Existence Of Solutions Research Articles

A theoretical and computational study of heteroclinic cycles in Lotka–Volterra systems

In general, global attractors are composed of isolated invariant sets and the connections between them. This structure can possibly be highly complex, encompassing attraction basins, repeller sets and invariant sets that, collectively, form a dynamical landscape. Lotka–Volterra systems have long been pivotal as preliminary models for dynamics in complex networks exhibiting pairwise interactions. In scenarios involving Volterra–Lyapunov (VL) stable matrices, the dynamics is simplified in such a way that the positive solutions converge to a single, globally asymptotically stable stationary point as time tends to infinity, thereby excluding the existence of periodic solutions. In this work, we conduct a systematic study on the emergence of heteroclinic cycles within Lotka–Volterra systems characterized by Volterra–Lyapunov stable matrices. Although VL stability of the matrix implies that ω-limit sets of solutions are always stationary points, our analysis of α-limit sets reveals finite sets of stationary points interconnected by global trajectories, forming structures referred to as heteroclinic cycles. Our findings indicate that even within the framework of VL stable matrices, such structures are more prevalent than previously thought in literature, driven by the interplay between the symmetric and antisymmetric components of the model matrix. This understanding also reinforces our comprehension of the classical three-dimensional May–Leonard model, which is known to be the unique case exhibiting heteroclinic cycle within the VL framework in dimension three, while also pointing to a surprising richness in the dynamics of these structures in higher dimensions.

The Existence of Positive Solutions for a p-Laplacian Tempered Fractional Diffusion Equation Using the Riemann–Stieltjes Integral Boundary Condition

In this paper, we focus on the existence of positive solutions for a class of p-Laplacian tempered fractional diffusion equations involving a lower tempered integral operator and a Riemann–Stieltjes integral boundary condition. By introducing certain new local growth conditions and establishing an a priori estimate for the Green’s function, several sufficient conditions on the existence of positive solutions for the equation are derived by using a fixed point theorem. Interesting points are that the tempered fractional diffusion equation contains a lower tempered integral operator and that the boundary condition involves the Riemann–Stieltjes integral, which can be a changing-sign measure.

Riemann–Liouville fractional stochastic evolution equations driven by mixed sub-fractional Brownian motion

Abstract This paper is concerned with the existence of mild solutions for stochastic differential equations driven by mixed sub-fractional Brownian motion with Riemann–Liouville fractional derivative. The results are obtained by the fixed point theorem. An example is given to illustrate the obtained results.

On the existence and uniqueness of solutions to natural equations with non-Lipschitz conditions

Abstract In this paper, we concentrate on a prominent financial model known as the natural model. Specifically, this model is represented by a stochastic differential equation called the natural equation, which plays a crucial role in our investigation. More precisely, our primary aim is to establish the existence and uniqueness of the solution to the natural equation under a condition weaker than the Lipschitz condition, termed the non-Lipschitz condition. Our investigation is grounded in certain inequalities, such as Gronwall’s inequality, Bihari’s inequality, and Burkholder–Davis–Gundy (BDG) inequality. Additionally, the mean square stability of the solutions to the natural equation is obtained in this research.

Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs

Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs

Second order Lp estimates for subsolutions of fully nonlinear equations

We obtain new Lp estimates for subsolutions to fully nonlinear equations. Based on our Lp estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations, critical exponents of Lp estimates and maximum principles, and the existence and uniqueness of solutions to fully nonlinear equations on the torus with free terms in the Lp spaces or in the space of Radon measures.

Nonlinear Coupled Equations Free-Boundary Problem in Atherosclerosis: Symmetry Analysis of Solutions and Numerical Simulation

Atherosclerosis, a chronic inflammatory cardiovascular disease closely related to plaque formation during arteriosclerosis, poses a significant threat to global health. To deepen the understanding of the multifaceted interactions driving atherosclerosis progression and provide theoretical support for designing targeted therapeutic strategies, this study establishes a nonlinear coupled atherosclerotic free-boundary model integrating inflammatory immune cells, cytokines, and oxidized low-density lipoprotein. By applying the compression mapping principle, the local and global existence and uniqueness of solutions are proven, while revealing certain symmetries in the model’s solution structure. Under specific assumptions, a quasi-steady-state approximate model is derived, and the existence of its solution is demonstrated. Through numerical simulations of the quasi-steady-state approximate model using the finite difference method, the temporal and spatial evolution of pro-inflammatory macrophages and oxidized low-density lipoprotein is analyzed. The findings highlight the model’s strength in capturing the intricate dynamics of atherosclerosis, uncovering underlying mechanisms and identifying therapeutic targets. By evaluating inflammatory dynamics across plaque types and stenosis levels, the experimental design further validated the model’s ability to replicate clinical processes and reinforced its predictive accuracy. Notably, in the process of model analysis and solution, symmetries in the equations and boundary conditions play a crucial role in determining the solution properties. However, the current one-dimensional model has limitations. Future research should focus on developing higher-dimensional models and integrating more influencing factors to enhance the model’s clinical applicability and deepen the understanding of this complex disease.

Exploring Optimisation Strategies Under Jump-Diffusion Dynamics

This paper addresses the portfolio optimisation problem within the jump-diffusion stochastic differential equations (SDEs) framework. We begin by recalling a fundamental theoretical result concerning the existence of solutions to the Black–Scholes–Merton partial differential equation (PDE), which serves as the cornerstone for subsequent analysis. Then, we explore a range of financial applications, spanning scenarios characterised by the absence of jumps, the presence of jumps following a log-normal distribution, and jumps following a distribution of greater generality. Additionally, we delve into optimising more complex portfolios composed of multiple risky assets alongside a risk-free asset, shedding new light on optimal allocation strategies in these settings. Our investigation yields novel insights and potentially groundbreaking results, offering fresh perspectives on portfolio management strategies under jump-diffusion dynamics.

A new result on the existence of positive solutions for a class of fractional Kirchhoff-type systems with multiple parameters

This paper is devoted to the problem of existence of positive weak solutions for a class of fractional Kirchhoff-type systems with multiple parameters. By using the sub-supersolution method and some analysis techniques, we prove a new existence result. To the best of our knowledge, our results are new in the study of fractional Kirchhoff-type systems.

Accelerating mosquito population replacement: a two-dimensional model with periodic release of Wolbachia-infected males

This study investigates strategies for accelerating mosquito population replacement through the periodic release of Wolbachia-infected male mosquitoes using a two-dimensional time-switching model. We obtain the existence and stability of periodic solutions within this framework and establish several sufficient conditions for eradicating wild mosquitoes by varying the release amounts of infected males. Our results indicate that accelerated replacement is feasible as long as the Wolbachia infection is favorable. Despite Wolbachia infection brings a fitness costs, effective population replacement can still be achieved if the release waiting time does not exceed a certain threshold and the release amount is sufficient to match or surpass the capacity of wild mosquitoes. However, when the release waiting time exceeds this threshold, we identify a continuous, unbounded curve that separates two boundary equilibrium points, with solutions originating from either side converging to their respective boundary equilibrium points. Numerical simulations are presented to validate our theoretical findings, highlighting the potential of this approach for effective mosquito population control.

Nonlocal separable elliptic and parabolic equations and applications

The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted Lp spaces. The equations include the variable coefficients and abstract operator function A = A (x) in a Banach space E in leading part. We find the sufficient growth assumptions on A and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. Byusing these results, the existence and uniqueness of maximal regular solution of the nonlinear nonlocal anisotropic elliptic equation is obtained in weighted Lp spaces. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation in mixed Lp norms, the boundary value problem for anisotropic elliptic convolution equation, the Wentzel-Robin type boundary value problem for degenerate integro-differential equation and infinite systems of degenerate elliptic integro-differential equations are obtained.

Fractional differential equations of Riemann–Liouville of variable order with anti-periodic boundary conditions

PurposeThe purpose of this paper is to investigate the existence, uniqueness and stability of solutions to a class of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order (R-LFDEAPBCVO). The study utilizes standard fixed point theorems (FiPoTh) to establish the existence and uniqueness of solutions. Additionally, the Ulam-Hyers-Rassias (Ul-HyRa) stability of the considered problem is examined. The obtained results are supported by an illustrative example. This research contributes to the understanding of fractional differential equations with variable order and anti-periodic boundary conditions, providing valuable insights for further studies in this field.Design/methodology/approach This paper (1) defines the Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order (R-LFDEAPBCVO); (2) discusses the existence and uniqueness of solutions to these equations using standard FiPoTh; (3) investigates the stability of the considered problem using the Ul-HyRa stability concept (Ul-HyRa); (4) provides a detailed explanation of the design and methodology used to obtain the results and (5) supports the obtained results with a relevant example.FindingsThe authors confirm that no funds, grants or any other form of financial support were received during the preparation of this manuscript.Originality/valueThe originality/value of our paper lies in its contribution to the field of fractional differential equations. Specifically, we address the existence, uniqueness and stability of solutions to a class of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order. By utilizing standard FiPoTh and investigating Ul-HyRa stability, we provide novel insights into this problem. The results obtained are supported by an example, further enhancing the credibility and applicability of your findings. Overall, our paper adds to the existing knowledge and understanding of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions, making it valuable to the scientific community.

Orbital stability of bifurcating solutions in the nonlinear Schrödinger system with quadratic interaction

In this paper, we study the existence of bifurcation solutions and the orbital stability of a class of coupled elliptic systems with quadratic nonlinearities. These systems are relevant to Raman amplification in plasma. Using the Crandall-Rabinowitz local bifurcation theorem, we prove the existence of positive bifurcation solutions. Additionally, we calculate the Morse index and establish the orbital stability and instability of these bifurcation solutions. Notably, we extend our study to the bifurcation results at the bifurcation point, addressing an open question posed by M. Colin, T. Colin, and M. Ohta (Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), 2211–2226; SIAM J. Math. Anal. 44 (2012), 206–223). Furthermore, we discover that the bifurcation solution emerges from the semitrivial solution, indicating a stability exchange when the parameter is small in certain regions. This phenomenon is a particularly interesting finding in this context.

Periodic solutions in a class of periodic switching delay differential equations

In this paper, we explore the dynamical properties of a class of nonlinear systems governed by delay differential equations with multitime periodic switching. The systems incorporate piecewise-smooth birth and death functions to capture complex population dynamics under seasonal variations. Assuming monotonicity for both birth and death functions, we obtain a novel equivalence result: when the delay is a positive integer multiple of the switching period, the existence and stability of periodic solutions for the systems are equivalent to those in the nondelay case. To illustrate and validate the theoretical findings, a logistic model with seasonal switching is presented. Numerical simulations further confirm that the system exhibits consistent dynamical behaviors across varying delay values.

Existence of Solutions for a Differential System Involving Doubly Nonlinear Inclusions

Existence of Solutions for a Differential System Involving Doubly Nonlinear Inclusions

Dynamical behaviors of Gilpin–Ayala competitive model with periodic coefficients on time scales

In the present paper we study the existence and stability problems of positive periodic solutions to a Gilpin–Ayala competitive model with periodic coefficients on time scales. Firstly, based on Schauder’s fixed theorem, some sufficient conditions for the existence of positive periodic solution to the considered system are obtained. Furthermore, we establish asymptotic behavior by using the existence of periodic solutions. Since the considered system is based on an arbitrary time scale, our results are applicable to both discrete and continuous scenarios. We provide a specific example to verify the above results.

A comprehensive investigation of fractional glucose-insulin dynamics: existence, stability, and numerical comparisons using residual power series and generalized Runge-Kutta methods

The regulation of blood glucose levels involves complex interactions between glucose, insulin, and beta cells, where disruptions may lead to diabetes. Traditional integer-order models fail to capture the memory-dependent and hereditary properties of biological systems. This study develops a fractional-order glucose-insulin-beta cell model and investigates its dynamics using the Residual Power Series Method (RPSM) and the Generalized Runge-Kutta Method (GRKM). Theoretical analyses establish the model's existence, uniqueness, and boundedness of solutions, ensuring biological validity. Stability and bifurcation analyses reveal the critical role of fractional orders in shaping system dynamics. RPSM demonstrates computational efficiency with rapid convergence, while GRKM excels in stability and bifurcation studies. Comparative numerical simulations highlight the complementary strengths of these methods, providing a robust framework for predictive modeling in diabetes management. This work underscores the potential of fractional-order models to advance understanding and control of metabolic disorders.

Modeling and Pricing European‐Style Continuous‐Installment Option Under the Heston Stochastic Volatility Model: A PDE Approach

ABSTRACTInstallment options, as path‐dependent contingent claims, involve paying the premium discretely or continuously in installments, rather than as a lump sum at the time of purchase. In this paper, we applied the PDE approach to price European continuous‐installment option and consider Heston stochastic volatility model for the dynamics of the underlying asset. We proved the existence and uniqueness of the weak solution for our pricing problem based on the two‐dimensional finite element method. Due to the flexibility to continue or stop paying installments, installment options pricing can be modeled as an optimal stopping time problem. This problem is formulated as an equivalent free boundary problem and then as a linear complementarity problem (LCP). We wrote the resulted LCP in the form of a variational inequality and used the finite element method for the discretization. Then the resulting time‐dependent LCPs are solved by using a projected successive over relaxation iteration method. Finally, we implemented our numerical method. The numerical results verified the efficiency and accuracy of the proposed numerical method.

Analysis and computations of a stochastic Cahn–Hilliard model for tumor growth with chemotaxis and variable mobility

Abstract In this work, we present and analyze a system of PDEs, which models tumor growth by taking into account chemotaxis, active transport, and random effects. Tumor growth may undergo erratic behaviors such as metastases that cannot be predicted simply using deterministic models. Moreover, random perturbations are evident in models accounting for therapeutic treatment in terms of therapy uncertainty or parameter identification problems. The stochasticity of the system is modeled by Wiener noises that appear in the tumor and nutrient equations. The volume fraction of the tumor is governed by a stochastic phase-field equation of Cahn–Hilliard type, and the mass density of the nutrients is modeled by a stochastic reaction-diffusion equation. We allow a variable mobility function and nonincreasing growth functions, such as logistic and Gompertzian growth. Via approximation and stochastic compactness arguments, we prove the existence of a probabilistic weak solution and, in the case of constant mobilities, the well-posedness of the model in the strong probabilistic sense. Lastly, we propose a numerical approximation based on the Galerkin finite element method in space and the semi-implicit Euler–Maruyama scheme in time. We illustrate the effects of stochastic forcings in tumor growth in several numerical simulations.

Asymptotic behavior of the basic reproduction number for periodic nonlocal dispersal operators and applications.

This paper is concerned with asymptotic behavior of the basic reproduction number defined by next generation nonlocal (convolution) dispersal operators in a time-periodic environment and applications. First we investigate the influence of the frequency and dispersal rate on the basic reproduction number, and we obtain that the basic reproduction number is monotone on the frequency. In the nonautonomous situation, the basic reproduction number is not a monotone function of dispersal rate in general. We derive the monotonicity for large frequency or dispersal rate. Then we apply the obtained results to a time-periodic SIS epidemic model and establish the existence and asymptotic profiles of the endemic periodic solution. Since solution maps of nonlocal system lack compactness, the standard uniform persistence theory and topological degree theory are unapplicable to obtain the existence of the endemic periodic solution. To overcome this difficulty, we apply the asymptotic fixed point theorem with the help of the Kuratowski measure of noncompactness.