Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter

In this paper, we study the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. By using the properties of the Green’s function, u ₀-positive function and the fixed point index theory, we obtain some existence results of positive solution under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The method of this paper is a unified method for establishing the existence of multiple positive solutions for a large number of nonlinear differential equations of arbitrary order with any allowed number of non-local boundary conditions.     

LELONG-DEMAILLY NUMBERS IN TERMS OF CAPACITY AND WEAK CONVERGENCE FOR CLOSED POSITIVE CURRENTS

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity, where T is a closed positive current and CT is the capacity associated to T. This derived from some esimate on the growth of the CT-capacity of the sublevel sets of a weighted plurisubharmonic （psh for short） function. These estimates enable us to give another proof of the Demailly＇s comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality CT（K） ： fK T ∧ （dd^cu）^p, where K is a compact subset. Finally, we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT, for positive closed currents T, Tk and psh functions uk, u.     

Fourth order singular p-Laplacian differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions for fourth order singular p-Laplacian differential equations with integral boundary conditions and non-monotonic function terms. Firstly, we establish a comparison theorem, then we define a partial ordering in E ₀ and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C ²[0,1] as well as pseudo-C ³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x=0, y=0, t=0 and t=1. Finally, we give some the dual results for the other cases of fourth order singular integral boundary value problems and an example to demonstrate the corresponding main results.     

Multiple positive solutions for a critical quasilinear elliptic system

The existence of cat_Ω(Ω) positive solutions for the p-Laplacian system with convex and Sobolev critical nonlinearities is obtained by some standard variational methods, whose key is to construct homotopies between Ω and levels of the functional J_λ,μ, and some analytical techniques.     

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains

In this paper we are concerned with the existence and multiplicity of nodal solutions to the Dirichlet problem associated to the elliptic equation Δu+q(|x|)g(u)=0 in a ball or in an annulus in .The nonlinearity g has a superlinear and subcritical growth at infinity, while the weight function q is nonnegative in [0,1] and strictly positive in some interval [r₁,r₂]⊂[0,1].By means of a shooting approach, together with a phase-plane analysis, we are able to prove the existence of infinitely many solutions with prescribed nodal properties.     

Absolutely convergent Fourier series and function classes

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0<α?1, or to one of the Zygmund classes Λ_∗(1) and λ_∗(1). Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463-483] and one by Németh [J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291-304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205-208] on the existence and continuity of the derivative of f.     

On matrix measures and convex Liapunov functions

In this paper, we extend the concept of the measure of a matrix to encompass a measure induced by an arbitrary convex positive definite function. It is shown that this “modified” matrix measure has most of the properties of the usual matrix measure, and that many of the known applications of the usual matrix measure can therefore be carried over to the modified matrix measure. These applications include deriving conditions for a mapping to be a diffeomorphism on Rⁿ, and estimating the solution errors that result when a nonlinear network is approximated by a piecewise linear network. We also develop a connection between matrix measures and Liapunov functions. Specifically, we show that if V is a convex positive definite function and A is a Hurwitz matrix, then μ_V(A) < 0, if and only if V is a Liapunov function for the system $\text{x}\text{?}\text{= Ax}$. This linking up between matrix measures and Liapunov functions leads to some results on the existence of a “common” matrix measure μ_V(·) such that μ_V(A_i) < 0 for each of a given set of matrices A₁,…, A_m. Finally, we also give some results for matrices with nonnegative off-diagonal terms.     

Existence and uniqueness of positive periodic solutions of functional differential equations

In this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y′(t)=−a(t)y(t)+λh(t)f(y(t−τ(t))). By the eigenvalue problems of completely continuous operators and theory of α-concave or −α-convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution y_λ(t) of the above equation depends continuously on the parameter λ. Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model.     

Existence of positive solutions for non-positive higher-order BVPs

We shall provide conditions on non-positive function f(t,u₁,…,u_n−1) so that the boundary value problem

has at least one positive solution. Then, we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions.     

Existence of positive solutions for higher-order generalized p-Laplacian BVPs

In this paper, we provide suitable conditions on the function f(t, u₁, …, u_n−1) such that the higher-order generalized p-Laplacian boundary value problem

has at least one positive solution.Moreover, we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions.     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

Near invariance and kernels of Toeplitz operators

This paper makes a systematic study of kernels of Toeplitz operators on scalar and vector-valued H _p spaces (for 1 < p < ∞). The property of near invariance of a kernel for the backward shift is analysed and shown to hold in increased generality. In the scalar case, and in some vectorial cases, the existence of a minimal kernel containing a given function is established, and a symbol for a corresponding Toeplitz operator is determined; thus, for rational symbols, its dimension can be easily calculated. It is shown that every Toeplitz kernel in H _p is the minimal kernel for some function lying in it.     

Existence of positive solutions for a second-order ordinary differential system

In this paper we consider the existence of positive solutions for a second-order ordinary differential system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing a cone K₁×K₂ which is the Cartesian product of two cones in space C[0,1] and computing the fixed point index in K₁×K₂, we establish the existence of positive solutions for the system. We remark that, differently from the literature, we deal with our problem on the Cartesian product of two cones, in which the feature of two equations can be exploited better.     

Maximum principle and existence of solutions for non necessarily cooperative systems involving Schrödinger operators

In this paper, we obtain the necessary and sufficient conditions for having the maximum principle and existence of positive solutions for some cooperative systems involving Schrödinger operators defined on unbounded domains. Then, we deduce the existence of solutions for semi-linear systems. Finally we discuss the generalized maximum principle (gf _q -positivity) for non cooperative systems.     

Modifications of object sizes and box capacities to achieve a simultaneous fitting

Given a set of k objects of positive integral sizes {s_i} and a set of n boxes of positive integral capacities {b₁} we define a compatibility relation between each box and some subset of objects such that the condition $\text{s}{}_{\mathrm{i}}\text{? b}{}_1$ is necessary but not sufficient for object i to be compatible with box j. A simultaneous fitting of objects in boxes is one where every object is compatible to its box and no box has its capacity exceeded. In this paper, a necessary condition for the existence of such a fitting is stated. It is shown that this condition is sufficient for the existence of fittings under certain modifications of sizes or capacities. Two measures of the magnitude of the needed modification are introduced and examined.     

New properties of King’s operators

We prove the existence of a sequence of King’s operators which approximate each continuous function on [0, 1] and preserve the functions e ₀(x) = 1 and e _2i (x) = x ²ⁱ . Moreover, we construct a sequence of polynomial bounded positive linear operators possessing similar properties.     

Invariant manifolds of admissible classes for semi-linear evolution equations

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to certain classes of admissible function spaces.     

Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls

We study the existence and non-existence of positive singular solutions of second-order non-divergence type elliptic inequalities of the form $\sum\limits_{i,j = 1}^N {a_{ij} (x)\frac{{\partial ^2 u}} {{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^N {b_i (x)\frac{{\partial u}} {{\partial x_i }} \geqslant K(x)u^p ,} - \infty < p - \infty , $ with measurable coefficients in a punctured ball B _R \{0} of ? ^N , N ≥ 1. We prove the existence of a critical value p* which separates the existence region from the non-existence region. We show that in the critical case p = p*, the existence of a singular solution depends on the rate at which the coefficients (a _{i j} ) and (b _i ) stabilize at zero, and we provide some optimal conditions in this setting.     

Bahri-Coron type theorem for the scalar curvature problem on high dimensional spheres

We consider the existence and multiplicity results for the prescribed scalar curvature problem on the standard spheres of high dimension n ?? 7. Given a C ² positive function K, using the theory of critical points at infinity, we prove an existence result as Bahri-Coron theorem. Our case is a generalization of Li (J Differ Equ 120:319?C410, 1995). Indeed, here the function K is flat near some critical points as in Li (J Differ Equ 120:319?C410, 1995) and it can have some nondegenerate critical points with ?? K ?? 0. Furthermore, using some topological arguments, we prove another kind of result.