Advances in Analysis

5

Authors: Cai-Mei Yan, Jin-Lin Liu

Let An denote the class of functions of the form f(z) = z + P1 k=n+1 akzk, which are analytic in the open unit disk U = {z : |z| < 1}. In this note we shall find max|z|=r<1 Re{f0(z) + zf00(z)} under the condition f0(z)  1+Az 1+Bz for f 2 An.

10

Authors: Mian Bahadur Zada, Muhammad Sarwar, Panda Sumati Kumari

In this article, we establish common fixed point theorems for two and four self-mappings in the context of complex valued metric spaces. The derived results generalize and extend some well known results in the literature. Examples are also given to demonstrate the validity of our results.

15

Authors: Yupin Wang, Shurong Sun, Zhenlai Han

In this paper, we investigate the existence and uniqueness of solution to boundary value problems for a class of nonlinear fuzzy factional differential equations involving the fuzzy gH- fractional Caputo derivative. By means of the Schauder fixed point theorem in semi-linear spaces and integral inequality technique, some qualitative results of solutions are obtained. An example is provided from which new results are found.

8

Authors: K.P.R. Rao, A.Som Babu, Md.Mustaq Ali

In this paper, we obtain a Presic type fixed point theorem for two pairs of jointly 2k-weakly compatible maps in fuzzy metric spaces.We also give an example to illustrate our main theorem. We obtain two corollaries for three maps and two maps.

10

Authors: Adela Ionescu, Mario Lefebvre, Florian Munteanu

Using the feedback linearization method, a state feedback control for the two-dimensional Kermack-McKendrick model for the spread of epidemics is obtained. This form of the dynamical system is suitable to carry out a qualitative analysis of the model. An optimal control problem for a stochastic version of the model is also set up and solved explicitly in a particular instance.

29

Authors: V.S. Guliyev, E.J. Ibrahimov, S.Ar. Jafarova

In this paper we consider some problems of the theory of approximation of functions on interval [0, ∞) in the metric of Lp,λ with weight sh2λ x using generalized Gegenbauer shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Gegenbauer shifts. We establish the equivalence of the modulus of smoothness and K-functional, defined in terms of the space of the Sobolev type corresponding to the Gegenbauer differential operator. We define function spaces of Nikol’skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use some functions classes of spectrum. In these classes, we prove analogues of Bernstein’s inequality and others for the Gegenbauer differential operator. Our results are analogues of the results for generalized Bessel shifts obtained in the work [30].

6

Authors: Yaé Ulrich Gaba

In this article, we solve an open problem initially suggested in [2], namely: Let (X,d) be a Hausdorff left K-complete T0-quasi-pseudometric space, φ : X → R be a bounded from above function and  the preorder induced by φ. Let F : X × X → X and Gi : X → X;i = 1, 2, · · · , N for N > 2 be N + 1 d-sequentially continuous mapping on X such that the pairs {F; Gi};i = 1, 2, · · · , N are weakly left-related. Problem: 1. Can we prove that F, G1, · · · , GN have a common coupled fixed point in X? 2. Alternatively, what could be a correct formulation of the statement, using the induced preorder and the weakly left-related property that guarantees a positive answer? We answer this question by the affirmative. In fact we prove that a more general result holds when F : X n → X for a natural number n > 2.

8

Authors: Yang Li, Yongshun Liang

In the present paper, upper bound estimation of upper Box dimension of RiemannLiouville fractional integral of order v of any continuous functions on a closed interval has been proved to be no more than 2 − v when 0 < v < 1. If a continuous function which satisfies α-Hölder condition on a closed interval, upper Box dimension of its Riemann-Liouville fractional integral is no more than 2 − α when 0 < α < 1. Upper bound of upper Box dimension of Riemann-Liouivlle fractional integral of certain type of fractal functions has been proved to be no more than Box dimension of functions themselves.

13

Authors: R.Sathya, K.Balachandran

In this paper we study the controllability of damped second order neutral impulsive stochastic functional differential system with infinite delay in Hilbert spaces. Sufficient conditions for controllability results are obtained by using the theory of cosine families of bounded linear operators and fixed point technique. An example is provided to illustrate the theory.

17

Authors: Chenkuan Li, Changpin Li, Bailey Kacsmar, Roque Lacroix, Kyle Tilbury

This paper is to study the following generalized Abel’s integral equation g(x) = 1 Γ(1 − α) Z x 0 f(ζ) (x − ζ) α dζ, where α ∈ R and its variant in the distributional (Schwartz) sense based on fractional calculus of distributions. We obtain a number of interesting and new results which are not achievable in the classical sense.

5

Authors: Shanming Ji, Jingxue Yin, Jian Deng

This paper deals with the existence of positive periodic solutions of the superlinear heat equation with inhomogeneity. We first prove the existence of positive periodic solutions to the equation with an indispensable nontrivial and globally small inhomogeneity; and then the non-existence for a locally large inhomogeneity with possibly small support is obtained.

6

Authors: Tinggang Zhao

In this paper, we consider some interpolations of Birkhoff-type with fractional-order derivative. The Birkhoff interpolations is related with collocation method for the corresponding initial or boundary value problems of differential equations of fractional-order. The solvability of the interpolation problems is studied. For Gauss-type interpolating points, error of interpolation approximation is deduced.

10

Authors: Umar Yusuf Batsari

Let E be a uniformly smooth and uniformly convex real Banach space and C be a nonempty, closed and convex subset of E. In this paper, it is shown that {xn} obtained from Batsari’s[21] CQ algorithm with relatively nonexpansive maps converges strongly to a point xb which is also a common fixed point of some finite relatively nonexpansive mappings and solves a system of equilibrium problems in E. The result obtained improves some existing results in the literature.

7

Authors: Artyem Yefimushkin

The existence of nonclassical solutions is proved for the Neumann and Poincare problems for generalizations of the Laplace equation in anisotropic and nonhomogeneous media in almost smooth domains with arbitrary boundary data that are measurable with respect to logarithmic capacity. Moreover, it is shown that the spaces of such solutions have the infinite dimension.

9

Authors: Dan Yu, Shujing Gao, Youquan Luo, Feiping Xie

In this paper, a mathematical model for SIV S including impulsive vaccination strategy and general incidence is proposed and analyzed. By applying the comparison theorem, scaling method techniques for inequalities and stroboscopic map, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the disease are established, that is, the disease dies out if R1 < 1, whereas the disease persists if R2 > 1.

15

Authors: Rauf Kh AMIROV

Sturm-Liouville problem with boundary and discontinuity conditions was studied. For the solution inverse problem necessary and sufficient condition was obtained by the classical Gelfand-Levitan-Marchenko (GLM) type main integral equation and also algorithm was already given.