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We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.

Our main interest in this manuscript is to explore the main positive solutions (PS) and the first implications of their existence and uniqueness for a type of fractional pantograph differential equation using Caputo fractional derivatives with a kernel depending on a strictly increasing function $ \Psi $ (shortly $ \Psi $-Caputo). Such function-dependent kernel fractional operators unify and generalize several types of fractional operators such as Riemann-Liouvile, Caputo and Hadamard etc. Hence, our investigated qualitative concepts in this work generalise and unify several existing results in literature. Using Schauder's fixed point theorem (SFPT), we prove the existence of PS to this equation with the addition of the upper and lower solution method (ULS). Furthermore using the Banach fixed point theorem (BFPT), we are able to prove the existence of a unique PS. Finally, we conclude our work and give a numerical example to explain our theoretical results.

In this paper, we describe a new generalization of the multivariate Mittag-Leffler (M-L) function in terms of generalized Pochhammer symbol and study its properties. We provide a few differential and fractional integral formulas for the generalized multivariate M-L function. Furthermore, by using the generalized multivariate M-L function in the kernel, we present a new generalization of the fractional integral operator. Finally, we describe some fundamental characteristics of generalized fractional integrals.

By making use of the conformable integrals, we establish some new results on Cerone's and Bellman's generalization of Steffensen's integral inequality. In fact, we provide a variety of generalizations of Steffensen's integral inequality by using conformable calculus.

The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as $ w^2 $-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on $ w^2 $-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature.

The boundary value problem (BVP) for a nonlinear non positone or semi-positone multi-point Caputo-Hadamard fractional differential pantograph problem is addressed in this study. $ \begin{equation*} \mathfrak{D}_{1}^{\upsilon}x(\mathfrak{t})+\mathrm{f}(\mathfrak{t}, x( \mathfrak{t}), x(1+\lambda\mathfrak{t})) = 0, \ \mathfrak{t}\in(1, \mathfrak{b}) \end{equation*} $ $ \begin{equation*} x(1) = \delta_{1}, \ x(\mathfrak{b}) = \sum\limits_{i = 1}^{m-2}\zeta_{i}x(\mathfrak{\eta } _{i})+\delta_{2}, \ \delta_{i}\in\mathbb{R}, \ i = 1, 2, \end{equation*} $ where $ \lambda\in\left(0, \frac{\mathfrak{b-}1}{\mathfrak{b}}\right) $. The novelty in our approach is to show that there is only one solution to this problem using the Schauder fixed point theorem. Our results expand some recent research in the field. Finally, we include an example to demonstrate our findings.

In this paper, we propose and prove an extension and generalization, which extends and generalizes the Darbo's fixed point theorem (DFPT) in the context of measure of noncompactness (MNC). Thereafter, we use DFPT to investigate the existence of solutions to mixed-type fractional integral equations (FIE), which include both the generalized proportional (κ,τ)-Riemann-Liouville and Hadamard fractional integral equations. We've included a suitable example to strengthen the article.

In this article, we define the generalized exponential function on arbitrary time scales in the conformable setting and develop its fundamental characteristics. We address the fundamental theory of a conformable fractional dynamic equation on time scales, subject to the local and non-local initial conditions. We generalized the Grönwall type inequalities in a conformable environment. The generalized exponential function and the Grönwall's inequalities are indispensable for the study of the qualitative aspects of the local initial value problem. We developed some criteria related to global existence, extension and boundedness, as well as stability of solutions.

In this article we have introduced a metric named complex valued controlled metric type space, more generalized form of controlled metric type spaces. This concept is a new extension of the concept complex valued b-metric type space and this one is different from complex valued extended b-metric space. Using the idea of this new metric, some fixed point theorems involving Banach, Kannan and Fisher contractions type are proved. Some examples togetheran application are described to sustain our primary results.

In this manuscript, we consider a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo fractional derivative with respect to another monotonic function ϑ with antiperiodic boundary conditions. The existence and uniqueness results are proved for the suggested problem. Our approach is relying on properties of ϑ-Caputo's derivative, and implementation of Krasnoselskii's and Banach's fixed point theorem. At last, we discuss the Ulam-Hyers stability criteria for a nonlinear fractional Langevin equation. Some examples justifying the results gained are provided. The results are novel and provide extensions to some of the findings known in the literature.

In this paper, we propose a new family of interval valued ($\mathrm{IV}$) convex functions termed as generalized modified $(p,h)$-convex $\mathrm{IV}$ functions. We obtain the counterpart of Hermite-Hadamard $H\cdot H$ inequality by extending the $\mathrm{IV}$ fractional integral to the $\mathrm{IV}$ $\psi_{k}$-Riemann-Liouville ($\psi_{k}-RL$) fractional integrals. Also, several inequalities using extended operations on the newly defined class of convex $\mathrm{IV}$ functions are given.

In this manuscript, we examine the SIR model under convex incidence rate. We first formulate the famous SIR model under the aforesaid incidence rate. Further, we develop some sufficient analysis to examine the dynamical behavior of the model under consideration. We compute the basic reproductive number $\mathcal{R}_0.$ Also we study the global attractivity results via using Dulac function theory. Further, we also provide some information about the stability of the endemic and disease free equilibria for the considered model. In addition, we use nonstandard finite difference scheme to perform numerical simulation of the considered model via using Matlab. We provide different numerical plots for two different values of contact rate and taking various initial values for compartments involved in the considered model.