# Publications of Morosanu, Gheorghe

Asymptotically periodic solutions to some second order evolution and difference equations

In this paper we examine the asymptotic periodicity of the solutions of some second-order differential inclusions on [0,∞)associated with maximal monotone operators in a Hilbert space H, whose forcing terms are periodic functions perturbed by functions from L^1(0,∞; H; tdt). It is worth pointing out that strong solutions do not exist in general, so we need to consider weak solutions for this class of evolution inclusions. Similar second-order difference inclusions are also addressed. Our main results on asymptotic periodicity represent signiﬁcant extensions of the previous theorems proved by R.E. Bruck (1980) and B. Djafari Rouhani (2012).

Existence for second order difference inclusions on R+ governed by monotone operators

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u"(t) + q(t)u'(t) ∈ Au(t) + f (t) for a.a. t ∈ R+ = [0,∞), subject to the condition u(0) = x ∈ Cl {D(A)}, where A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p, q ∈ L∞(R+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0. Eq. (E) has been investigated under various assumptions by Barbu [1]-[2], Brezis [4], V´eron [9]-[10] and others. Recently (see Morosanu [8]), we proved existence and uniqueness of a strong solution to Eq. (E) subject to u(0) = x ∈ D(A) in the weighted space X = L^2_b(R+; H), where b(t) = a(t)/p(t), a(t) = exp(\int_0^t q(s)/p(s) ds), under our weak conditions on p and q (see above). Here we extend this result to the general case x ∈ Cl {D(A)}, while keeping the other assumptions unchanged, thus solving a long standing problem. In addition, the proof of our recent result, which is the starting point of the present paper, is considerably simplified. Furthermore, some qualitative properties of solutions are pointed out, an application to a minimization problem is given, and some open problems are formulated.

Strong and weak solutions to second order differential inclusions governed by monotone operators

In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u″(t) ∈ Au(t) + f(t), where A is a maximal monotone operator in a Hilbert space H. We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf(t) ∈ L^ 1(0, ∞; H), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f(t) ∈ u′(t) + Au(t) which has already been well developed.

Existence results for second-order monotone differential inclusions on the positive half-line

Consider in a real Hilbert space H the differential equation (inclusion) (E ): p(t)u"(t)+q(t)u′(t)∈Au(t)+f(t) a.e. in (0,∞), with the condition (B): u(0)∈ Cl {D(A)}, where A:D(A)⊂H→H is a (possibly set-valued) maximal monotone operator whose range contains 0; p,q∈L^∞(0,∞), with ess sup p >0, and q+∈L^1(0,∞). More than four decades ago, V. Barbu established the existence of a unique bounded solution to (E ), (B), in the particular case p≡1, q≡0 and f≡0. Subsequently the existence and uniqueness of bounded solutions in the homogeneous case (f≡0) have been further investigated by H. Brezis (1972), N. Pavel (1976), L. Véron (1974–1976), and by E.I. Poffald and S. Reich (1986) when A is an m-accretive operator in a Banach space. The non-homogeneous case has received less attention from this point of view. In this paper, we prove existence and uniqueness of bounded solutions to (E), (B) in the general case of non-constant functions p, q satisfying the mild conditions above, thus compensating for the lack of existence theory for such kind of second order problems. Note that our results open up the possibility to apply Lions' method of artificial viscosity towards approximating the solutions of some nonlinear parabolic and hyperbolic problems, as shown in the last section of the paper.

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space $H$ the Cauchy problem $(P_{0})\colon u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0\leq t \leq T; \, u(0)=u_{0}$, where $-A$ is the infinitesimal generator of a $C_0$-semigroup of contractions, $B$ is a nonlinear monotone operator, and $f$ is a given $H$-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem $(P_{0})$ the following regularization: $(P_{\varepsilon})\colon -\varepsilon u^{\prime \prime}(t)+u^{\prime}(t)+Au(t)+Bu(t)=f(t), \, 0\leq t \leq T; \,u(0)=u_{0}, \, u^{\prime}(T)=u_{T},$ where $\varepsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem $(P_{\varepsilon})$. Then we establish asymptotic expansions of order zero, and of order one, for the solution of $(P_{\varepsilon})$. Problem $(P_{\varepsilon})$ turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of $C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of $L^{2}(0,T;H)$.

A multiplicity result for an elliptic anisotropic differential inclusion involving variable exponents

In this paper we are concerned with the study of a class of quasilinear elliptic differential inclusions involving the anisotropic $\overrightarrow{p}(\cdot)$-Laplace operator, on a bounded open subset of $IR^n$ which has a smooth boundary. The abstract framework required to study this kind of dierential inclusions lies at the interface of three important branches in analysis: nonsmooth analysis, the variable exponent Lebesgue-Sobolev spaces theory and the anisotropic Sobolev spaces theory. Using the concept of nonsmooth critical point we are able to prove that our problem admits at least two non-trivial weak solutions.

An existence result for a nonhomogeneous problem in R^2 related to nonlinear Hencky-type materials

This paper investigates a nonlinear and non-homogeneous system of partial differential equations. The motivation comes from the fact that in a particular case the problem discussed here can be used in modeling the behavior of nonlinear Hencky-type materials. The main result of the paper establishes the existence of a nontrivial solution in an adequate functional space of Orlicz-Sobolev type by using Schauder’s fixed point theorem combined with adequate variational techniques.

Strong convergence of the method of alternating resolvents

Abstract. In this paper, we present a generalization of the method of alter- nating resolvents introduced in the previous authors' paper [On the method of alternating resolvents, Nonlinear Anal. 74 (2011), 5147-5160]. It is shown that the sequence generated by this method converges strongly under weaker condi- tions on the control parameters. Concerning the error sequences, many more conditions are used here as compared to the above quoted paper.

A dimension-depending multiplicity result for a perturbed Schrödinger equation

We consider the Schrodinger equation $$ \Delta u + V (x)u = \lambda K(x)f(u) + \mu L(x)g(u) \mbox{ in } R^N; \ u\in H^1(R^N), \eqno{(P)} $$ where $N\ge 2$, $\lambda , \mu \ge 0$ are parameters, $V,K,L : R^N\rightarrow R$ are radially symmetric potentials, $f : R\rightarrow R$ is a continuous function with sublinear growth at infinity, and $g : R\rightarrow R$ is a continuous sub-critical function. We first prove that for $\lambda $ small enough no non-zero solution exists for $(P)$, while for $\lambda $ large and $\mu $ small enough at least two distinct non-zero radially symmetric solutions do exist for $(P)$. By exploiting a Ricceri-type three-critical points theorem, the principle of symmetric criticality and a group-theoretical approach, the existence of at least $N-3$ ($N$ mod 2) distinct pairs of non-zero solutions is guaranteed for $(P)$ whenever $\lambda $ is large and $\mu $ is small enough, $N\neq 3$, and $f, g$ are odd.

Multi parameter proximal point algorithms

The aim of this paper is to prove a strong convergence result for an algorithm introduced by Y. Yao and M. A. Noor in 2008 under a new condition on one of the parameters involved. Further, convergence properties of a generalized proximal point algorithm which was introduced in [5] are analyzed. The results in this paper are proved under the general condition that errors tend to zero in norm. These results extend and improve several previous results on the regularization method and the proximal point algorithm.

A generalization of the regularization proximal point method

This paper deals with the generalized regularization proximal point method which was introduced by the authors in [Four parameter proximal point algorithms, Nonlinear Anal. 74 (2011), 544-555]. It is shown that sequences generated by it converge strongly under minimal assumptions on the control parameters involved. Thus the main result of this paper unifies several results related to the prox-Tikhonov method, the contraction proximal point algorithm and/or the regularization method as well as some results of the above quoted paper.

The method of alternating resolvents revisited

The purpose of this article is to prove a strong convergence result associated with a generalization of the method of alternating resolvents introduced by the authors in "Strong convergence of the method of alternating resolvents" [4] under minimal assumptions on the control parameters involved. Thus, this article represents a significant improvement of the article mentioned above.

A contraction proximal point algorithm with two monotone operators

It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. For the case when the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature.

Elliptic-like regularization of semilinear evolution equations

Consider in a real Hilbert space the Cauchy problem (P0): u′(t)+Au(t)+Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, where −A is the generator of a C_0-semigroup of linear contractions and B is a smooth nonlinear operator. We associate with (P_0) the following problem: (Pε): −εu′′(t) + u′(t) + Au(t) + Bu(t) = f (t), 0 ≤ t ≤ T ; u(0) = u_0, u(T ) = u_1, where ε > 0 is a small parameter. Existence, uniqueness and higher regularity for both (P0) and (Pε) are investigated and an asymptotic expansion for the solution of problem (Pε) is established, showing the presence of a boundary layer near t = T .

Four parameter proximal point algorithms

Several strong convergence results involving two distinct fourparameterproximalpointalgorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximalpointalgorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.

Eigenvalues of the Laplace operator with nonlinear boundary conditions

An eigenvalue problem on a bounded domain for the Laplacian with a nonlinear Robin-like boundary condition is investigated. We prove the existence, isolation and simplicity of the ﬁrst two eigenvalues.

Inexact Halpern-type proximal point algorithm

We present several strong convergence results for the modified, Halpern-type, proximal point algorithm x_{n+1}=a_nu+(1−a_n)J_{b_n}x_n+e_n (n = 0,1, . . .; u, x_0 given, and J_{b_n}=((I+b_nA)^{−1}, for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on a_n and b_n. These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.

On the method of alternating resolvents

The work of H. Hundal (Nonlinear Anal. 57 (2004), 35-61) has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. This paper seeks to design strongly convergent algorithms by means of alternating the resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeroes of A and B. In particular, methods of alternating projections which generate sequences that converge strongly are obtained. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals under less restrictive conditions on the regularization parameters involved.

Equations involving a variable exponent Grushin-type operator

In this paper we define a Grushin-type operator with a variable exponent growth and establish existence results for an equation involving such an operator. A suitable function space setting is introduced. Regarding the tools used in proving the existence of solutions for the equation analysed here, they rely on the critical point theory combined with adequate variational techniques.

Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions

We study a boundary value problem of the type in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in (N≥ 3) with smooth boundary and the functions are of the type with , (i = 1, …, N). Combining the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle we show that under suitable conditions the problem has two non-trivial weak solutions.

A proximal point algorithm converging strongly for general errors

In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14:877–898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used sofar for different versions of PPA. Thus this Note provides a lutiontoalongstandingopenproblemandinparticularoffersnewpossibilitiestowards the approximation of the minimum points of convex functionals.

On a class of boundary value problems involving the p-biharmonic operator

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p > 1. It describes various problems in the theory of elasticity, e.g., the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p − 1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular boundary conditions which are frequently considered in the literature.

Eigenvalue problems for anisotropic elliptic equations: an Orlicz-Sobolev space setting

The paper studies a class of anisotropic eigenvalue problems involving an elliptic, nonhomogeneous di®erential operator on a bounded domain from RN with smooth boundary. Some results regarding the existence or non-existence of eigenvalues are obtained. In each case the competition between the growth rates of the anisotropic coe±cients plays an essential role in the description of the set of eigenvalues.

New competition phenomena in Dirichlet problems

We study the multiplicity of nonnegative solutions to the problem, (Pλ) where Ω is a smooth bounded domain in RN, f:[0,∞)→R oscillates near the origin or at infinity, and p>0, λ∈R. While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving up may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem (Pλ) is strongly influenced by up and depends on λ whenever one of the following two cases holds: •p⩽1 and f oscillates near the origin; •p⩾1 and f oscillates at infinity (p may be critical or even supercritical). The coefficient a∈L∞(Ω) is allowed to change its sign, while its size is relevant only for the threshold value p=1 when the behaviour of f(s)/s plays a crucial role in both cases. Various - and L∞-norm estimates of solutions are also given.

A necessary and sufficient condition for input identifiability for linear time-invariant systems

A necessary and sufficient condition for input identifiability for linear autonomous systems is given. The result is based on a finite iterative process and its proof relies on elementary arguments involving matrices, finite dimensional linear spaces, Gronwall’s lemma, and linear differential systems. Our condition is equivalent to the classical condition involving the geometrical concept of controlled invariant [V. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, NJ, 1992, p. 237] and the dimension reduction algorithm that we propose seems to be useful in designing deconvolution methods.

Multiplicity results for double eigenvalue problems involving the p-Laplacian

The existence of multiple nontrivial solutions for two types of double eigenvalue problems involving the p-Laplacian is derived. To prove the existence of at least two nontrivial solutions we use a Ricceri-type three critical point result for non-smooth functions of S. Marano and D. Motreanu \cite{MarMot}. The existence of at least three nontrivial solutions is shown by combining a result of B. Ricceri \cite{Ricceri} and a Pucci-Serrin mountain pass type theorem of S. Marano and D. Motreanu \cite{MarMot}.

Eigenvalue problems in anisotropic Orlicz–Sobolev spaces

We establish sufficient conditions for the existence of solutions to a class of nonlinear eigenvalueproblems involving nonhomogeneous differential operators inOrlicz–Sobolevspaces. To cite this article: M. Mihăilescu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

An extension of the Jordan-von Neumann theorem

The purpose of this Note is to present an extension of the classical Jordan-von Neumann (JN) theorem [3] - which is recalled below - to the case of a normed space over the skew field IH of quaternions. It is known that this extension is valid in a more general framework (see [5], [6], [7], [8]), but our approach is based on elementary arguments only. So, this result may be of interest for students, applied researchers, etc. We think this extension could be applied to control theory, mechanics and other areas.