Existence of solutions for fractional impulsive differential equations with p-Laplacian operator

We investigate the boundary value problems for nonlinear fractional impulsive differential equations with p-Laplacian operator. By applying some standard fixed point theorems, we obtain new results on the existence and uniqueness of solutions. Examples are given to show the applicability of our results.     

Existence and approximation of solutions to nonlinear operator equations

Summary We consider the approximation of solutions to a nonlinear operator equation of the formLu+N(u)=f in a Hilbert space by the Rayleigh-Ritz-Galerkin method. Using a variant of Cesari's alternative method we determine when the existence of an approximation of orderk determines the existence of a solution to the equation and give a method to determine error bounds on the approximation.

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Zusammenfassung Wir benutzen das Rayleigh-Ritz-Galerkin'sche Verfahren um Lösungen der nichtlinearen Operatorgleichung vom TypLu+N (u)=f in einem Hilbertraum zu gewinnen. Eine Umformulierung der Cesari'schen alternativen Methode wird angewendet um die Frage zu lösen ob die Existenz einer Approximation der Ordnungk die Existenz einer exakten Lösung mit sich bringt. Eine Methode um Fehlerschranken für die Approximationen zu bestimmen wird angegeben.

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Supported in part by the DAAD/CONICYT, professor exchange program     

Structure of positive solution sets of sub-linear semi-positone operator equations

In this paper, first we obtain some results on the structure of the positive solution set of a nonlinear operator equation. Then using these results, we obtain some existence results for positive solutions of nonlinear operator equations. We use global bifurcation theories to show our main results.     

Existence of periodic solutions for nonlinear differential equations with a p-Laplacian-like operator

The existence of periodic solutions for nonlinear differential equations with a $p$-Laplacian-like operator is studied by applying a new generalization of polar coordinates.     

Existence of periodic solutions for Hamiltonian systems with super-linear and sign-changing nonlinearities

In this paper, we consider the existence of periodic solutions for the super quadratic second order Hamiltonian system, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some existed results in the literature.     

Existence of periodic solutions for neutral functional differential equations with nonlinear difference operator

In this paper, the authors consider the problem of existence of periodic solutions for a second order neutral functional differential system with nonlinear difference D-operator. For such a system, since the possible periodic solutions may not be differentiable, our method is based on topological degree theory of condensing field, not based on Leray Schauder topological degree theory associated to completely continuous field.     

Existence of solutions of two generalized riccati operator equations and their representation

The solutions of two generalized Riccati operator equations are discussed in terms of two critical parameter values, which are related to the application of optimal control under unknown disturbances. Explicit formulas for calculating these two critical parameters as well as the closedform solutions of these two generalized Riccati operator equations are given. The connection between these two parameters and a zero-sum differential game is also investigated.     

Existence of solutions for the equations

We introduce a concept of weak solution for a boundary value problem modelling the interactive motion of a coupled system consisting in a rigid body immersed in a viscous fluid. The fluid, and the solid are contained in a fixed open bounded set of R3. The motion of the fluid is governed by the incompresible Navier-Stokes equations and the standard conservation's laws of linear, and angular momentum rules the dynamics of the rigid body. The time variation of the fluid's domain (due to the motion of the rigid body) is not known apriori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary     

Existence of solutions for set differential equations involving causal operator with memory in Banach space

This paper concerns with the IVP for set differential equations involving causal operators with memory. By using the techniques of the measure of non-compactness, the existence of solutions has been established. The maximal and minimal solutions are obtained by means of Arzela-Ascoli Theorem.     

带p-Laplacian算子的分数阶微分方程非局部边值问题解的存在性

利用经典的Schaefer不动点定理,研究了一类带有p-Laplacian算子的非线性分数阶微分方程非局部边值问题,在适当的假设条件下,得到了解的存在性结果.并举例说明结果的应用.     

On sign-changing solutions for nonlinear operator equations

In this paper, the existence of sign-changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory. The main theorems are some new three-solution theorems which are different from the famous Amann's and Leggett-Williams' three-solution theorems as well as the results in [F. Li, G. Han, Generalization for Amann's and Leggett-Williams' three-solution theorems and applications, J. Math. Anal. Appl. 298 (2004) 638-654]. These three solutions are all nonzero. One of them is positive, another is negative, and the third one is a sign-changing solution. Furthermore, the theoretical results are successfully applied to both integral and differential equations.     

Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We develop index theories for linear selfadjoint operator equations and investigate multiple solutions for asymptotically linear operator equations. The operator equations consist of two kinds: the first has finite Morse index and can be used to investigate second order Hamiltonian systems and elliptic partial differential equations; the second may have infinite Morse index and can be used to investigate first order Hamiltonian systems.     

Existence of solutions for nonlinear elliptic degenerate equations

In this paper, we study the problem$\text{−div}\text{a(x,u,}\text{∇}\text{u)−div}\text{φ(u)+g(x,u)=f}\text{in}\text{Ω}$in the setting of the weighted sobolev space $\text{W}{}_0{}^{\mathrm{1,p}}\text{(Ω,ν)}$. The main novelty of our work is L^∞ estimates on the solutions, and the existence of a weak and renormalized solution.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.