The Existence of Solutions for Kirchhoff-Type Equations with General Singular Terms

We study the existence of solutions for Kirchhoff-type equations.Firstly,we use the Sobolev inequality and the weakly lower semi-continuity of the norm to prove that the corresponding function can reach the global minimum.Then,we use the variational method and some analytical techniques to obtain the existence of the positive solution of the equation whenλis small enough.

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

AN EXISTENCE THEOREM OF POSITIVE SOLUTIONS FOR ELASTIC BEAM EQUATION WITH BOTH FIXED END-POINTS

By using the degree theory on cone an existence theorem of positive solution for a class of fourth-order two-point BVP's is obtained. This class of BVP's usually describes the deformation of the elastic beam with both fixed end-points.

CLASSIFICATION AND EXISTENCE OF POSITIVE SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DELAY DEPENDING ON THE UNKNOWN FUNCTION

A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M ： R＋ → R＋ and V ： RN →R＋ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION

In the present paper,we consider the nonlocal Kirchhoff problem-(ε^2a+εb∫|■u|^2)Δu+u=Q(x)u^p,u>0 in R^3,,where a,b>0,1<p<5 andε>0 is a parameter.Under some assumptions on Q(x),we show the existence and local uniqueness of positive multi-peak solutions by LyapunovSchmidt reduction method and the local Pohozaev identity method,respectly.

On the Existence of Positive Solutions for Impulsive Neutral Differential Equations

Aim To investigate the existence of positive solutions for impulsive neutral differential equations. Methods The Banach contraction principle was used to establish our results. Results and Conclusion The results of the existence of positive solutions for impulsive neutral differential equations are obtained.

SYMMETRY OF POSITIVE SOLUTIONS FOR THE FRACTIONAL HARTREE EQUATION

In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.

Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation X^s+ A~*X^(-t) A = Q

The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.

On the Existence of Positive Solutions of Neutral Differential Equations

Consider the first order neutral delay differential equation with positive and negative coefficients:[x(t)-c(t)x(t-γ)]+p(t)x(t-τ)-Q(t)x(t-δ)=0,t≥t 0,(1)where c,p,Q∈C((t 0,∞),R +),R +=(0,∞),γ】0,t】δ≥0. We obtain the sufficient condition for the existence of positive solutions of Eq.(1). As a corollary, we improve the correspondent result in by removing the condition∫ ∞ c 0 (t) d t=∞,where (t)=p(t)-Q(t-τ+δ)≥0.

On the positive nonoscillatory solutions of the difference equation Xn＋1=α＋（xn-k/xn-m）^p

The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α＋1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.

On the Behavior of the Positive Solutions of the System of Two Higher-Order Rational Difference Equations

We study the convergence of the positive solutions of the system of the following two difference equations: where K is a positive integer, the parameters?A,B,?α, β? and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].

The Existence of Positive Periodic Solutions for a Kind of Delay Logistic Equations

In the present paper, we consider the existence of positive periodic solutions for a kind of delay Logistic equations. By using a fixed point theorem in cones, we give some new existence results of single and multiple positive periodic solutions for a kind of delay Logistic equations. Some biomathematical models are presented to illustrate our results.

SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

Positive Solutions of a Semilinear Elliptic Equation with Coefficient That Changes Sign

We prove the existence of a positive solution to the problem-Δu=a(x)f(u), x∈Ω, u(x)=0,x∈Ω,where Ω is a bounded domain in R n with smooth boundary, a(x) is allowed to change sign.

An Existence Theorem of Positive Solution for Fourth-order Superlinear Semipositone Eigenvalue Problems

In this paper we investigate the existence of positive solution for a class of fourth_order superlinear semipositone eigenvalue problems. This class of problems usually describes the deformation of the elastic beam whose both end_points are fixed.

Asymptotic Behavior and Nonexistence of Positive Solutions of Delay Difference Equations *L

Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION

In the present paper, we consider the problem {-△u=u^(β_(1))|■u|^(β_(2)),in Ω,u=0,on ■Ω,u>0,in Ω,(0.1) where β_(1), β_(2) > 0 and β_(1) + β_(2) < 1, and Ω is a convex domain in R~n. The existence, uniqueness,regularity and (2-β_(2))/(1-β_(1)-β_(2))-concavity of the positive solutions of the problem(0.1) are proven.

POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRODINGER EQUATIONS

In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].