Narrow Results

In this paper, we investigate a nonlocal multi-point and multi-strip coupled boundary value problem of nonlinear fractional Langevin equations. The standard fixed point theorems (Leray–Schauder’s alternative and Banach’s fixed point theorem) are applied to derive the existence and uniqueness results for the given problem. We also discuss the Ulam–Hyers stability for the given system. Examples illustrating the obtained results are presented. Some new results appearing as special cases of the present ones are also indicated.

In this paper, we develop a non-local mathematical model describing cancer cell invasion and movement as a result of integrin-controlled cell–cell adhesion and cell–matrix adhesion, and transforming growth factor-beta (TGF-$\beta$) effect on cell proliferation and adhesion, for two cancer cell populations with different levels of mutation. The model consists of partial integro-differential equations describing the dynamics of two cancer cell populations, coupled with ordinary differential equations describing the extracellular matrix (ECM) degradation and the production and decay of integrins, and with a parabolic PDE governing the evolution of TGF-$\beta$ concentration. We prove the global existence of weak solutions to the model. We then use our model to explore numerically the role of TGF-$\beta$ in cell aggregation and movement.

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.

In this paper, we investigate the existence and uniqueness of solutions for Riemann–Liouville fractional integro-differential equations equipped with fractional nonlocal multi-point and strip boundary conditions in the weighted space. The methods of our study include the well-known tools of the fixed point theory, which are commonly applied to establish the existence theory for the initial and boundary value problems after converting them into the fixed point problems. We also discuss the case when the nonlinearity depends on the Riemann–Liouville fractional integrals of the unknown function. Numerical examples illustrating the main results are presented.

In this paper, we introduce and investigate a new class of coupled fractional $q$-integro-difference equations involving Riemann–Liouville fractional $q$-derivatives and $q$-integrals of different orders, equipped with $q$-integral-coupled boundary conditions. The given problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed-points coincide with solutions of the problem at hand. The existence and uniqueness results for the given problem are, respectively, derived by applying Leray–Schauder nonlinear alternative and Banach contraction mapping principle. Illustrative examples for the obtained results are constructed. This paper concludes with some interesting observations and special cases dealing with uncoupled boundary conditions, and non-integral and integral types nonlinearities.

The primary goal of this paper is to study a nonlinear fuzzy fractional dynamic system (FFDS) involving a time-dependent variational inequality. We use the monotone argument and Knaster–Kuratowski–Mazurkiewicz (KKM) theorem to prove that the variational system of FFDS is solvable and its solutions become a bounded, closed and convex set. Employing this result together with Bohnenblust–Karlin fixed point theorem and Filippov implicit function, we show the existence of a mild solution to FFDS.

In this paper, we study the question of the existence and nonexistence of solutions for some fractional equations with variable exponents. This paper generalizes some analog results in the classical fractional one. As we know, there are no previous results on the nonexistence of solutions for nonlinear equations with fractional $p(\cdot ,\cdot )$-Laplacian.

Public health awareness programs have been a crucial strategy in mitigating the spread of emerging and re-emerging infectious disease outbreaks of public health significance such as COVID-19. This study adopts an Susceptible–Exposed–Infected–Recovered (SEIR) based model to assess the impact of public health awareness programs in mitigating the extent of the COVID-19 pandemic. The proposed model, which incorporates public health awareness programs, uses ABC fractional operator approach to study and analyze the transmission dynamics of SARS-CoV-2. It is possible to completely understand the dynamics of the model’s features because of the memory effect and hereditary qualities that exist in the fractional version. The fixed point theorem has been used to prove the presence of a unique solution, as well as the stability analysis of the model. The nonlinear least-squares method is used to estimate the parameters of the model based on the daily cumulative cases of the COVID-19 pandemic in Nigeria from March 29 to June 12, 2020. Through the use of simulations, the model’s best-suited parameters and the optimal ABC fractional-order parameter $\tau$ may be determined and optimized. The suggested model is proved to understand the virus’s dynamical behavior better than the integer-order version. In addition, numerous numerical simulations are run using an efficient numerical approach to provide further insight into the model’s features.

In this paper, we use Krasnoselskii’s fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative

In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP)

In this paper, we analyzed the chaotic complexity of a financial mathematical model in terms of a new generalized Caputo fractional derivative. There are three components in this nonlinear financial model: price indexes, interest rates, and investment demand. Our analysis is based on applying the fixed point hypothesis to determine the existence and uniqueness of the solutions. The bifurcation of the proposed financial system has been analyzed at various parameters of the system. Dynamical phase portraits of the proposed financial model are demonstrated at various fractional-order values. We investigated the possibility of finding new complex dynamical behavior with generalized Caputo fractional derivative. This economic model is solved numerically using a predictor–corrector (PC) algorithm with a generalized Caputo derivative. This algorithm can be viewed as a non-integer extension of the classical Adams–Bashforth–Moulton (ABM) algorithm. Additionally, this numerical algorithm has been studied for stability. A number of diverse dynamic behaviors have been observed in numerical simulations of the system, including chaos. Over a broad range of system parameters, bifurcation diagrams indicate that the system behaves chaotically.

In this paper, we present new definitions of generalized fractional integrals and derivatives with respect to another function and derive some of their properties, such as their inter-relationship and semigroup law. Caputo-type generalized fractional derivative with respect to another function is also defined and its properties are derived. A Cauchy problem involving the new Caputo-type generalized fractional derivative is also studied. We also provide an expansion formula for Caputo-type derivative and apply it to solve a fractional-order problem.

This paper designates two important properties of the existence-uniqueness of the mild solution for a fractional controlled fuzzy evolution equation involving the Caputo-derivative by using nonlocal conditions, where $\beta \in (1,2)$, and it is formulated as

In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron $\mathrm{\Delta }$-integrals defined in Banach spaces. We also provide a variation-of-constant formula for a nonhomogeneous linear dynamic equations on time scales and we establish results on controllability for linear dynamic equations. Since we work in the framework of Perron $\mathrm{\Delta }$-integrals, we can handle functions not only having many discontinuities, but also being highly oscillating. Our results require weaker conditions than those in the literature. We include some examples to illustrate our main results.

We consider a class of nonhomogeneous elliptic equations with fractional Laplacian and nonlinear gradient terms, namely ${(-\mathrm{\Delta })}^{\frac{\alpha }{2}}u=V(x)u+g(u,\nabla u)+f$ in ${ℝ}^n$, where $0<\alpha <n$, $g$ is the nonlinearity, $V$ the potential and $f$ is a forcing term. Some examples of nonlinearities dealt with are $u|u{|}^{\rho -1}$, $|\nabla u{|}^{\rho }$ and $\left|u{|}^{{\rho }_1}\right|\nabla u{|}^{{\rho }_2}$, covering large values of $\rho ,{\rho }_1,{\rho }_2$, and particularly variational supercritical powers for $u$ and super-$\alpha$ ones for $|\nabla u|$ (superquadratic if $\alpha =2$). Moreover, we are able to consider some exponential growths, $g$ belonging to certain classes of power series, or $g$ satisfying some conditions in the Lipschitz spirit. We obtain results on existence, uniqueness, symmetry, and other qualitative properties in a new framework, namely modulation-type spaces based on Lorentz spaces. For that, we need to develop properties and estimates in those spaces such as complex interpolation, Hölder-type inequality, estimates for product, convolution and Riesz potential operators, among others. In order to handle the nonlinearity, other ingredients are estimates for composition operators in our setting.

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann–Liouville-type $q$-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on $q$-fractional differential equations. Some interesting special cases are also discussed.