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In this paper, we mainly study the following boundary value problems of fractional discontinuous differential equations with impulses: $ \hskip 3mm \left\{ \begin{array}{lll} _{t}^{C} \mathcal {D}^{\mathfrak{R}}_{0^{+}}\Lambda(t) = \mathcal {E}(t)\digamma(t, \Lambda(t)), \ a.e.\ t\in Q, \\ \triangle \Lambda|_{t = t_{{\kappa}}} = \Phi_{{\kappa}}(\Lambda(t_{{\kappa}})), \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ \triangle \Lambda'|_{t = t_{{\kappa}}} = 0, \ {\kappa} = 1, \ 2, \ \cdots, \ m, \\ {\vartheta} \Lambda(0)-{\chi} \Lambda(1) = \int_{0}^{1}\varrho_{1}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \\ {\zeta} \Lambda'(0)-\delta \Lambda'(1) = \int_{0}^{1}\varrho_{2}({\upsilon})\Lambda({\upsilon})d{\upsilon}, \end{array}\right. $ where $ {\vartheta} > {\chi} > 0, \ {\zeta} > \delta > 0 $, $ \Phi_{{\kappa}}\in C(\mbox{ $\mathbb{R}$ }^{+}, \mbox{ $\mathbb{R}$ }^{+}) $, $ \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \geq 0 $ a.e. on $ Q = [0, 1] $, $ \mathcal {E}, \ \varrho_{1}, \ \varrho_{2} \in L^{1}(0, 1) $ and $ \digamma:[0, 1]\times \mbox{ $\mathbb{R}$ }^{+}\rightarrow \mbox{ $\mathbb{R}$ }^{+} $, $ \mbox{ $\mathbb{R}$ }^{+} = [0, +\infty) $. By using Krasnosel skii's fixed point theorem for discontinuous operators on cones, some sufficient conditions for the existence of single or multiple positive solutions for the above discontinuous differential system are established. An example is given to confirm the main results in the end.

HIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through nonsingular fractional operator. The non-integer mathematical model of HIV-1 infection under the Caputo-Fabrizio derivative is presented in this paper. The concept of Picard-Lindelof and fixed-point theory are used to address the existence of a unique solution to the HIV-1 model under the suggested operator. Also, the stability of the suggested model is proved through the Picard iteration and fixed point theory approach. The model's approximate solution is constructed through three steps Adams-Bashforth numerical method. Numerical simulations are provided for different values of fractional-order to study the complex dynamics of the model. Lastly, we provide the oscillatory and chaotic behavior of the proposed model for various fractional orders.

This paper is mainly concerned with the existence of multiple solutions for the following boundary value problems of fractional differential equations with generalized Caputo derivatives: $ \hskip 3mm \left\{ \begin{array}{lll} ^{C}_{0}D^{\alpha}_{g}x(t)+f(t, x) = 0, \ 0 0 $, $ f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+}) $, $ g' > 0 $, $ h\in C([0, 1], \mathbb{R}^{+}) $, $ \mathbb{R}^{+} = [0, +\infty) $. Applying the fixed point theorem on cone, the existence of multiple solutions for considered system is obtained. The results generalize and improve existing conclusions. Meanwhile, the Ulam stability for considered system is also considered. Finally, three examples are worked out to illustrate the main results.

In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.

In present paper, we introduce a new extension of the double controlled metric-like spaces, so called double controlled quasi metric-like spaces "assuming that the self-distance may not be zero". Also, if the value of the metric is zero, then it has to be "a self-distance". After that, by using this new type of quasi metric spaces, we generalize many results in the literature and we prove fixed point theorems along with some examples illustrating.

In this paper, we discuss the existence and uniqueness of boundary value problems for sequential ψ-Hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions. The existence results are obtained via the well known Krasnoselskii's fixed point theorem while the uniqueness is demonstrated by using the Banach's contraction mapping principle. Some examples are also given to demonstrate the application of the main results.

In this article, a novel semi-discrete model for stochastic competitive neural networks (SCNNs) is proposed. At first, taking advantage of some famous inequalities and fixed point theory, a few conditions are obtained for the existence of 2p-th mean almost periodic sequence (MAPS) of the semi-discrete stochastic model. In the next palace, 2p-th moment global exponential stability (MGES) of the above model is discussed as well. The research findings exhibit the stochastic and delayed effects on the mean dynamics of the semi-discrete stochastic networks. In the end, some numerical illustrations are presented to visually expound the feasibility of the works in this paper. The methods in this article could be applied to investigate other models in the areas of science and engineering.

In this paper we present a global existence theorem of a positive monotonic integrable solution for the mixed type nonlinear quadratic integral equation of fractional order\[x(t) = p(t) + h(t,x(t)) \int_{0}^{t} k(t,s)( f_1( s, I^\alpha f_2(s, x(s)))+g_1( s, I^\beta g_2(s, x(s))))ds,~t\in [0,1],\alpha,\beta>0\]by applying the technique of measures of weak noncompactness. As an application, we consider an initial value problem of arbitrary (fractional) order differential equations.