In the direction of three shared set problems, we have answered a question of Yi providing a best possible combination of three sets with least cardinalities. We have also exhibited some examples assuring the sharpness of our results.

In this paper, using the concept of measure of noncompactness we present some results on the existence of $n$-fixed points for a class of operators. Also as an application, we derive the sufficient conditions for the existence of solutions for $n$-dimensional fractional order functional boundary value problems.

In this paper, we investigate the determining of meromorphic functions when two differential polynomials share a set of the roots of unity. Our results extended some results of Yang and Hua, Bhoosnurmath, Dyavanal, Lahiri and Sinha.

In this paper, we consider the discrete kinetic Broadwell system. The Broadwell system is the kinetic Boltzmann equation, and for this model momentum is conserved but energy is not. For the first time, new traveling wave solutions are found using the extended tanh-function method and the $\exp(-\varphi(\xi))$-expansion method. The extended tanh-function method is applied for the classical system and $\exp(-\varphi(\xi))$ for the fractional order. With the help of computerized symbolic computation, we obtain new solutions. Similarly, it is possible to find exact solutions for other kinetic models.

We construct the relation between the 5-dimensional complex Lie algebra $\it{\mathcal{K}_5}$ and certain special functions. In this paper, we discuss the new three variable models of the irreducible representations of the Lie algebra $\it{\mathcal{K}_5}$. We use a two-fold Euler type integral transformation to obtain the new transformed models and hence various new recurrence relations in terms of Kampe de Feriet function are derived.

In 2010, J.Kaczorowski and K.Wiertelak considered the Volterra integral equation of second type for the remainder term in the asymptotic formula for the Euler totient function. The author found that the consideration made by them holds for other remainder terms in the asymptotic formula having certain common properties. In this paper, we first consider the pair of complex-valued arithmetical functions $(a(n),b(n))$ satisfying $b(n) \hspace{-0.1cm} = \sum_{d|n} a(d)n/d$. We prove that the solution of the Volterra integral equation of second type for the error term in the asymptotic formula for $b(n)$ can be obtained when $a(n)$ satisfies some special condition.

In this paper, we give a novel representation of numerical solution for fractional Bratu-type equation using a recent and reliable method, namely the general fractional residual power series method (GFRPSM). The advantage of this method compare with the existing methods is, that method solves the nonlinear problems without using linearization and any other restriction. Three numerical examples are given and the results are compared with the exact solutions both numerically and graphically. The results reveal that GFRPSM is very effective simple and efficient technique to handle a wide range of nonlinear fractional differential equations.

This paper deals with the parabolic type Kirchhoff equation with logarithmic nonlinearity in a bounded domain. Firstly, we prove the local and global existence of solutions. Later, we investigate the decay of solutions. This improves and extend some previous studies.

In this paper, we prove a typical fixed point theorem for a couple of expansive mappings in complex partial b-metric space. Some of the well-known results in the literature are generalized and expanded by the proved results. An example is being given to help our results. In addition, the usability of the results obtained is proved here by an approach to integral equations.

This paper deals with some approximation properties of Stancu-Kantorovich variant of Sz\'{a}sz-Mirakjan operators based on B\'{e}zier basis functions with shape parameter $\lambda\in\lbrack-1,1]$. We compute several preliminary results such as moments and central moments. Later, we introduce a Korovkin-type convergence theorem and discuss the order of approximation in terms of the modulus of continuity and for the elements belong to Lipschitz-type class and Peetre's $K$-functional, respectively. Also, we prove a Voronovskaya type asymptotic theorem. Lastly, we present the comparison of the convergence of constructed operators to the certain functions with some graphical illustrations for different values of $m$, $\alpha$, $\beta$ and $\lambda$ parameters.

In this study, we create hypersurface families from each non-null isogeodesic, with non-null Frenet vectors, given in the Minkowski 4-space. We obtain parametric representations for hypersurface families whose members have the same curves as the given isogeodesic curves. By using the Frenet frame of each given geodesic curve, we create the hypersurfaces as a linear combination of this Frenet frame and obtain the necessary and sufficient conditions for these curves to be isogeodesic. Also, we give some examples so that the method presented is clear and understandable. In addition, the graphs of the surfaces formed by projecting the surfaces given with parametric equations to 3-dimensional space are drawn.

We have studied some qualitative properties such as boundedness of solutions, stability character and periodic nature of the third-order rational difference equation \begin{equation*} z_{n+1}=\delta +\dfrac{1}{z_{n}z_{n-1}z_{n-2}},\;\;\;\ \ n=0,1,... \end{equation*} with parameter $\delta $ and with arbitrary initial conditions for which the denominator is always positive. It is worth to mention that our problem is a natural extension of the Open Problem 8.2 and confirm the Conjecture 8.1 declared by A.M. Amleh, E. Camouzis and G. Ladas in On the Dynamics of a Rational Difference Equations I (International Journal of Difference Equations, Volume 3, Number 1, 2008, pp. 1-35).