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The purpose of the paper is to study the uniqueness of meromorphic functions sharing a small function with weight. The results of the paper improve and extend some recent results due to Banerjee and Sahoo (Sarajevo J Math 20:69–89, 2012), which in turn radically improve, extend and supplement some results of Dyavanal (J Math Anal Appl 372(1):252–261, 2010; 374(1):334, 2011; 374(1):345–355, 2011).

We address in this paper the Fredholm and compactness issues for the variational problem (J,C β ), Bahri (Pitman Research Notes in Mathematics Series No. 173. Scientific and Technical, London, 1988), Bahri (C. R. Acad. Sci. Paris 299, Serie I, 15, 757–760, 1984). We prove that the intersection operator restricted to periodic orbits of the Reeb vector-field ∂per does not mix with the intersection operator ∂∞ of the critical points at infinity. The Fredholm issues are extensively discussed in the Introduction and solved in Bahri (Arab J Maths, 2014). We also address in this paper the issue of existence of periodic orbits for three-dimensional Reeb vector-fields, the Weinstein conjecture (Weinstein, J Differ Equ 33:353–358, 1979) on S3, solved in dimension 3 throughout the works of Rabinowitz (Commun Pure Appl Math 31:157–184, 1978) and Hofer (Invent Math 114:515–563, 1993); see also Hutchings (Proc. 2010 ICM 46:1022–1041, 2010) and Taubes (Geom Topol 11:2117–2202, 2007) for the full Weinstein conjecture in dimension 3. Following our previous work (Bahri, Adv Nonlinear Stud 8:1–17, 2008), we devise a new method to find these periodic orbits when they are of odd index. We conjecture that this method, when combined with the other results described above about the intersection operator, gives rise to a homology that is specific of the contact structure and that is invariant by deformation. The existence result, as derived here, is weaker than the one announced by Taubes (Geom Topol 11:2117–2202, 2007). After appropriate generalization, it provides a new proof, via variational theory, of the Weinstein conjecture on three-dimensional closed contact manifolds with finite fundamental group.

In this paper, we investigate commutativity of rings with involution in which derivations satisfy certain algebraic identities on Jordan ideals. Moreover, we extend some results for derivations of prime rings to Jordan ideals. Furthermore, an example is given to prove that the ∗-primeness hypothesis is not superfluous.

We introduce two generalizations, the first of which generalizes the concept of multiresolution analysis. We define the irregular generalized multiresolution analysis (IGMRA). This structure is defined taking translations on sets that are not necessarily regular lattices, for which certain density requirements are required, and without using dilations, also allows each subspace of IGMRA to be generated by outer frames of translations of different functions. The second generalization concerns the concept of association of wavelets to these new structures. We take frames of translations of a countable set of functions, which we called generalized wavelets, and define the concept of association of these generalized wavelets to those previously defined IGMRA. In the next stage, we prove two existence theorems. In the first theorem, we prove existence of IGMRA, and in the second existence of generalized wavelets associated with it. In the latter, we show that we are able to associate frames of translations with optimal localization properties, to IGMRA. In the last section of this paper, concrete examples of these structures are presented for L 2 ( R ) and for L 2 ( R 2 ) .

In this paper, some Gruss-type results via Pompeiu’s-like inequalities are proved. Mathematics Subject Classification 26D15 · 25D10

In this paper, we introduce some new kinds of generalized convexity, which include (semistrict) G-semipreinvexity and (semistrict) G-semipreincavity. Moreover, we establish the relations with common generalized convexity, present properties of (semistrictly) G-semipreinvex and (semistrictly) G-semipreincave functions, and also give characterizations of the classes of G-semipreinvex and G-semipreincave functions. Moreover, we deal with programming involving G-semipreinvex functions. Our results extend the existing ones in the literature.

In this paper, we shall give new examples on meromorphic functions that share one value with their first derivative and also give the solution for Riccati differential equation.

In this paper, we investigate separability of CP-graded ring extensions. With restrictions neither to graded fields nor to grading by torsion–free groups, we show that some results on graded field extensions given in Hwang and Wadsworth [Commun Algebra 27(2):821–840, 1999] hold.