In this paper, for a modified Gross–Pitaevskii energy functional with critical parameter a=a^* , as the parameter of higher order interaction strength vanishes, we obtain the precise blow-up rate and locate the blow-up point of the minimizers of the functional using the cut-off function technique. Furthermore, through a detailed analysis of the limit behavior of these minimizers, the uniqueness of minimizers is established as the parameter of higher order interaction strengths vanishes, by constructing some suitable Pohozaev identities.
In this paper, we construct a Lagrangian cobordism in the symplectic mirror of ℙ^n which we propose as the mirror of the Euler exact sequence 0 ⟶Ω ^1_ℙ^n⟶𝒪_ℙ^n(-1)^⊕ (n+1)⟶𝒪_ℙ^n⟶ 0 . (1) This cobordism produces an immersed Lagrangian object L_Ω whose virtual Floer-theoretic weight invariant W̌(L_Ω ,σ ) , defined in the graded Fukaya category via Euler characteristics of Floer cohomology with the standard linking disks at each toric fixed point σ , agrees with the Klyachko weight system of the cotangent sheaf Ω ^1_ℙ^n . Our construction is based on Biran–Cornea’s theory of Lagrangian cobordisms and is motivated by a question of Suen about mirrors of higher-rank bundles on toric Fano varieties. In particular, it provides a concrete candidate for a Lagrangian mirror of Ω ^1_ℙ^n∈𝒟^b(ℙ^n) , extending the description of line bundles to the first nontrivial T-equivariant vector bundle.
In the present paper, we study the inverse problem for the two-dimensional convective Brinkman–Forchheimer (CBF) equations with the integral overdetermination condition and discontinuity at the initial moment of time. By CBF equations, we mean the Navier–Stokes equations with an absorption term -(αv+β |v|^r-1v), α , β >0, r∈ [1,+∞ ). Specifically, we approximate the discontinuous overdetermination condition with a continuous one. Following the classical results on the initial layer of singularly perturbed differential equations, we scale the problem in an infinitesimal neighborhood [0,1/n] of the point t=0 , using the fast time variable t= tn∈ [0,1] . In the limit, we obtain ordinary differential equations that provide a new initial velocity for the inverse limit problem in the rest domain.
In this paper, we study the existence of multiple normalized solutions to the following fractional Kirchhoff equation with prescribed mass {[ ( a+b∫ _ℝ^3|(-Δ )^s/2u|^2dx) (-Δ )^su+V(ϵ x)u=λ u+f(u), in ℝ^3,; ∫ _ℝ^3|u|^2dx=m^2, ]. where s ∈ (3/4,1) , ϵ , a, b, m>0 , λ∈ℝ appears as an unknown Lagrange multiplier, the potential V: ℝ^3→ [0,+∞ ) is a continuous function, and f is a continuous function with L^2 -subcritical growth. We relate the number of normalized solutions with the topology of the set where V attains its minimum value. The main result is established using minimization techniques and the Ljusternik–Schnirelmann category.
We study the split common solution problem with multiple output sets in real Hilbert spaces. To solve this problem, we propose two new self-adaptive iterative algorithms based on hybrid and shrinking projection methods. Unlike existing approaches, the proposed methods require no prior knowledge of the norms of the transfer operators or the inverse strong monotonicity constants of the associated operators. Under suitable conditions, we establish strong convergence of the generated sequences to a solution of the problem. As applications, we show that our results can be used to solve several related problems, including the split common fixed point problem and the split minimum point problem with multiple output sets. Numerical experiments are presented to illustrate the effectiveness and computational performance of the proposed algorithms.
In this work, we study the existence, uniqueness, and exponential decay of asymptotically almost periodic mild solutions to non-autonomous parabolic equations with Lipschitz continuous functional coefficients in admissible spaces. Under the assumptions that the associated evolution family admits an exponential dichotomy and that the Green function is exponentially almost periodic, we establish a Massera-type principle: if the right-hand side is an asymptotically almost periodic function, then the corresponding linear equation admits a unique asymptotically almost periodic mild solution. The well-posedness of such solutions for semilinear equations is proved using fixed point arguments, taking into account that the nonlinear part on the right-hand side is φ -Lipschitz, where φ belongs to an admissible space. In addition, the exponential decay of these solutions is obtained via Gronwall’s inequality. Finally, we apply the abstract results to a non-autonomous parabolic equation and to certain second-order differential equations, such as wave-type equations.
In this article, we study the finite-approximate controllability of a class of non-autonomous semilinear neutral impulsive evolution systems with state-dependent delays in a separable Hilbert space. Our approach is based on constructing a nonlinear solution operator and proving the existence of a fixed point using Schauder’s fixed-point theorem together with the compactness of the associated evolution family. This fixed-point framework is combined with a variational control formulation, which allows us to treat impulses, neutral terms, and memory effects in a unified way under mild assumptions. The resulting criteria guarantee that the system can be steered arbitrarily close to a desired terminal state while matching prescribed finite-dimensional constraints. An application to a semilinear parabolic equation with memory illustrates the effectiveness of the method.
For a general domain Ω⊂ℝ^d in higher dimensions d≥ 5 , we examine the initial value problem for a class of nonlinear fourth-order reaction–diffusion equations with variable coefficients. We prove the local-in-time well-posedness (local existence, regularity estimate, continuous dependence on initial data and continuation) of the solutions in the time-weighted Sobolev space. Moreover, we investigate the global-in-time theory of the solutions (global existence or finite-time blow-up) in the Sobolev space.
In this paper, we present new multiplicity fixed point theorems for operators acting on Cartesian products of two normed linear spaces. We show that Leggett–Williams type conditions in each component of the system guarantee the existence of nine distinct fixed points, of which four of them are coexistence fixed points, i.e., points with all components nontrivial. In addition, a hybrid approach combining Leggett–Williams conditions in one component with Krasnosel’skiĭ compression–expansion conditions in the other allows us to obtain three fixed points. As an application, we establish the existence of multiple positive solutions for nonlinear systems of second-order equations with two-point boundary conditions.
In this paper, we study the normalized solutions for the following Chern–Simons–Schrödinger system with the Choquard type nonlinearity and the local nonlinear perturbation: {[ - Δ u + λ u + ( A_0 + ∑ _j = 1^2 A_j^2) u = (I_α *|u|^α/2 + 1)|u|^α/2-1u + μ| u | ^p - 2u, x∈ℝ^2,; ∂ _1A_2 - ∂ _2A_1 = - 1/2| u | ^2,∂ _1A_1 + ∂ _2A_2 = 0,; ∂ _1A_0 = A_2| u | ^2,∂ _2A_0 = - A_1| u | ^2,; ∫ _ℝ^2| u | ^2dx = c>0, ]. where λ∈ℝ is known as the Lagrange multiplier, μ >0 , 2
In this paper, we study existence and summability of distributional solutions in H_0^1(Ω ) or H^1_loc(Ω ) to a class of quasilinear stationary Schrödinger equations with double singularities and a quadratic convection term. The results obtained depend on the summability of a datum g(x) (which belongs to a Lebesgue space) and on a parameter λ of a singular term. In particular, we establish the existence of distributional solutions in H_0^1(Ω ) for weak and strongly singular nonlinearities.
We are concerned with the initial-boundary-value problem for a coupled system of viscoelastic equations with variable exponents. Under assumptions on initial data and log-Hölder continuous exponents m(x), r(x) , we prove local existence of weak solutions to the initial-boundary-value problem using the fixed point theory, the Galerkin method, the auxiliary system approach, and other techniques. Moreover, we estimate the upper bound of the lifespan of the blow-up solutions to the coupled system using energy methods, differential inequalities, and new estimation techniques.
One of the most important tools in the study of the dynamics of an iterated function system (IFS) is the canonical projection defined between the code space and the attractor of the system. The canonical projection is obtained as the fixed point of the H-S operator associated with an IFS having a unique attractor. In this paper, we generalize the H(S )operator for a new class of IFSs for which the component functions are endowed with weaker contractivity conditions, so the attractor of such a system is not necessarily unique. We prove that the generalized operator is continuous and weakly Picard. Also, we use this generalization to prove that the Markov operator associated with such a system endowed with probabilities is weakly Picard.
This study focuses on the initial-boundary value problem of a class of coupled Klein–Gordon systems, which incorporate Kirchhoff terms, dispersive terms, damping terms, and nonlinear source terms. Initially, the local existence and uniqueness of the system’s weak solutions are established by applying the Galerkin method in conjunction with priori estimates. Afterward, within the framework of the potential well theory and under the subcritical initial energy condition, three key results are derived: the global existence of weak solutions, their polynomial decay behavior, and the finite-time blow-up phenomenon. Finally, upper bound estimates for the blow-up time are obtained. In addition, we extend the conclusions of global existence, energy decay and blow-up in the case of critical initial energy.
This paper is concerned with the life span of solutions for a semilinear pseudo-parabolic equation with inhomogeneous source term λ f(x) . First, by using Kaplan’s first eigenvalue method, an upper bound estimate for the life span T_λ is obtained. Based on this, by means of the properties of the pseudo-parabolic kernel, we establish the asymptotic behavior of the life span T_λ as λ→ 0 for the case where f(x) is a radially decreasing function. Finally, employing the comparison principle, we obtain the asymptotic behavior of the life span T_λ as λ→ 0 .
In this paper, we consider the class of Lipschitz maps on the unit ball B_X of a Banach space X, and the question we deal with is whether for any λ >1 there exists a λ -Lipschitz fixed-point free mapping T:B_X→ B_X with d(T,B_X)=0 . We also consider its Hölder version. New related results are obtained. We show that if X has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in [7]. In the general case, using a recent approach of Medina [33] concerning Hölder retractions of (r_n) -flat closed convex sets, we show that for any decreasing null sequence (r_n)⊂ℝ and α∈ (0,1) , there exists a fixed-point free mapping T on B_X so that ‖ T^nx - T^n y‖≤ r_n(‖ x - y‖ ^α +1) for all x, y∈ B_X and n∈ℕ .
We consider the discrete systems of prescribed mean curvature equations with Lane-Emden type nonlinearities in Minkowski spaces {[ ∇ (Δ u(x)/√(1-|Δ u(x)|^2) )+λ _1μ _1(x)u^p_1v^q_1=0, x∈ [1, N-1]_ℤ,; ∇ (Δ v(x)/√(1-|Δ v(x)|^2) )+λ _2μ _2(x)u^p_2v^q_2=0, x∈ [1, N-1]_ℤ,; u(0)=u(N)=0, v(0)=v(N)=0, ]. where λ _1, λ _2>0 are real parameters, and μ _1, μ _2: [1, N-1]_ℤ→ [0,∞ ) are continuous functions. Based on the lower and upper solutions method and fixed point index, we prove that there exists a continuous curve Γ , such that the first quadrant is divided into two disjoint unbounded open sets 𝒪_1 and 𝒪_2 by this curve, and the discrete system has no positive solutions if (λ _1, λ _2)∈𝒪_1 , at least one positive solution if (λ _1, λ _2)∈Γ or at least two positive solutions if (λ _1, λ _2)∈𝒪_2 .
We are concerned with the following fractional elliptic equation with almost critical non-power non-linearity {[ (-Δ )^s u =|u|^2_s^*-2u/[ln (e+|u|)]^ε in Ω ,; u= 0 on ∂Ω , ]. where Ω is a bounded smooth domain in ℝ^n with n≥ 2 s+1 , s∈ (1/2,1) , (-Δ )^s is the spectral fractional Laplacian operator with zero Dirichlet boundary condition, 2_s^*=2n/n-2s is the fractional critical Sobolev exponent, ε >0 is a small parameter. By employing the Lyapunov-Schmidt reduction argument, we prove that this problem admits a sign-changing solution behaving like a superposition of bubbles blowing-up at minimum points of the Robin function with different rates of concentration as ε goes zero.
One of the most important tools in the study of the dynamics of an iterated function system (IFS) is the canonical projection defined between the code space and the attractor of the system. The canonical projection is obtained as the fixed point of the H_S operator associated with an IFS having a unique attractor. In this paper, we generalize the H_S operator for a new class of IFSs for which the component functions are endowed with weaker contractivity conditions, so the attractor of such a system is not necessarily unique. We prove that the generalized operator is continuous and weakly Picard. Also, we use this generalization to prove that the Markov operator associated with such a system endowed with probabilities is weakly Picard.
This paper investigates a class of ϕ -Laplacian singular differential equations with an indefinite weighted parameter (ϕ (x'))'+g(t)/x^ρ=h(t)x^δ+s· e(t), where e∈ C(ℝ/Tℤ;ℝ) is sign-changing, s∈ℝ is a parameter, and T>0 . By employing the method of upper and lower solutions and Leray-Schauder degree, we establish an Ambrosetti-Prodi type result for the equation in the cases where the weight function e is either strictly positive or sign-changing. In addition, we analyze the asymptotic behavior of the periodic solutions as the parameter tends to infinity.
