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The current study establishes neutral fractional differential equation (NFDE) of Sobolev with finite delay. The probability density function (PDF) and Fixed Point Technique (FPT) are used to demonstrate the existence of the equation with nonlocal circumstances. The fractional theory is used to illustrate the convergence. We then use approximation techniques to arrive at a unique result. We also provide an application for analyzing theoretical results.

This paper resolves around relative controllability of ψ−fractional delayed differential equations in finite dimensional space. The Mittag Leffler type of ψ−delayed perturbation matrix function with two parameters exhibits the Grammian matrix of fractional delay system. Based on this Grammian matrix we derived the necessary and sufficient conditions for the linear system which is relatively controllable. The fixed point approach is applied for obtaining a controllability result for semilinear system. This concept can be applied to some examples in order to illustrate the efficacy of our results.

In this article, we analyse the controllability criteria on sobolev-type Volterra-Fredholm functional Integro-differential Equations (SVFIDE) via Atangana–Baleanu Caputo (ABC) derivative. The primary outcomes was established by utlising the concepts on the measure of noncompactness, semigroup theory, the contraction principle and Darbo fixed point technique with nonlocal condition. An illustrative example to demonstrate the analytical result is provided at the end.

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.

Stability analysis over a finite time interval is a well-formulated technique to study the dynamical behaviour of a system. This article provides a novel analysis on the finite-time stability of a fractional-order system using the approach of the delayed-type matrix Mittag-Leffler function. At first, we discuss the solution’s existence and uniqueness for our considered fractional model. Then standard form of integral inequality of Gronwall’s type is used along with the application of the delayed Mittag-Leffler argument to derive the sufficient bounds for the stability of the dynamical system. The analysis of the system is extended and studied with impulsive perturbations. Further, we illustrate the numerical simulations of our analytical study using relevant examples.

A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by $\chi _2(G) $. We denote the cartesian product of G and H by $G\Box H $. In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph $K_{r} \Box K_{s} $, complete graph with complete bipartite graph $K_n \Box K_{1,s} $ and wheel graph with complete graph $W_l \Box K_n $.

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.

The notion of equitable colorability was introduced by Meyer in $1973$ \cite{meyer}. In this paper we obtain interesting results regarding the equitable chromatic number $\chi_{=}$ for the corona graph of a simple graph with a wheel graph $G\circ W_n$. Some extensions into $l$-corona products are also determined.

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number Xs (G) of G is the least number of colors needed to star color G. In this paper, we determine the star chromatic number of lexicographic product of complete graph with complete graph Km[Kn], complete graph with wheel graph Km[Wn], complete graph with path Km[Pn] and path with path Pm[Pn]. [ABSTRACT FROM AUTHOR]

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number (G) of G is the least number of colors needed to star color G. In this paper, we obtain the star chromatic number of modular product of two graphs G and H, denoted by G-H. We consider the graph G-H, where G and H be the path graphs, the cycle graphs, the star graphs and Petersen graphs. [ABSTRACT FROM AUTHOR]

For any graph class G of any two positive integers i and j, the Ramsey number RG(i; j) is the smallest integer such that every graph in G on atleast RG(i; j) vertices has a clique of size i or an independent set of size j. In this paper, we found the Ramsey numbers for the graph class of edgeless graphs, complete graphs, star graphs and class of all edgeless and complete graphs. [ABSTRACT FROM AUTHOR]

Finite time stability practically examines the trajectories of a system which converge to equilibrium state in a short period of time. This notion requires predefined bounds on system parameters and bounded time interval. Considering the idea that many practical system often operates over time interval being finite rather than infinite, we explore the finite time stability concept of damped fractional system with neutral conditions and impulsive effects. The desired bounds for the stability of the system is derived by implementing Gronwall's inequality conditions. Further, the finite time stability conditions of the proposed fractional linear model is extended to nonlinearity term with disturbance. Finally, numerical simulations are given to show the effectiveness of the derived results., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 ISA. Published by Elsevier Ltd. All rights reserved.)

Metformin (MET) is a common treatment for type II diabetes. Here, we demonstrate the anticancer activity of a polymeric metformin derivative. We successfully synthesized the polypeptide (poly-l-lysine [PLL]) derivative of metformin (LysMET) and demonstrated its capacity as an anticancer therapeutic and gene carrier. miRNA-320a was loaded into the cationic LysMET and enveloped in a lipid bilayer, and a MUC1-specific aptamer was conjugated to the surface (A-Lipo@mLysMET). The LysMET-containing guanidine moiety was more tolerable than the secondary amine-containing PLL. LysMET showed similar efficacy to MET in the induction of HT-29 tumor suppression, indicating the importance of the biguanide moiety. The synergistic effect of miRNA-320a and LysMET treatment significantly decreased cell viability compared with LysMET treatment alone, which was attributed to the role of miRNA in the β-catenin pathway. A-Lipo@mLysMET showed excellent antitumor efficacy and significantly reduced the tumor burden in all groups. AMPKα phosphorylation was markedly increased by LysMET compared with the control, with significant inhibition of the mTOR pathway. The TUNEL assay showed that apoptosis was the main mechanism responsible for cancer cell death and that A-Lipo@mLysMET resulted in the highest proportion of TUNEL-positive cells (∼36%). No noticeable organ damage was observed after treatment with either LysMET or A-Lipo@mLysMET, confirming the excellent safety profile of guanide-modified polymers. Overall, we demonstrated the feasibility of LysMET for the effective control of tumor progression as well as its dual role, as both a drug and a gene carrier.

In this study, we developed pH and redox-responsive crosslinked polypeptide-based combination micelles for enhanced chemotherapeutic efficacy and minimized side effects. The stability and drug release properties of the polypeptide micelles were efficiency balanced by the corona-crosslinking of the triblock copolymer, poly(ethylene glycol)-b-poly(aspartic acid)-b-poly(tyrosine) (PEG-b-pAsp-b-pTyr) with coordinated redox and pH dual-sensitivity by introducing disulfide crosslinkages. Because of the crosslinking of the middle shell of the triblock polypeptide micelles, their robust structure was maintained in strong destabilization conditions and exhibited excellent stability. GSH concentrations were significantly higher in tumor tissue than in normal tissue, which formed the basis for our design. Drug release was elevated under redox and low acidic conditions. Furthermore, crosslinked micelles showed a superior anticancer effect compared to that of non-crosslinked micelles. Incorporation of docetaxel (DTX) and lonidamine (LND) in crosslinked polypeptide micelles increased the intracellular reactive oxygen species (ROS) level and oxidative stress and caused damage to intracellular components that resulted in greater apoptosis of cancer cells than when DTX or LND was used alone. The combination of DTX and LND in crosslinked micelles exhibited efficacious inhibition of tumor growth with an excellent safety profile compared to that reported for drug cocktail combinations and non-crosslinked micelles. Overall, redox/pH-responsive polypeptide micelles could be an interesting platform for efficient chemotherapy., Statement of Significance: We have synthesized a biodegradable polypeptide block copolymer to construct a facile pH and redox-responsive polymeric micelle asan advanced therapeutic system for cancer therapy. We have designed a corona-crosslinked triblock copolymer (poly (ethylene glycol)-b-poly(aspartic acid)-b-poly(tyrosine) (PEG-b-pAsp-b-pTyr)) micelles co-loaded with docetaxel and lonidamine (cl-M/DL). The corona of triblock polymer was crosslinked to maintain its structural integrity in the physiological environment. The mitochondrial targeting LND is expected to generate ROS, oxidative stress and thereby synergize the chemotherapeutic efficacy of DTX in killing cancer cells. Consistently, cl-M/DL exhibited excellent antitumor efficacy in xenograft tumor model with remarkable tumor regression. Overall, we demonstrated the construction of bioreducible nanosystem for the effective synergistic delivery of DTX/LND in tumor tissues towards cancer treatment., (Copyright © 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.)