In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}x^{k}W_{mk+j}^{2}$ for generalized balancing numbers are presented. As special cases, we give sum formulas of balancing, modified Lucas-balancing and Lucas-balancing numbers.

The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena. This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.

In this paper, we study the existence of a continuous solution for a nonlinear integral equation of a product type. The analysis uses the techniques of measures of noncompactness and Darbos fixed point theorem. Our results are obtained under rather general assumptions. Moreover, the method used in the proof allows us to obtain the asymptotic stability of the solutions.

In this work, we show that the system of difference equations xn+1=(ayn-2xn-1yn+bxn-1yn-2+cyn-2+d)/(yn-2xn-1yn), yn+1=(axn-2yn-1xn+byn-1xn-2+cxn-2+d)/(xn-2yn-1xn), where n belongs to the set of positive integer numbers, x-2, x-1, x0, y-2, y-1 and y0 are arbitrary nonzero real numbers, and the parameters a, b, c and d are arbitrary real numbers with d nonzero can be solved in a closed form. We will see that when a = b = c = d = 1, the solutions are expressed using the famous Tetranacci numbers. In particular, the results obtained here extend those in our recent work.

Recently, we have introduced the notion of standard single valued neutrosophic (SSVN) metric space as a generalization of the notion of standard fuzzy metric spaces given by J.R. Kider and Z.A. Hussain. In this paper, we study the fundamental properties of standard single valued neutrosophic metric spaces. Furthermore, we introduce the notion of continuous mapping and uniformly continuous mapping in standard single-valued neutrosophic metric spaces. To that end, we give a number of properties and characterizations of these notions.

In this work, we study a nonlinear inverse problem for an elliptic partial differential equation known as the Calderón problem or the inverse conductivity problem. We give a short survey on the reconstruction question of conductivity from measurements on the boundary by covering the main currently known results regarding the isotropic problem with full data in two and higher dimensions. We present Nachman’s reconstruction procedure and summarize the theoretical progress of the technique to more recent results in the field. An open problem of significant interest is proposed to check whether it is possible to extend the method for Lipschitz conductivities.

In this paper, we first introduce the conformable fractional Laplace transform. Then, we give its generalization for higher-order. Finally, as an application, we solve a non-homogeneous conformable fractional differential equation with variable coefficients and a system of fractional differential equations.