In this paper, we investigate the existence of positive solutions to a class of $n$-th order nonlocal differential equations with convolution coefficients and derivative dependence. By constructing nonstandard cone and specific open set, we establish the existence result of positive solutions by means of fixed point index theory. Two examples are also given to illustrate the main result.
Given a finite and connected two-dimensional complex $K$ and a homomorphism $\beta\in\hom(\pi_1(K);\mathbb Z_2)$, we consider the function $\Phi_{\beta}\colon [K;\RP^2]^{\ast}_{\beta}\to H^2(K;{}_{\beta}\mathbb Z)$ defined by $[f]^{\ast}\mapsto f^{\ast}(\nu)$, where $\nu$ is a preferred generator of the twisted cohomology group $H^2(\mathbb{R}\mathrm{P}^2;{}_{\varrho}\Z)$. We prove that $\Phi_{\beta}$ is a $\kappa_{\beta}$-to-one function, where $\kappa_{\beta}$ is the order of the kernel of the multiplication by $2$ on $H^2(K;{}_{\beta}\mathbb Z)$, and we present necessary and sufficient conditions for both: $\Phi_{\beta}$ to be injective and $\Phi_{\beta}$ to be surjective. Furthermore, we prove that $\Phi_{\beta}$ detects strong surjectivity if and only if either $0\notin{\rm im}(\Phi_{\beta})$ or the set $[K;\mathbb{R}\mathrm{P}^2]^{\ast}_{\beta}$ contains $\kappa_{\beta}$ classes having a representative given by a composite $K\to S^1\hookrightarrow\mathbb{R}\mathrm{P}^2$.
This article focuses on the study of the following Choquard-Kirchhoff type critical multiphase problem: \begin{alignat*}2 -M& (\varphi_{\h} (\p{u}))\\ &\times\text{div} \big(\p{u}^{p(x)-2}\nabla u +a_1(x)\p{u}^{q(x)-2}\nabla u +a_2(x)\p{u}^{r(x)-2}\nabla u\big)\hidewidth \\ =&\e g(x)\ve{u}^{\gamma(x)-2}u+\theta B(x,u) \\ &+\kappa \left(\int_{\q}\frac{F(y,u(y))}{\ve{x-y}^{d(x,y)}} dy\right) f(x,u) &\quad\hskip2.6cm & \text{in } \q,\\ u&=0 &\quad & \text{on } {\partial \Omega}. \end{alignat*} Here, the nonlinearity $B(x,u)$ exhibits critical growth. To handle this critical growth, we present the concentration compactness principle in the space $ W_0^{1,\h}(\q)$. To address the Choquard term, we prove the Hardy-Littlewood-Sobolev-type inequality in the framework of the generalized Sobolev space $ W_0^{1,\h}(\q)$. These tools, combined with variational methods, are used to establish the existence and multiplicity of weak solutions.
In this paper, we deal with the following p-Monge-Ampère system: \begin{equation*} \begin{cases} \text{det}(D(|Du_{1}|^{p-2}Du_{1}))=f_{1}(|x|,-u_{2}), & x\in B,\\ \text{det}(D(|Du_{2}|^{p-2}Du_{2}))=f_{2}(|x|,-u_{1}),& x\in B,\\ u_{1}=u_{2}=0, & x\in\partial B, \end{cases} \end{equation*} where $B=\{x\in\mathbb{R}^{n}:|x|< 1\}$ and $f_{i}$ $(i=1,2)$ are continuous and nonnegative functions. Based on the fixed-point theory, some results regarding existence of radial solutions are established when $f_{i}$ $(i=1,2)$ satisfy some new growth conditions.
In this paper, we introduce a class of locally convex spaces $(X, \Lambda)$ endowed with a topology generated by a family $\Lambda$ of $K$-seminorms, along with a family of generalized non-compact measures $\mathcal{M}_\Lambda$ taking values in the corresponding family of ordered sets $(R_p, \preceq_p)_{p \in \Lambda}$. We establish several fixed point theorems for $Q$-$\Lambda$-g-Lipschitz operators, $\mathcal{M}_\Lambda$-g-condensing/contraction operators, and mappings defined as the sum of such operators. These results are applied to address the existence and uniqueness of nonlocal solutions to generalized integrodifferential equations. Through illustrative examples, the paper demonstrates the utility of the family of $K$-seminorms and the family of generalized non-compact measures $\mathcal{M}_\Lambda$ in studying the existence of fixed points.
In this paper, we consider the family of problems \begin{equation*} \begin{cases} - \Delta_{H, p} u = f_{\lambda}(x, u) & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 2$ , $\lambda$ is a real parameter and the anisotropic Finsler $p$-Laplacian operator $\Delta_{H, p}$ with $1< p< N$, is defined as $\Delta_{H, p} u: = \text{div}(H(\nabla u)^{p-1} \nabla_{\eta} H(\nabla u)).$ The non-linear term $f_\lambda \colon \Omega \times \mathbb{R} \to \mathbb{R} $ is the sum of a sublinear and a superlinear term. We establish the strong comparison principle and extend the well-known Brezis-Nirenberg result concerning local minimization in $C_0^1$ and $W_0^{1,p}$ to the framework of the Finsler $p$-Laplace operator. Under various growth assumptions on the non-linear term $f_\lambda (x, s)$ and the real parameter $\lambda$, we establish the existence, non-existence and multiplicity of solutions.
Let $\Omega$ be a bounded smooth domain in $\R^N$ $ (N \geq 5)$ with $0 \in \Omega$. Non-trivial nonnegative weak solutions of Steklov problems of the fourth order Hénon-Hardy equation are obtained via variational methods and new embeddings related to new Caffarelli-Kohn-Nirenberg inequalities. Meanwhile, the further regularities of the weak solutions are studied, especially, the optimal regularity at the singular point $x=0$ of the solutions is obtained.
We establish existence results for a generalized Schrödinger equation on a smooth bounded domain in $\mathbb{R}^N$, which appears naturally in several applications of mathematical physics. The nonlinearity considered in the equation depends on a concave and sublinear-nonlocal terms that may be concave-convex. In such a case, variational methods cannot be applied. Our approach is based on a suitable change of variables, which transforms the original problem into an equivalent semilinear one. The positive solutions to semilinear equations are then presented using the Galerkin method together with a variation of the fixed point theorem. We also study the asymptotic behavior of the solutions with respect to the parameters. An important feature is that there are few works in the literature for the type of problem considered here, and the Galerkin method was not used to consider generalized quasilinear Schrödinger equations with nonlocal terms.
Muoi and Wong proved that a finite and commuting family of weakly continuous pointwise eventually nonexpansive mappings from $E$ into itself has a common fixed point in $E$ whenever $E$ is a nonempty weakly compact convex subset of a Banach space $B$. Without the assumption of weak continuity of pointwise eventually nonexpansive mappings, we prove the existence of common fixed points for a commuting family of pointwise eventually nonexpansive mappings in uniformly convex in every direction Banach spaces, $k$-uniformly rotund Banach spaces, and nearly uniformly convex Banach spaces. This improves the result of Muoi and Wong and extends the fixed point result of Dom\'inguez Benavides and Lorenzo Ram\'irez to a commuting family of pointwise eventually nonexpansive mappings. Moreover, we prove that a left reversible semigroup of pointwise eventually nonexpansive mappings from $E$ into itself has a common fixed point in $E$ whenever $E$ is a nonempty weakly compact convex subset of a uniformly convex in every direction Banach space $B$. This extends the fixed point result of S. Rajesh to a left reversible semigroup of pointwise eventually nonexpansive mappings.
We discuss the existence and nonexistence of positive decaying solutions for a semilinear elliptic systems considered in an exterior domain. Applying the subsolutions and supersolutions method and the Sattinger's iteration schema we prove that our problem possesses solutions with minimal growth and finite energy in a neighborhood of infinity. We also formulate necessary conditions for the existence of such solutions for a certain class of nonlinearities. Finally, some nonexistence results are formulated.
The paper deals with a semi-linear elliptic equation with non-local boundary conditions. In order to tackle problem under consideration we employ the classical Schauder theorem and the theory of convergent matrices. Firstly, we examine the solvability of the auxiliary problem -Delta u(x) = f (x, u(x)) Zfor x E Q, u(x) = K(x, ) h( , u( )) d for x E partial derivative Q, Omega where the growth of f, h and the kernel K are limited and where Q is open and bounded set with a Lipschitz bondary. Next, under the assumption of a sublinear growth for f and g, and using the approximation methods, the solvability of the problem of the form ( -Delta u(x) = f (x, u(x)) for x E Q, u(x) = alpha(x) g(u(p1), ... , u(pm)) for x E partial derivative Q, is examined in the H2-sense, where Q is an open and bounded subset of R2 or R3 with a C2 boundary and where p1, ... , pm E Q. We focus mainly on having the nonlo cal condition fixed at one point and then proceed to the multipoint case.
. In the 1960s Arnold conjectured that a Hamiltonian diffeomorphism of a closed connected symplectic manifold (M, omega) should have at least as many contractible fixed points as a smooth function on M has critical points. Such a conjecture can be seen as a natural generalization of Poincar's last geometric theorem and represents one of the most famous problems in symplectic geometry still open today in its full generality. In this paper, we build on a recent approach of the authors and Izydorek to the Arnold conjecture on CPn to show that the (degenerate) Arnold conjecture holds for Hamiltonian diffeomorphisms phi of T2m & times; CPn, m, n >= 1, which are C0-close to the identity in the CPn-direction, namely that any such phi has at least CL(T2m & times; CPn) + 1 = 2m + n + 1 contractible fixed points.
In this paper, we investigate a boundary case of the classical prescribed curvature problem. We focus on a prescribing scalar curvature which is equal to a given function K and a boundary mean curvature which is equal to a given function H on the standard ball (B-n, g). Our analysis extends previous studies by considering the scenario where the curvatures K and H are close to constants K-0 > 0 and H-0 > 0. Using a perturbative approach and leveraging the ansatz introduced by Han-Li [20], we establish new existence results for the conformal metric when the prescribed curvatures are near constants.
In this paper we are interested in the study of a two-phase problem equipped with the-Laplacian operator (Diu := div phi(|del u|) del u , |del u| where (s) = es2-1 and phi = '. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most (N-1).
For N >= 3 we study the following semipositone problem -Delta gamma u = g(z)fa(u) in R-N, where Delta(gamma) is the Grushin operator Delta(gamma)u(z) = Delta(x)u(z) + |x|(2 gamma)Delta(y)u(z) (gamma >= 0), z = (x, y) is an element of R-N = R-m & times; R-l, g is an element of L-1(RN) boolean AND L infinity (R-N) is a positive function, a > 0 is a parameter and fa is a continuous function on R that coincides with f (t) - a for t is an element of R+, where f is a continuous function with sub critical and Ambrosetti-Rabinowitz type growth and which satisfies f (0) = 0. Depending on the range of a, we obtain the existence of mountain pass solutions in a suitable Sobolev space associated to the Grushin operator, extending the results found in [5] to the Grushin operator.
The main goal of this paper is to formulate a full variational principle for measure-theoretic entropy for multivalued upper semicontinuous maps in a compact metric space. To this end, we introduce a new variant of measure-theoretic entropy, called natural extension entropy, which is consistent with the standard one in the single-valued case and satisfies an analogue of the classical variational principle formulated by Dinaburg and Goodman in 1971. In our principle, the key role is played by the topological entropy for multivalued maps invented by Gromov in 1977 and rediscovered 40 years later by Kelly and Tennant. Several natural properties of this new concept are established here. As far as we know, there is no such result dealing with a full variational principle for general upper semicontinuous maps. The only attempts made so far concern only the half-variational (only one inequality) principles for iterated function systems generated by single-valued continuous maps.
The main aim of this paper is to generalize and improve our earlier results in [J. Differential Equations 317 (2022), 365-386; 367 (2023), 783-803] and our joint result with Pavel Ludv & imath;k in [Internat. J. Bifur. Chaos Appl. Sci. Engrg. 33 (2023), no. 9, 2350113]. The theoretical part concerns the topological entropy of nonautonomous multivalued dynamical systems, studied by means of the asymptotic Nielsen theory. The practical part deals with the application of theoretical results via the associated Poincare translation operators to impulsive differential inclusions on tori.
We study a quasilinear elliptic problem involving the $p$-Laplacian operator and a slightly subcritical nonlinearity with a sign-changing weight. We assume that the slightly subcritical nonlinearity is a regularly varying function at zero and at infinity, which are not necessarily asymptotic to a power at infinity. We state sufficient conditions which guarantee a Palais-Smale condition. We also provide a bifurcation theorem for these nonlinearities, which allow us to state the existence of a bifurcated branch of positive solutions, containing a turning point, and multiplicity of solutions.
First we provide short, elementary and self-contained proofs of all known results concerning the lower bounds of the Banach-Mazur distances between the space $c_0$ of sequences converging to $0$ and other $\ell_1$-preduals isomorphic to $c_0$. Then, we use our technique to obtain lower bounds for the Banach-Mazur distances between any two $\ell_1$-preduals $X$ and $Y$. Our estimate depends only on the smallest radiuses $r^*(X)$ and $r^*(Y)$ of the closed balls in $\ell_1$ containing, respectively, all $\sigma(\ell_1,X)$-cluster points and all $\sigma(\ell_1,Y)$-cluster points of the set of all extreme points of the closed unit ball in $\ell_1$, and for any values of $r^*(X)$ and $r^*(Y)$ it is sharp. We apply this result to show that for every $\ell_1$-predual $X$ with $r^*(X)< 1$, every $\ell_1$-predual $Y$ with the distance from $X$ strictly less than $\frac{3-r^*(X)}{1+r^*(X)}$ induces a weak$^*$ topology on $\ell_1$ such that $\ell_1$ has the $\sigma(\ell_1,Y)$-fixed point property for nonexpansive mappings. If we additionally assume that the standard basis in $\ell_1$ is $\sigma(\ell_1,X)$-convergent, then the estimate is precise. The same holds if the standard basis in $\ell_1$ has a finite number of $\sigma(\ell_1,X)$-cluster points and each of them has a finite number of non-zero coordinates. It should be emphasized that the value of this constant was known only for the space $c_0$ so far.
We prove that a mapping $u \colon \mathcal{M}'\to \mathcal{N}$, where $\mathcal{M}'$ and $ \mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U \colon \mathcal{M}' \times [0, 1) \to \mathcal{N}$ if and only if it is on some open covering of $\mathcal{M}'$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u \colon \partial \mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U \colon \mathcal{M} \to \mathcal{N}$ are purely local.
