The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^{n}\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.
This paper studies a natural one-parameter extension of the Hardy-Hilbert integral inequality. The proposed generalization introduces a parameter that interpolates between different forms. This allows us to establish a hierarchy among a family of related double integrals. We provide sharp upper bounds expressed in terms of the integral norms of the functions involved. In doing so, we extend a classical result while maintaining the optimality of the constant in the original inequality.
This article concerns the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation with \(p-\)Laplacian \[\Big(\left|u''(t)\right|^{p-2}u''(t)\Big)''-\omega\Big(\left|u'(t)\right|^{p-2}u'(t)\Big)'+V(t)\left|u(t)\right|^{p-2}u(t)=f(t,u(t)),\] where \(p>1\), \(\omega\) is a constant, \(V\in C(\mathbb{R},\mathbb{R})\) is noncoercive and \(f\in C(\mathbb{R}^{2},\mathbb{R})\) is of subquadratic growth at infinity. Some results are proved using variational methods, the minimization theorem and the generalized Clark’s theorem. Recent results in the literature are extended and improved.
In this paper we find viscosity solutions to a system with two parabolic obstacle-type equations that involve two normalized \(p-\)Laplacian operators. We analyze a two-player zero-sum game played on two boards (with different rules in each board), in which at each board one of the two players has the choice of playing in that board or switching to the other board and then play. We prove that the game has a value and show that these value functions converge uniformly (when a parameter that controls the size of the steps made in the game goes to zero) to a viscosity solution of a system in which one component acts as an obstacle for the other component and vice versa. In this way, we find solutions to the parabolic two-membranes problem.
The paper considers real valued stretched graphons defined on the Lebesgue measure space \(([0,\infty),{ m})\). The topological space of these graph functions is equipped with the core Hopf algebra to assign renormalized values to unbounded stretched graphons.
In this paper, the dynamical behaviors of a stochastic competition system with a saturation effect are analyzed. The existence and uniqueness of globally positive solution are proved in detail, and the sufficient conditions for stochastic permanence, strong persistence in the mean, weak persistence, and extinction are obtained respectively. Then the existence and uniqueness of stationary distribution are also obtained under some appropriate assumptions. Finally, several numerical simulations are provided to justify the analytical results.
It is proved that every infinite-dimensional Banach space \(X\) of cardinality \(m\) admits both a strictly descending chain and a strictly ascending chain of dense linear subspaces of length \(m\).
In this article, we extend a key integral inequality established by Pachpatte in 2002 by introducing a new convexity-based approach. Specifically, we incorporate a general convex function to create a flexible framework that can be adapted to various mathematical contexts. The resulting techniques are original and reusable, offering potential for further innovations in related analytical frameworks. We present several examples to illustrate the theory and demonstrate the versatility of the approach.
We construct a semigroup of composition operators on a subspace of the Dirichlet space of the upper half-plane. We then determine both the semigroup and spectral properties of the composition semigroup. Finally, we represent the resolvents of the infinitesimal generator as integral operators and obtain their norm and spectra.
Fractional differential equations is a rapidly growing field of mathematical analysis with a wide and robust applicability in several areas of physics and geometry. Picone identity is a powerful tool which has been applied extensively in the study of second order elliptic equations. In this paper we prove some nonlinear anisotropic Picone type identities and give its applications to deriving Sturmian comparison principle and Liouville type results for anisotropic conformable fractional elliptic differential equations and systems.
