -convexity, Pacific J. Optim. 1 (2005) 13–30]. We show here that all the basic results related to fixed point theorems are available in -convexity. Ky Fan inequality, existence of Nash equilibria and existence of equilibria for abstract economies are established in the framework of -convexity. Monotone analysis, or analysis on Maslov semimodules [V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and Its Applications, Math. Appl., vol. 401, Kluwer Academic, 1997; V.P. Litvinov, V.P. Maslov, G.B. Shpitz, Idempotent functional analysis: An algebraic approach, Math. Notes 69 (2001) 696–729; V.P. Maslov, S.N. Samborski (Eds.), Idempotent Analysis, Advances in Soviet Mathematics, Amer. Math. Soc., Providence, RI, 1992], is the natural framework for these results. From this point of view Max-Plus convexity and -convexity are isomorphic Maslov semimodules structures over isomorphic semirings. Therefore all the results of this paper hold in the context of Max-Plus convexity. [Copyright &y& Elsevier]

Abstract: We prove a uniqueness theorem for two non-constant meromorphic functions sharing three values which improves a recent result of T.C. Alzahary. As a consequence of our main result we also improve a theorem of G. Brosch. [Copyright &y& Elsevier]

Abstract: In this paper we present a sandwich-type theorem for Carathéodory multifunctions. We obtain our result by means of the Marczewski function. [Copyright &y& Elsevier]

Abstract: We report on a result of upper–lower solutions for nonlinear elliptic systems without the assumption of quasi-monotonicity. An application is described involving the existence of positive steady states of a certain interaction system arising in biology and medical sciences. [Copyright &y& Elsevier]

Abstract: We use a unified approach to obtain several integrability theorems of Boas [R.P. Boas, Integrability of trigonometric series, I, Duke Math. J. 18 (1951) 787–793]. In particular, we settle two conjectures of Móricz [F. Móricz, On the integrability of double cosine and sine series, II, J. Math. Anal. Appl. 154 (1991) 466–483] concerning double trigonometric series. [Copyright &y& Elsevier]

Abstract: It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then . The main result is an extension of the classical Cauchy–Goursat Theorem: the equality conjectured holds (with no boundary condition on ) under the additional hypothesis that the winding numbers of Γ define an function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if .) The proof uses approximations of a rectifiable path by piecewise linear paths. [Copyright &y& Elsevier]

Abstract: In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on . These results are not restricted to positive solutions. [Copyright &y& Elsevier]

Abstract: Of concern is a class of abstract semilinear integrodifferential equations with nonlocal initial conditions. Under some suitable hypotheses, we establish some new theorems about the existence of asymptotically almost automorphic solutions to the integrodifferential equations. Moreover, an example is given to illustrate our results. [Copyright &y& Elsevier]

Abstract: In this paper we obtain a very general intersection theorem for the values of a map. From this we derive an analytic alternative, a new type of minimax inequality and an alternative theorem concerning three abstract equilibrium problems. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with a uniqueness theorem of two meromorphic functions that have three weighted sharing values and a sharing set with two elements. The results in this paper improve those given by G. Brosch, K. Tohge, T.C. Alzahary and H.X. Yi and other authors. [Copyright &y& Elsevier]

Abstract: In this paper, the stability of functional differential equations (FDE) with impulses is investigated. Some comparison theorems are given. Several Lyapunov–Razumikhin functions of partial components of the state variable x, which can be much easier constructed, are used so that the conditions ensuring that stability are simpler and less restrictive. The results improve and generalize the ones in the literature. An example is also given to illustrate the importance of our results. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with a nondifferentiable minimax fractional problem with inequality constraints. We introduce a new class of generalized convex function, that is, nonsmooth generalized -d-univex function. In the framework of the new concept, we derive Kuhn–Tucker type sufficient optimality conditions and establish weak, strong and converse duality theorems for the problem and its three different types of dual problems. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the existence and multiplicity of nonzero periodic solutions with saddle point character for the following nonautonomous second order systems: Some new existence and multiplicity theorems are obtained by using saddle point reduction method and a three-critical-point theorem proposed by Brezis and Nirenberg. [Copyright &y& Elsevier]

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equation (a(t)x′)′+b(t)g(x)=0. Here g(x) satisfied the signum condition xg(x)>0 if x≠0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Lie´nard type. We also explain an analogy between our results and an oscillation criterion of Kneser–Hille type for linear differential equations. [Copyright &y& Elsevier]

In this paper, we establish two theorems of alternative with generalized subconvexlikeness. We introduce two dual models for a generalized fractional programming problem. Theorems of alternative are then applied to establish duality theorems and a saddle-point type optimality condition. [Copyright &y& Elsevier]

In this paper, we improve some previous results about multiple p-summing multilinear operators by showing that every multilinear form from L1 spaces is multiple p-summing for 1&les;p&les;2. The proof is based on the existence of a predual for the Banach space of multiple p-summing multilinear forms. We also show the failure of the inclusion theorem in this class of operators and improve some results of Y. Mele´ndez and A. Tonge about dominated multilinear operators. [Copyright &y& Elsevier]

Some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory. [Copyright &y& Elsevier]

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T :K→E be an asymptotically nonexpansive nonself-map with sequence {kn}n&ges;1⊂[1,∞), limkn=1, F(T):={x∈K: Tx=x}≠∅. Suppose {xn}n&ges;1 is generated iteratively by x1∈K, xn+1=P(1−αn)xn+αnT(PT)n−1xn, n&ges;1, where {αn}n&ges;1⊂(0,1) is such that ∊<1−αn<1−∊ for some ∊>0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑n&ges;1(kn2−1)<∞ and T is completely continuous, strong convergence of {xn} to some x&ast;∈F(T) is proved. If T is not assumed to be completely continuous but E also has a Fre´chet differentiable norm, then weak convergence of {xn} to some x&ast;∈F(T) is obtained. [Copyright &y& Elsevier]

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map ϕ from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that ϕ(f)(y)=ω(y)fh(y) for all f∈C(X) and y∈Y. The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that ϕ(a)=ωψ(a) for all a∈A. We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map ϕ from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional. [Copyright &y& Elsevier]

By means of the abstract continuation theory for k-contractions, some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation: dN/dt=N(t)a(t)−β(t)N(t)−∑lower limit j=1, upper limit n bj(t)Nt−σj(t)−∑lower limit i=1, upper limit m ci(t)N′t−τi(t). [Copyright &y& Elsevier]

There exists a set S with 3 elements such that if f is a non-constant entire function satisfying E(S,f)=E(S,f′), then f≡f′. The number 3 is best possible. The proof uses the theory of normal families in an essential way. [Copyright &y& Elsevier]