In this paper, first we introduce new subclasses of analytic and bi-univalent functions defined by fractional derivative operator in the open unit disk and then for the functions belongs to these classes obtain upper bounds for the initial coefficients.

Let $G$ be a simple finite undirected and an isolate-free graph. A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in $S$ is within distance 2 of another vertex of $S$. The semitotal domination number, $\gamma_{t2}(G)$, is the minimum cardinality of such a set. In this paper, we study the semitotal domination number for Harary graphs, which was first introduced by Frank Harary. Since Harary graphs have the maximum possible connectivity with the minimum number of edges, many researchers are interested in studying its stability properties.

The goal of this paper is to derive the fractional Green's function for the first time in the distributional space for the fractional-order integro-differential equation with constant coefficients. Our new technique is based on Babenko's approach, without using any integral transforms such as the Laplace transform along with Mittag-Leffler function. The results obtained are not only much simpler, but also more generalized than the classical ones as they deal with distributions which are undefined in the ordinary sense in general. Furthermore, several interesting applications to solving the fractional differential and integral equations, as well as in the wave reaction-diffusion equation are provided, some of which cannot be achieved by integral transforms or numerical analysis.

In this paper, we present new existence criterion of solutions for partial discrete Dirichlet problems. More precisely, under appropriate assumptions on the nonlinearities that possesses the appropriate behaviors related to the eigenvalues at infinity, the existence result of infinitely many solutions is discussed.

With the help of Nevanlinna theory, we investigate the existence of meromorphic solution of certain class of q-difference differential equations and consequently we are able to show that no transcendental meromorphic function with some conditions will satisfy such equations.

In this paper, we define a new class of functions called generalized $\varphi$-convex function. Several variants of Hermite-Hadamard type fractional integral inequalities are presented. This ideas and techniques used in this paper may open new avenues of research and motivate the reader to explore the application of generalized $\varphi$-convex functions in various branches of pure and applied sciences.

The object of the present paper is to study quasi conformal curvature tensor and conformal curvature tensor on Lorentzian $\beta$-Kenmotsu manifold with a semi-symmetric semi-metric connection. Moreover, we consider quasi conformally flat, $\xi$-quasi conformally flat, $\phi$-conformally semisymmetric and conformally flat Lorentzian $\beta$-Kenmotsu manifold with a semi-symmetric semi-metric connection and obtain the scalar curvature $r$ in each case.

The problems of third order differential subordination as well as superordination for functions analytic in the open unit disk seem to be new and have been of interest after the seminal work of Antonino and Miller, and Tang. In the present paper by considering suitable classes of admissible functions associated with Hohlov operator, various third order differential subordination as well as differential superordination results are obtained. As a consequence, the dual problems which gives the sandwich-type relations are presented. Upon suitable choice of the parameters, the results obtained in this paper include some classical as well as recently studied results. An attempt has also been made to illustrate the applications of the new results in the context of electromagnetic cloaking problem.

In this study, Smarandache curves are defined according to the Sabban frame by means of the spherical indicatrix curve of the Salkowski curve. Then using these curves we calculated geodesic curvatures. Finally each curve is drawn with maple program.

The main goal of this article is to propose a new reliable method to solve nonlinear second-order fractional differential equations in particular, nonlinear fractional Bratu-type equation. This method called the modified Taylor fractional series method (MTFSM). The fractional derivative is defined in the Liouville-Caputo sense. Simplicity, rapid convergence, and high accuracy are the advantages of this method. In addition, the MTFSM reduces the size of calculations by not requiring linearization, discretization, perturbation or any other restriction. Three numerical examples are exhibited to demonstrate the reliability and efficiency of the proposed method, and the solutions are considered as an infinite series that converge rapidly to the exact solutions. The results display that the MTFSM is very effective and accurate to solve this type of nonlinear fractional problems.

Combining the new concept of $F-$contraction and Hausdorff metric, we give some generalized results for integral type mapping on weak partial metric space.

For a $2$-generated semigroup $S$ we define the invariant $\lambda (S)$ as the minimum of the Fibonacci lengths over all generating pairs of the semigroup $S$, where the Fibonacci length with respect to a generating pair $(x, y)$ is the fundamental period (if exist) of the truncated periodic sequence $z_0=x, z_1=y, z_k=z_{k-2}z_{k-1}, (k\geq 2)$ of the elements of $S$. We name this invariant as the Fibonacci invariant of $S$. Our used notation is the same as of the celebrated work of D.L. Johnson in $2005$ on infinite groups. In this paper we examine two classes of semigroups for existence of this invariant. The considered semigroups are the finite semigroup $S=\langle a, b\mid a^{p^\alpha}=a, b^{q^\beta}=b,ab=a\rangle$ of order $p^{\alpha}q^{\beta}-1$ and the infinite semigroup $T=\langle a, b\mid a^{p^\alpha}=a, ab=a\rangle$, for all integers $\alpha, \beta \geq 2$ and all distinct primes $p$ and $q$. We prove the existence of the Fibonacci invariants of these semigroups, for all parameters. As a numerical result we show that $\lambda (T)\leq \lambda (S)=p^{\alpha +1}$, if $p$ is even.

The paper is devote to the study of bronze Riemannian structures. An integrability condition and curvature for bronze Riemannian structure are investigated via the Tachibana operator which is applied to pure tensor fields. After twin bronze Riemannian metric is studied. Examples of bronze structures are given.

In this paper, existence of unique periodic solutions for two classes of third order delay differential equations are considered. The two equations are expressed in their integral equivalents with which suitable Green’s function is constructed and its associated properties stated. The main tool adopted to establish the existence of unique periodic solutions to the classes of delay differential equations is the Krasnoselskii’s fixed point theorem due to its applicability to the combination of a compact and contraction mappings which occur in dealing with perturbed differential operators.