An extension of Ezeilo's result

Summary In a recent paper[1] Ezeilo considered the nonlinear third order differential equation x‴ + ω(x′)x″ + ω(x)x′ + ϑ(x, x′, x″)=p(t). He proved the ultimate boundedness of the solutions on rather general conditions for the nonlinear terms ϕ, ϕ, ϑ. These conditions (in a little weaker form) are also sufficient in order to prove the existence of forced oscillations in the case when the excitation is ω-periodic. For this purpose the Lerag-Schauder principle in a form suggested by G. Güssefeldt[2] is applicable. Dedicated to ProfessorKarl Klotter on his 70th birthday Entrata in Redazione il 21 ottobre 1971.     

Stability of lurie-type non-linear equations

Summary Conditions are given for the indirect control system x′=a(x)+bμ, μ′=φ(σ), σ=c^Tx−ϱμ, to be absolutely stable. These conditions reduce to LaSalle and Lefschetz's in the linear case: a(x)=Ax. The conditions obtained for the stability of the direct control system x′=a(x)+bφ(σ), σ=c^Tx, reduce also to Lurie's condition in the linear case. The case of the direct control system x′=a(x, t)+bφ(σ), σ=c^Tx is also investigated. Entrata in Redazione il 18 febbraio 1976.     

On the equationx″+f(x)h(x′)x′+g(x)=e(t)

Summary We consider the equation x″+f(x)h(x′)x′+g(x)=e(t) in which f, g, and h are continuous, e is sectionally continuous and absolutely integrable, h(u)>0, xg(x)>0 if x ≠ 0, and f(x)≥0. Necessary and sufficient conditions are given for boundedness of all solutions and their derivatives. When f(0)>(0) we give necessary and sufficient conditions for all solutions and their derivatives to converge to zero. Entrata in Redazione il 14 giugno 1969.     

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

Abstract. In this Paper, the existence and uniqueness of solutions for boundary valueproblem     

EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS

This paper is concerned with the following second-order vector boundary value problem ：x^R=f（t,Sx,x,x＇）,0〈t〈1,x（0）=A,g（x（1）,x＇（1））=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.     

Sviluppo delle funzioni implicitamente definite da un sistema di equazioni nel campo reale

Sunto Per la funzione reale del punto P(x, y): F[ξ(P), η(P)], ove u=ξ(P), v=η(P) sono implicitamente definite nel campo reale dal sistema delle due equazioni u−x+ϕ(u, v)=0, v−y+φ(u, v)=0 si dà uno sviluppo che estende quello ottenuto dalLevi-Civita per la funzione F[y(x)], ove y(x) è definita dalla equazione y−x+ϕ(y)=0. Ad Antonio Signorini nel suo 70^mo compleanno.     

Theoremes de Viabilite Pour les Inclusions Differentielles du Second Ordre

This paper gives conditions ensuring the existence for an initial value (x ₀,v ₀) of a solution to the second order differential inclusionx″(t) ∈F[x(t),x′(t)],x(0)=x ₀,x′(0)=v ₀ such thatx(t) ∈K for allt whereK is a nonempty given subset ofR ⁿ .      

Differential sobordination and Bazilevič functions

LetM(z)=z ⁿ +…,N(z)=z ⁿ +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ. In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.     

A Cauchy problem for the heat equation

Summary Let u(x, t) satisfy the heat equation in 0<x<1, 0<t≤T. Let u(x, 0)=0 for 0<x<1 and let |u(0, t)|<ε, | u_x(0, t) |<ε, and | u(1, t) |<M for 0≤t≤T. Then, , where M₁ and β(x) are given explicitly by simple formulas. The application of the a priori bound to obtain error estimates for a numerical solution of the Cauchy problem for the heat equation with u(x, 0)=h(x), u(0, t)=f(t), and u_x(0, t)=g(t) is discussed. Work performed under the auspices of the U. S. Atomic Energy Commission.     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Homogenization of quasi-linear equations with natural growth terms

In this paper we deal with the limit behaviour of the bounded solutions u_ε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|^p. Under these assumptions on a and H we prove that there exists a function H⁰=H⁰(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (u_ε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|^p-2u = H⁰(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.     

Singular solutions of some nonlinear parabolic equations

We consider the existence and uniqueness of singular solutions for equations of the formu ₁=div(|Du|^p−2 Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2. Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the existence of very singular solutions, i.e. singular solutions such that ∫_|x|≤r u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result. In the case ϕ(u)=u ^q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal. Dedicated to Professor Shmuel Agmon     

On the Hausdorff dimensions of certain attractors

For any a,b∈R let ϕ_a,b(x)=ax+b(x∈R). Suppose 0<a<1. Let C_a,b be the generalized a-Cantor set with generating iterated function systme {ϕ_a,0, ϕ_a,b; ϕ_a,l}. Then we prove the Hausdorff dimension of C_a,c² C_{a,c^2 } is \fracln(3 - ?5 - ln2lna\frac{{ln(3 - \sqrt 5 - ln2}}{{lna}} when 0<a≤2 cos 80°.     

Continuability,boundness and asymptotic behavior of solutions ofx″ +q(t)f(x)=r(t)

Summary In addition to obtaining sufficient conditions for continuability of solutions of x″ + q(t)f(x)=r(t), some sufficient conditions and some necessary and sufficient conditions for boundedness are obtained. The asymptotic behavior of solutions is studied through examination of r(t)/q(t) as t → ∞. Supported by Mississippi State University Biological and Physical Sciences Research Institute. Entrata in Redazione il 6 febbraio 1973.     

Bounded perturbations of forced harmonic oscillators at resonance

Summary Let e be continuous and 2π-periodic, h continuous and bounded, and n>0 an integer. Sufficient conditions for the existence of 2π-periodic solutions of x″+n²x+h(x)= =e(t) are given. The proofs are based on a modification of Cesari's method and the Schauder fixed point theorem. Author is partially supported by N. S. F. under Grant 7447. Entrata in Redazione il 26 agosto 1968.     

On the existence of solutions for nonlinear impulsive periodic viable problems

In this paper we prove the existence of periodic solutions for nonlinear impulsive viable problems monitored by differential inclusions of the type x′(t)∈F(t,x(t))+G(t,x(t)). Our existence theorems extend, in a broad sense, some propositions proved in [10] and improve a result due to Hristova-Bainov in [13].     

Asymptotic stability and boundedness for functional differential equations

We study a system(D)x′=F(t,x _t) of functional differential equations, together with a scalar equation(S)x′=−a(t)f(x)+b(t)g(x(t−h)) as well as perturbed forms. A Liapunov functional is constructed which has a derivative of a nature that has been widely discussed in the literature. On the basis of this example we prove results for (D) on asymptotic stability and equi-boundedness. Supported in part by NSF of China, Key Project # 19331060     

Estimates on the existence region for periodic solutions of equations involving a small parameter. Il: Critical cases

Summary The equation x′=A(t)x+f(t, x, ε) is investigated for periodic solutions in the critical case. Explicit estimates for the existence region in ε for these solutions are obtained by employing implicit function theorem techniques. This paper is based on a Ph. D. thesis rubmitted by the authour to the faculty of the Graduate School of the University of Minnesota. The research was paid for in part by U. S. Army Research Contract No. DA-ARO-D-31-124G737. Entrata in Redazione il 15 febbraio 1971.     

On the boundedness of solutions of nonlinear differential equations in Hilbert spaces

LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t ₀, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.

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Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t ₀, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.

     

非线性二阶微分系统正解的存在性

考虑二阶微分系统边值问题[JB({]x″(t)+λ f(t,x(t),y(t))=0,\=y″(t)+μ g(t,x(t),y(t))=0,\ 00, f, g:［0,1］×［0,∞)×［0,∞)→R连续. 突破了以往文献要求非线性项 f, g非负的限制，运用锥上的一个不动点定理,在半正的情形下建立了问题正解的存在性