We study nonlinear Volterra integral equations with two-variable non-convolution kernels. Under mild structural hypotheses and a Carathéodory nonlinearity satisfying linear growth and monotonicity on an interval generated by an ordered pair of functions acting as bounds, we adapt the method of upper and lower solutions and the monotone iteration scheme, and we obtain the existence of solutions confined to a prescribed ordered interval, which play the role of minimal and maximal solutions in the above-mentioned ordered interval. A compactness mechanism follows from equicontinuity provided by diagonal integrability and uniform boundedness provided by the order interval, via the Arzelà- Ascoli theorem. On the other hand, the positivity of the kernel ensures order preservation. We provide quantitative sufficient conditions that make the construction of ordered bounds explicit. Illustrative examples—power-law, positivity-restricted Bessel-type, and a genuinely non-convolution kernel—demonstrate the scope of the theory. Finally, a worked non-convolution example admitting a closed form offers a practical benchmark for numerical methods.

In this research paper, we utilize an analytical technique to investigate the behavior of the Drinfeld-Sokolov-Wilson equation of arbitrary order. The implemented technique is an adequate composition of the Kharrat-Toma transform and the q-homotopy analysis approach. Here, a regularized form of the Hilfer-Prabhakar derivative of arbitrary order is used to formulate the problem. The Drinfeld-Sokolov-Wilson equation of arbitrary order is utilized to model the dispersive water waves and plays a very significant role in fluid dynamics. The results of the discussed model are presented graphically to show the efficiency and reliability of the obtained results.

This paper deals with time-fractional stochastic Navier-Stokes equations, which are characterized by the coexistence of stochastic noise and a fractional power of the Laplacian. We establish sufficient conditions for the existence and approximate controllability of a unique mild solution to time-fractional stochastic Navier-Stokes equations. Using a fixed point technique, we first demonstrate the existence and uniqueness of a mild solution to the equation under consideration. We then establish approximate controllability results by using the concepts of fractional calculus, semigroup theory, functional analysis and stochastic analysis.

This study aims to introduce a notion of the degree of nondensifiability b-Banach spaces and to develop variations of fixed point theorems, such as those by Darbo and its generalizations, Krasnoselskii, and Dhage. These theorems are applicable to operators in Strong b-Banach spaces (quasi-Banach spaces) that are not necessarily compact. Furthermore, the study applies these theorems to perturbed conformable differential equations and supports the findings with an illustrative example.

In this paper, the controllability of a stochastic impulsive integro-differential system involving nonlocal conditions and conformable derivatives is analyzed. The solution of the system is derived by Duhamel’s formula using Laplace and inverse Laplace transforms. The controllability result for the linear system is proved by using controllability Grammian matrix, and for the nonlinear integro-differential system, fixed point techniques are used. The applicability of the system is verified by means of an example.

This paper presents the formulation and analysis of a mathematical model for pest control with the aim of controlling its natural pests using awareness-based interventions such as the use of biopesticides with nutrients applications. Biopesticides are used to control biologically. In addition, a balanced application of nutrients is critical to increase crop growth and high yield. In the model, a time delay due to the time it takes the farmer to respond to the awareness campaign is considered. Nonnegativity and boundedness are shown to verify the plausibility of the delay model. The dynamics of the system has been analyzed with and without delay by finding the equilibrium points and their stability nature. The incidence of Hopf bifurcation is studies for both delayed and nondelayed systems. This study shows that the nutrient that is required for plant growth leads to higher yield of crops. An excess amount of nutrients causes instability in the system which is not favorable for crop growth and ultimately the yield is reduced. However, the rate of application of biopesticides stabilizes the system. Hopf bifurcation is also seen when the delay parameter crosses its critical value, indicating that people should respond to awareness campaigns with tolerable time delay.

The general notion of fractional derivative consists of a convolution with some kernel function. In the context of a recently introduced probabilistic operator that extends and unifies certain fractional derivatives of this type (L-fractional, normalized Caputo, scaled Hadamard, etc.), we prove the Peano–Sard theorem on the representation of continuous linear functionals. Thus, we continue the methodology developed by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024). After some additional theory on the operator is built, different versions of the theorem are included, depending on the ψ -integrator, the domain of the function, and the order of differentiation.

We investigate fractional Peano kernels for continuous linear functionals, in the context of differintegral operators with Mittag-Leffler kernel. New bounds for polynomial interpolation are obtained and numerical computations are shown, indicating improvements.

With the fractional logistic differential equation and its power-series solution, we show the problems when modeling with the Riemann-Liouville calculus. We propose a modification, in terms of the Beta probability distribution, that improves deficiencies of the Riemann-Liouville and Caputo operators. It is related to the L-fractional normalization. Fractional Euler numbers, connected with the famous Riemann zeta function, are computed. A Mittag-Leffler function is introduced.

The aim of this article is to establish some new extensions and variants of Jensen’s discrete and Simic-type inequalities for HA-convex and uniformly HA-convex functions. We introduce uniformly HA-convex functions, which are the generalized class of HA-convex (harmonic-arithmetic)-convex functions and provide some examples too. As an application point of view for the new Jensen’s bounds, we present some new improved bounds for Shannon’s entropy. Finally, some analysis is given by graphical illustration to prove the improvements of bounds.

A Caputo type coupled system of nonlinear fractional Langevin equations supplemented with a new class of nonlocal multi-point and multi-strip coupled boundary conditions is investigated. The existence and uniqueness results for the given problem are proved by applying the Leray–Schauder’s alternative and Banach’s fixed point theorem, respectively. Examples illustrating the obtained results are presented. The Ulam–Hyers stability for the given problem is also discussed. Some new results appearing as special cases of the present ones are also recorded.

. The present work examines the solvability of a tripled system of fractional Langevin differential equations with cyclic antiperiodic boundary conditions. The Krasnoselskii fixed point theorem, the Banach contraction mapping theorem, and specific properties of the Mittag-Leffler functions are employed to establish sufficient conditions for the existence and uniqueness of solutions. The feasibility of the primary findings is illustrated through the discussion of several numerical examples.

This article investigates the stabilization property for the modeled memristive bidirectional associative memory neural networks with time-varying delay when the faulty signals received from the fluctuated controller. The non-fragile output-feedback controller is taken into account to counteract the impact of gain perturbations to end up with robust fault-tolerant setup. To tackle the weak signals in the actuator received from the fluctuated controller, control gain matrices encompass situations intended to memory non-fragile output-feedback controller. Based on the Lyapunov stability theory, differential inclusion theory, and congruence transformation, the sufficient condition for the global asymptotic stabilization property for the designed fault-tolerant memristive bidirectional associative memory neural network model is obtained in terms of linear matrix inequality by utilizing Wirtinger's inequality. Finally, numerical examples are approached with the state performance plots of the proposed memristive bidirectional associative memory neural network model with respect to the time-domain plane, to confirm the stabilization results and it illustrates the working mechanism of the designed controller.

In this work, we present a mathematical model based on Stieltjes differential equations to analyze the spread of Vespa Velutina. To this end, we start by defining a zero-dimensional model, which we later generalize to a two-dimensional model with a diagonalizable spatial differential operator. The advantage of considering Stieltjes differential equations lies in the fact that they allow us to naturally handle reproductive impulses due to the hatching of individuals and periods of inactivity resulting from hibernation. Finally, we present some numerical results obtained using real nest position data.

In this study, we define a new fractional derivative using the normalization t e^ρ -1/ρt by means of the proportionality parameter ρ , which allows us to proportionally adjust the concept of the memory effect, one of the most powerful aspects of the fractional derivative. Then, we consider the Malthusian and Verhulst equations using this derivative and solve them using the power series method. We show that the series solutions are convergent and numerically estimate the radii of convergence of these series. We analyze the behavior of the solution series via rich numerical calculations.

This article presents a mathematical model to study malaria transmission dynamics influenced by awareness campaigns and optimal control strategies. Protected human class is included as a model variable that can be increased by social media awareness and treatment, as a result, infected cases are reduced. The nonnegativity and boundedness of the solutions of the model are analyzed. The proposed model possesses two equilibria, namely the disease-free equilibrium and the endemic equilibrium point. The disease-free equilibrium point is stable when the basic reproduction number, ℛ_0<1 and endemic equilibrium exists when ℛ_0>1 . Consequently, forward bifurcation occurs at ℛ_0=1 . Sensitivity analysis is provided which identifies the more important parameters of the model system related to the disease transmission. Finally, optimal control theory has been applied to maximize the number of protected human and minimize the cost of malaria management. The optimal problem comprises of three control parameters, and it is solved using the ‘maximum principle.’ This study concluded that public awareness is important for malaria control, and a combination of awareness-based interventions such as use of bed nets, spraying insecticides, and treatment can optimally increase the protected human as well as minimize the malaria cases cost-effectively.

Stochastic functional differential equations driven by mixed fractional Brownian motion are often used to describe many systems involving random noise and time delays. In this paper, we give an estimate for the stochastic convolution operator and then give space-time regularity results of the mild solution under the global Lipschitz conditions and linear growth conditions. In particular, when 0

This paper investigates the problems of Mittag–Leffler projective synchronization (MLPS) and asymptotic adaptive projective synchronization (AAPS) in fractional quaternion-valued inertial neural networks (FQVINNs) subject to parametric uncertainties. To facilitate the analysis, the original FQVINN model is reformulated into an equivalent fractional system through an appropriate variable transformation. Two synchronization strategies are proposed: a quaternion-valued feedback controller is designed to realize MLPS, while a fractional adaptive controller is developed to achieve AAPS. By employing tools from fractional differential inequality theory and Lyapunov stability analysis, sufficient conditions for the synchronization of FQVINNs are rigorously established. The effectiveness of the proposed control schemes is demonstrated through a numerical example, which confirms the theoretical predictions and highlights the practical applicability of the methods.

Over the last 250 years, anthropogenic activity has increased atmospheric carbon dioxide by nearly 40%. This increase is mainly caused by human fossil fuel combustion and deforestation, which are the main causes of global warming. Phytoplankton of the world's oceans synthesizes half of the carbon dioxide of the total Earth's photosynthetic activity. Thus, phytoplankton plays a crucial role in controlling Earth's climate. To study this scenario, we propose and analyze a mathematical model for the carbon-phytoplankton-zooplankton interaction dynamics. Positivity, boundedness, existence, and stability of biologically possible equilibrium points are studied. The system exhibits Hopf bifurcation with respect to the carbon capture coefficient and the criteria of Hopf bifurcation is established around the coexisting equilibrium. Complex spatiotemporal dynamics and patchy pattern formation are observed in the spatially explicit model. The proposed carbon-phytoplankton-zooplankton system incorporates the effect of global warming, and our simulation shows shifts in plankton seasonal dynamics.