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This paper reviews the old and new landmark extensions of the famous intermediate value theorem (IVT) of Bolzano and Poincare to a set-valued operator \({\Phi : E \supset X \rightrightarrows E}\) defined on a possibly non- convex, non-smooth, or even non-Lipschitzian domain X in a normed space E. Such theorems are most general solvability results for nonlinear inclusions: \({\exists x_{0} \in X}\) with \({0 \in \Phi (x_{0}).}\) Naturally, the operator Φ must have continuity properties (essentially upper semi- or hemi-continuity) and its values (assumed to be non-empty closed sets) may be convex or have topological properties that extend convexity. Moreover, as the one-dimensional IVT simplest formulation tells freshmen calculus students, to have a zero, the mapping must also satisfy “direction conditions” on the boundary ∂X which, when \({X = [a,b] \subset E = \mathbb{R}}\), Φ (x) = f(x) is an ordinary single-valued continuous mapping, consist of the traditional “sign condition” f (a) f (b) ≤ 0. When X is a convex subset of a normed space, this sign condition is expressed in terms of a tangency boundary condition \({\Phi (x) \cap T_{X}(x) \neq \emptyset}\), where TX(x) is the tangent cone of convex analysis to X at \({x \in \partial X}\). Naturally, in the absence of convexity or smoothness of the domain X, the tangency condition requires the consideration of suitable local approximation concepts of non-smooth analysis, which will be discussed in the paper in relationship to the solvability of general dynamical systems. Open image in new window

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 derive some subordination and superordination results for certain p-valent analytic functions in the open unit disc, which are acted upon by an integral operator. Relevant connection of the results, which are presented in this paper with various known results are also considered. Open image in new window

This paper is concerned with stability analysis problem for uncertain stochastic neural networks with interval time-varying delays. The parameter uncertainties are assumed to be norm bounded and the delay is assumed to be time varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Both the cases of the time-varying delays which may be dif- ferentiable and may not be differentiable are considered in this paper. Based on the Lyapunov-Krasovskii functional and stochastic stability theory, delay/interval-dependent stability criteria are obtained in terms of linear matrix inequalities. Some stability criteria are formulated by means of the feasibility of a linear matrix inequality (LMI), by introducing some free-weighting matrices. Finally, three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed LMI conditions.

In this paper, we prove that the sequence {xn} generated by modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al. [J Appl Math Comput 29:383–389, 2009] converges strongly to a fixed point of a nonexpansive mapping T in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Furthermore, we present an example that illustrates our result in the setting of a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. The results of this paper extend and improve several results presented in the literature in the recent past. Open image in new window

In this paper, we consider the compound Poisson risk model involving two types of dependent claims, namely main claims and by-claims. The by-claim is induced by the main claim with a certain probability and the occurrence of a by-claim may be delayed depending on associated main claim amount. Using Rouche's theorem, both of the survival probability with zero initial surplus and the Laplace transform of the survival probability are obtained from an integro-differential equations system. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. An exact representation for the solution of this equation is derived through an associated compound geometric distribution. For exponential claim sizes, we present an explicit formula for the survival probability. We also illustrate the influence of model parameters in the dependent risk model on the survival probability by numerical examples.

In this paper, we obtain the solution of a new generalized reciprocal type functional equation in two variables and investigate its generalized Hyers–Ulam stability in non-Archimedean fields. We also present the pertinent stability results of Hyers–Ulam–Rassias stability, Ulam–Gavruta–Rassias stability and J. M. Rassias stability controlled by the mixed product-sum of powers of norms.

A dynamical system model is presented in this paper for genetic regulatory networks with hybrid regulatory mechanism. The sufficient conditions for the stability of the proposed model are established based on the Lyapunov functional method and linear matrix inequality techniques. To test the effectiveness and correctness of our theoretical results, illustrative examples regarding modified repressilator and modified 5-node genetic network models are also presented.

In this paper we show that if a ring R has finite Goldie dimension, then every finitely generated ideal of R consisting of zero-divisors has non-zero annihilator. We also construct an example of a ring of infinite Goldie dimension such that above condition does not hold. Mathematics Subject Classification (2010) 16P60 · 16D25

Let R be a Noetherian ring. We denote by G(R) the simple graph whose vertices are elements of R and in which two distinct vertices x and y are joined by an edge if x − y is a zero-divisor of R. Let \({\overline{G}(R)}\) be the complementary graph to G(R) and \({\overline{\chi}(R)}\) be the chromatic number of the graph \({\overline{G}(R)}\). In this paper, we determine the chromatic number \({\overline{\chi}(R)}\). Let P1, . . . , Pt be all maximal prime divisors of (0). If \({\overline{\chi}(R)}\) is finite, then $$\overline{\chi}(R) = {\rm min} \{|R/P_{i}|; i = 1, 2, \ldots, t\}$$ where \({|R/P_{i}|}\) denotes the number of the set R/Pi.

In this paper, we investigate a boundary value problem for fractional differential equations with fractional derivative condition. Some new existence results are obtained using Banach contraction principle and Leray–Schauder nonlinear alternative.

Ultrafunctions are a particular class of functions defined on a Non Archimedean field R^{*}\supset R. They have been introduced and studied in some previous works ([1],[2],[3]). In this paper we introduce a modified notion of ultrafunction and we discuss sistematically the properties that this modification allows. In particular, we will concentrated on the definition and the properties of the operators of derivation and integration of ultrafunctions., 29 pages, Keywords: Ultrafunctions, Delta function, distributions, Non Archimedean Mathematics, Non Standard Analysis. arXiv admin note: text overlap with arXiv:1302.7156

In this paper, we introduce (p, q)-Bernstein Durrmeyer operators. We define (p, q)-beta integral and use it to obtain the moments of the operators. We obtain uniform convergence of the operators by using Korovkin’s theorem. We estimate direct results of the operators by means of modulus of continuity and Peetre K-functional. Finally, we find Voronovskaya-type theorem for the operators.

An ordered set P has the fixed point property iff every order-preserving self-map of P has a fixed point. This paper traces the chronological development of research on this property, including most recent developments and open questions. Open image in new window

In this paper, the numerical solutions of doubly perturbed stochastic delay differential equations driven by Levy process are investigated. Using the Euler–Maruyama method, we define the numerical solutions, and show that the numerical solutions converge to the true solutions under the local Lipschitz condition. As a corollary, we give the order of convergence under the global Lipschtiz condition. Open image in new window

In this paper, we investigate the set of solutions for nonlinear Volterra type integral equations in Banach spaces in the weak sense and under Henstock–Kurzweil–Pettis integrability. Moreover, a fixed point result is presented for weakly sequentially continuous mappings defined on the function space C(K, X), where K is compact Hausdorff and X is a Banach space. The main condition is expressed in terms of axiomatic measure of weak noncompactness.

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.

In this paper, we consider a one-dimensional linear Timoshenko system of thermoelasticity type III and prove a polynomial stability result for the non-equal wave-propagation speed case.

In this paper, we consider the regularity criterion for the 3D MHD equations and prove that if the gradient of the pressure belongs to $${L^\frac{2}{2-r}(0,T;\dot X_r(\mathbb{R}^{3}))}$$ with $${0\leq r\leq 1}$$ , then the solution is smooth. Notice that we extend the result given by Gala (Appl Anal 92:96–103, 2013).

In this paper the prediction problem is studied under members of a class \({\Im^{*}}\) of multivariate distributions, constructed by AL-Hussaini and Ateya (Stat Pap 46:321–338, 2005; J Egypt Math Soc 14(1):45–54, 2006). More attention is given to bivariate compound Rayleigh distribution, which is a member of this class, as illustrative example.

In this paper, we obtain a new characterization of p-nilpotent groups under the assumption that some maximal subgroups of Sylow subgroup are \({\mathcal{F}}\)-supplemented. As its applications, we generalize many known results.

In this paper, two pivotal statistics are introduced to construct prediction intervals for future lifetime of three parameters Weibull observations based on generalized order statistics, which can be widely applied in reliability theory and lifetime problems. The probability density functions as well as the explicit form of the distribution functions of our pivotal statistics are derived. Monte Carlo simulations are performed to demonstrate the efficiency of the proposed methods and a real data analysis is conducted for illustrative purposes.

We present a note on the paper by Brown and Wu (J Math Anal Appl 337:1326–1336, 2008). Indeed, we extend the multiplicity results for a class of semilinear elliptic system to the quasilinear elliptic system of the form: $$\left\{\begin{array}{ll}-\Delta_p u + m(x)\,|u|^{p-2}u = \frac{\alpha}{\alpha + \beta} \, |u|^{\alpha -2} \,u\,|v|^{\beta}, \quad\quad\quad\quad\quad\quad\quad \,\,\, x\in \Omega,\\ -\Delta_p v + m(x)\,|v|^{p-2}v = \frac{\beta}{\alpha + \beta} \, |u|^{\alpha} \,|v|^{\beta -2} \,v, \quad\quad\quad\quad\quad\quad\quad \,\,\, x\in \Omega,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial n} = \lambda\,a(x)|u|^{\gamma-2}u, \,\,\,|\nabla v|^{p-2} \frac{\partial v}{\partial n} = \mu\,b(x)|v|^{\gamma-2}v, \quad x\in \partial \Omega. \end{array} \right.$$ Here Δ p denotes the p-Laplacian operator defined by \({\Delta_{p}z=div\,(|\nabla z|^{p-2} \nabla z), p >2 , \Omega \subset \mathbb{R}^N}\) is a bounded domain with smooth boundary, \({\alpha >1 , \beta >1 , 2 p, p*=\infty {\rm if} N \,\leq p), \frac{\partial}{\partial n}}\) is the outer normal derivative, \({(\lambda, \mu) \in \mathbb{R}^{2} {\setminus} \{(0,0)\},}\) the weight m(x) is a positive bounded function, and \({a(x), b(x)\in C(\partial \Omega)}\) are functions which change sign in \({\overline{\Omega}.}\)

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 implement automatic procedures to enumerate all Nash maximal subsets of a bimatrix game and compute their dimensions. We propose a linear programming approach to identify extreme perfect Nash equilibria, enumerate all Selten maximal subsets and compute their dimensions. We present the Eχ-MIPerfect and the EEE-Perfect algorithms which enumerate all extreme perfect Nash equilibria. We finally report and comment computational experiments on randomly generated bimatrix games with different size and density.

We prove in this paper that the intersection numbers between periodic orbits have an intrinsic meaning for the variational problem (J,Cβ) {Bahri (Pseudo-Orbits of Contact Forms Pitman Research Notes in Mathematics Series No. 173, 1984), Bahri (C R Acad Sci Paris 299, Serie I 15:757–760, 1984), Bahri (Classical and Quantic periodic motions of multiply polarized spin-manifolds. Pitman Research Notes in Mathematics Series No. 378, 1998)}, corresponding to the periodic orbit problem on a sub-manifold of the loop space of a three dimensional compact contact manifold (M, α). Open image in new window

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.

Let R be a commutative ring. The unit graph of R, denoted by G(R), is a graph with all elements of R as vertices and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ U(R) where U(R) denotes the set of all units of R. In this paper, we examine the preservation of the connectedness, diameter, girth, and some other properties, such as chromatic index, clique number and planarity of the unit graph G(R) under extensions to polynomial and power series rings.

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.

Reversible difference sets have been studied extensively by many people. Dillon showed that reversible difference sets existed in groups (C2r ) 2 and C4. Davis and Polhill showed the existence of DRAD difference sets in the groups (C2 r ) 2 for r ≥ 2 and also for the group C4. This paper gives a construction technique utilizing character values, rational idempotents, and tiles to produce both reversible and DRAD Hadamard difference sets in the group C2r × C2r for r ≥ 2a nd inC4. We also show necessary conditions for both reversible and DRAD difference sets in abelian 2-groups.

In this paper we investigate a maximax optimization problem related to a homogeneous Dirichlet problem in two classes of rearrangements. We prove existence and representation of the maximizers.

Let C be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E. Let T: C→ C be relatively nonexpansive mapping and let Ai: C→ E* be Li-Lipschitz monotone mappings, for i = 1,2. In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a relatively nonexpansive mapping and the solution set of variational inequality problems for A1 and A2. Under some mild assumptions, we show that the proposed algorithm converges strongly to a point in \({F(T)\cap VI(C, A_1)\cap VI(C, A_2)}\). Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

This paper studies the adaptive synchronization of two complex networks with non-delayed and delayed couplings, in which the coupling configuration matrices are not necessarily symmetric or irreducible. Considering the case of identical and nonidentical network topological structures, we obtain several criteria for synchronization of two complex networks based on the Lyapunov stability theory. Numerical simulations are presented to demonstrate the effectiveness of the proposed criteria. Open image in new window