A further three critical points theorem

If X is a real Banach space, we denote by W_X the class of all functionals possessing the following property: if {u_n} is a sequence in X converging weakly to u∈X and lim inf_n→∞Φ(u_n)≤Φ(u), then {u_n} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C¹ functional, belonging to W_X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X^∗; a C¹ functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C¹ functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation

Φ^′(x)=λJ^′(x)+μΨ^′(x)     

Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form ${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$ , where ${L:X \rightarrow \mathbb R}$ is a sequentially weakly lower semicontinuous C ¹ functional, ${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X → Y, S : X → Z are linear and compact operators and λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 0 such that the functional ${\mathcal{E}_{\lambda^\ast}}$ possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some “energy functional” which satisfies the conditions required in our main result.     

On a three critical points theorem

In this paper, using a recent result by J. Saint Raymond ([6]), we improve the three critical points theorem established in [5].     

A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let (n≥3) be a bounded domain with smooth boundary and let .Then, for each ?>0 small enough, there exists λ_?>0 such that, for every compact interval , there exists ρ>0 with the following property: for every λ∈[a,b] and every continuous function satisfying for some , there exists δ>0 such that, for each ν∈[0,δ], the problem has at least three weak solutions whose norms in are less than ρ.     

Some remarks on a three critical points theorem

Some remarks on a strict minimax inequality, which plays a fundamental role in Ricceri's three critical points theorem, are presented. As a consequence, some recent applications of Ricceri's theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions.     

增算子的不动点和耦合不动点

Under some general continuous and compact conditions, the existence problems of fiked points andd coupled fixed points for increasing operators are studied. an application, we utilize the results obtained to study the existence of solutions for differential inclusions in Banach spaces.     

A non-smooth version of the Lojasiewicz-Simon theorem with applications to non-local phase-field systems

We consider a simple system modelling phase transition phenomena with long term interactions. It is shown that any solution converges with growing time to a single stationary state. To this end, a non-smooth version of the celebrated Simon-Lojasiewicz theorem is proved.     

An infinite-time relaxation theorem for differential inclusions

The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.

     

A general multi-valued hybrid fixed point theorem and perturbed differential inclusions

Combining three basic multi-valued versions of Banach, Schauder and Tarski fixed point theorems, a general hybrid fixed point theorem for multi-valued mappings in Banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed Lipschitz, compactness and monotone conditions.     

A multiplicity theorem for critical points of functionals on reflexive Banach spaces

In this paper we establish a multiplicity theorem for critical points of functionals on reflexive Banach spaces. Precisely, we deduce the main result using a general variational principle proved by Ricceri. Moreover, we present an application to a Neumann problem which gives a positive answer to some questions formulated by the previous author.Received: 6 February 2003     

A fixed point theorem for multi-valued mappings and its applications to integral inclusions

In this paper we prove an existence theorem for the common solutions for a pair of integral inclusions via a common fixed point theorem of Dhage et al. [B.C. Dhage, D. O’Regan, R.P. Agarwal, Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces, J. Appl. Math. Stoch. Anal. 16 (3) (2003) 243–248].