Periodic solutions for nonconvex differential inclusions

In this paper we prove the existence of periodic solutions for differential inclusions with nonconvex-valued orientor field. Our proof is based on degree theoretic arguments.

     

Maximal monotone differential inclusions with memory

In this paper we study maximal monotone differential inclusions with memory. First we establish two existence theorems; one involving convex-valued orientor fields and the other nonconvex valued ones. Then we examine the dependence of the solution set on the data that determine it. Finally we prove a relaxation theorem.     

Measures of noncompactness and fixed points of random operators

The existence of random solutions is established for a class of random Volterra integral inclusions in which the orientor field has a stochastic domain. The proof is based on a stochastic analog of the Tietze extension theorem and on a deterministic existence result which are established as well in the paper. In particular the deterministic existence theorem is stated and proved for nonconvex orientor fields     

Periodic Solutions for Second Order Differential Inclusions with Nonconvex and Unbounded Multifunction

In this paper we consider a second order multivalued periodic boundary value problem with a nonconvex and unbounded orientor field (set-valued vector field). Using a directionally continuous selector, through its Filippov regularization we produce a convex-valued, bounded multifunction and with this as orientor field we introduce a new multivalued periodic problem. Using the Leray-Schauder principle, we solve the convex problem and then we show that its solutions also solve the original nonconvex problem.     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

一个抽象的Kuhn—Tucker定理

本文用鞍点定理证明了一个抽象的Ｋｕｈｎ－Ｔｕｃｋｅｒ定理，即得到由算子形式给出的约束，定义在抽象空间上的函数的非线性规划问题的存在性的一个等价条件。     

On monotonic solutions of an integral equation of Volterra type with supremum

Using a technique associated with measures of noncompactness, we prove the existence of nondecreasing solutions of an integral equation of Volterra type in C[0,1].     

A global bifurcation phenomenon for second order singular boundary value problems

We establish an existence result of unbounded continuum of solutions from the trivial branch for a second order singular boundary value problem. As an application, we prove several existence and multiplicity results of positive solutions for certain singular boundary value problems.     

A note on the theorem of Leray-Schauder

Using elemental methods of Topology, theory of degree and theory of metric continua, we prove a new version of the theorem of Leray-Schauder. It provides the existence of arc-connected set of solutions. This result may have a lot of applications in a large variety of problems. Although the assumption of the theorem is not easy to verify in practice, this theorem could be an important tool to prove not only the existence of set of solutions but also the existence of a set of solution which is homeomorphic to the interval [0,1]⊂R.     

一类渐近线性双调和方程非平凡解的存在性

通过建立新空间H,证明了一个嵌入定理.进一步,利用该嵌入定理与山路引理得到了一类渐近线性双调和问题在空间H中非平凡解的存在性.     

Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

Ultimate boundedness and periodicity for some partial functional differential equations with infinite delay

In this work we study the existence of periodic solutions for some partial functional differential equation with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies the known Hille-Yosida condition. Firstly, we give some estimates of the solutions. Secondly, we prove that the Poincaré map is condensing which allows us to prove the existence of periodic solutions when the solutions are ultimately bounded.     

Existence of Nonoscillatory Solutions for a Rational Difference Equation of Higher Order

In this paper, we investigate a rational difference equation of higher order for the existence of nonoscillatory solutions. To prove our main results, we use an inclusion theorem stated and proved in \cite{L1}. In this way, we give an answer of an open problem formulated in \cite{FPB}.     

MULTIPLE SOLUTIONS TO AN ASYMPTOTICALLY LINEAR ROBIN BOUNDARY VALUE PROBLEM

Under some weaker conditions,we prove the existence of at least two solutions to an asymptotically linear elliptic problem with Robin boundary value condition,using truncation arguments.Our results are also valid for the case of the so-called resonance at infinity.     

Existence of solutions of neutral functional integrodifferential equation in Banach spaces

In this paper we prove the existence of mild solutions for neutral functional integrodifferential equation in a Banach space. The results are obtained by using the Schaefer fixed-point theorem.     

Periodic solutions for semi-linear evolution inclusions

In this paper, we study the periodic problem for semi-linear evolution inclusion. Using techniques from multivalued analysis and fixed point theorems, we establish existence theorems under the one-sided Lipschitz condition. First we prove existence theorems for nonconvex and convex problem. Second, we look for extremal periodic solutions and prove the relaxation theorem.     

Applications of Measure of Noncompactness and Darbo’s Fixed Point Theorem to Nonlinear Integral Equations in Banach Spaces

We prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear integral equation of fractional order and give some sufficient conditions for existence of solutions of this equation. We use fixed point theorems associated with the measure of noncompactness as the main tool. Our existence results include several results obtained in previous studies. Finally we present some examples which show that our results are applicable.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

脉冲中立型时滞微分方程的正解的存在性

本文证明了线性脉冲中立型微分方程 [y(t)+p(t)y(t-τ)-r(t)y(t-ρ)]′+q(t)y(t-σ)=0,t≥0,t≠t_k, y(t_k~+)-y(t_k)=b_ky(t_k),k=1,2,…正解的存在性等价于一类非脉冲中立型方程正解的存在性,应用这一结果,建立了此类线性脉冲中立型微分方程正解存在性的若干充分条件。     

On vector variational inequalities

In this paper, we introduce a general form of a vector variational inequality and prove the existence of its solutions with and without convexity assumptions.