In this paper, we are interested to study a nonlinear Volterra equation with conformable deriva- tive. This kind of such equation has various applications, for example physics, mechanical engineering, heat conduction theory. First, we show that our problem have a mild soltution which exists locally in time. Then we prove that the convergence of the mild solution when the parameter tends to zero.

In this paper, we are interested to study a nonlinear Volterra equation with conformable derivative. This kind of such equation has various applications, for example physics, mechanical engineering, heat conduction theory. First, we show that our problem have a mild soltution which exists locally in time. Then we prove that the convergence of the mild solution when the parameter tends to zero.

An arbitrary point is removed from a three-dimensional Euclidean space on a two-dimensional sphere. The new well-posed solvable boundary value problems for the corresponding Laplace-Beltrami operator on the resulting punctured sphere are presented. To formulate the well-posed problems some properties of Green's function of the Laplace-Beltrami operator on a two-dimensional sphere are previously studied in detail.

The problems of stability system that arises in the construction of different automatic systems of indirect control are considered. It is known that a given program is not always exactly performed, since there are always initial, constantly acting perturbations. Therefore, it is also reasonable to require the stability of the program manifold itself with respect to some function. In the first part, the stability being investigated of automatic indirect control systems with rigid and tachometric feedback. Necessary and sufficient conditions for the absolute stability of a program manifold are established separately. In the second part, the automatic systems of indirect control taking into account the external load are considered. The equations of the hydraulic actuator, taking into account the action of an external load, are presented in a convenient form for research. Then it reduces to studying the stability of the system of equations with respect to a given program manifold. By constructing LyapunovВ’s functions for the system in canonical form, sufficient conditions are obtained for the absolute stability of the program manifold. The results obtained can be used in the construction of stable automatic indirect control systems.

In this paper, we initiate the study of fixed points for interpolative mappings in $m$-metric spaces. We discuss three different cases: the sum of \textquotedblleft interpolative exponents" is less than, equal to or greater than 1. We support each of our result by examples in $m$% -metric spaces. In the last section, we obtain our results in $p$-metric spaces. Finally we note that our results generalize results of \cite{EY}, \cite{GH} and \cite{K} from ordinary metric to $m$- and $p$-metrics.

Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.

This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $\varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $\varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.

In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem in combination with the perturbation method, we prove the existence of solutions.

In our previous works, a Metatheorem in ordered fixed point theory showed that certain maximum principles can be reformulated to various types of fixed point theorems for progressive maps and conversely. Therefore, there should be the dual principles related to minimality, anti-progressive maps, and others. In the present article, we derive several minimum principles particular to Metatheorem and their applications. One of such applications is the Brøndsted-Jachymski Principle. We show that known examples due to Zorn (1935), Kasahara (1976), Brézis-Browder (1976), Taskovi¢ (1989), Zhong (1997), Khamsi (2009), Cobzas (2011) and others can be improved and strengthened by our new minimum principles.

Fixed point theory is very useful in nonlinear analysis, diferential equations, differential and random differen- tial inclusions. It is well known that different types of fixed points implies the existence of specific solutions of the respective problem concerning differential equations or inclusions. There are several classifications of fixed points for single valued mappings. Recall that in 1949 M.K. Fort [19] introduced the notion of essential fixed points. In 1965 F.E. Browder [12], [13] introduced the notions of ejective and repulsive fixed points. In 1965 A.N. Sharkovsky [31] provided another classification of fixed points but only for continous mappings of subsets of the Euclidean space R n . For more information see also: [15], [18]-[22], [3], [25], [27], [31]. Note that for multivalued mappings these problems were considered only in a few papers (see: [2]-[8], [14], [23], [24], [32]) - always for admissible multivalued mappings of absolute neighbourhood retracts (ANR-s). In this paper ejective, repulsive and essential fixed points for admissible multivalued mappings of absolute neighbourhood multi retracts (ANMR-s) are studied. Let as remark that the class of MANR-s is much larger as the class of ANR-s (see: [32]). In order to study the above notions we generalize the fixed point index from the case of ANR-s onto the case of ANMR-s. Next using the above fixed point index we are able to prove several new results concerning repulsive ejective and essential fixed points of admissible multivalued mappings. Moreover, the random case is mentioned. For possible applications to differential and random di?erential inclusions see: [1], [2], [8]-[11], [16], [25], [26].