Evolution Problems involving non-stationary Operators between two Banach Spaces I. Existence and Uniqueness Theorems

Synopsis

(for ‘Evolution Problems involving non-stationary Operators between two Banach Spaces I-II)

In this series of two papers the initial-value problem [B(t)u(t)' = A(t)u(t), Bu(0) = y, with A = A(t) and B = B(t) time-varying operators from one Banach space X to another Banach space Y, and y an arbitrary element of Y, is considered. By making use of the theory of B-evolutions and by integrating certain temporally inhomogeneous equations, a unique solution is obtained for any y in Y. The solution is formulated explicitly in terms of a certain solution operator which involves the B(t)-evolution generated by the closed pair >A(t),B(t)< of operators. Certain properties of the solution operator are also studied. The well-known results, obtained by making use of semigroup theory, for the evolution problem [u(t)]' = A(t)u(t), u(0) = u₀, where A is a closed operator in a Banach space with dense domain, may also be derived from our results.     

Second order linear Volterra integrodifferential equations

In this paper we study mild and classical solutions of the second order linear Volterra integrodifferential equation $$(VE^f )\left\{ {\begin{array}{*{20}c} {u''(t) = Au(t) + {\text{ }}\int_0^t {B(t - s)u(s)ds + f(t){\text{ }}for{\text{ }}t \in [0,T]} } \\ {u(0) = x{\text{ }}and{\text{ }}u'(0) = y,} \\ \end{array} } \right.$$ whereA is a closed linear operator whose domainD(A) is not necessarily dense in a Banach spaceX, and {B(t)|t≥0} is a family of bounded linear operators from the Banach space,D(A) endowed with the graph norm intoX. We also give two examples to illustrate the abstract results.     

Oscillatory and Asymptotic Behavior of First Order Functional Differential Equations

In this paper the author discusses the following first order functional differential equations: $x''(t)+[\int_a^b {p(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (1)$ $x''(t)+[\int_a^b {f(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0} \] (2)$ Some sufficient conditions of oscillation and nonoscillation are obtained, and two asymptotic properties and their criteria are given. These criteria are better than those in [1, 2], and can be used to the following equations: $x''(t)+[\sum\limits_{i = 1}^n {{p_i}(t)x[{g_i}(t)] = 0} \] (3)$ $x''(t)+[\sum\limits_{i = 1}^n {{f_i}(t)x[{g_i}(t)] = 0} \] (4)$     

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

Three symmetric positive solutions for second-order nonlocal boundary value problems

Using the Leggett-Williams fixed point theorem,we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u(t)+g(t)f(t,u(t))=0,0相似文献   

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

Existence of one-signed solutions of nonlinear four-point boundary value problems

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$ - u'' + Mu = rg(t)f(u),u(0) = u(\varepsilon ),u(1) = u(1 - \varepsilon ) $$ and $$u'' + Mu = rg(t)f(u),u(0) = u(\varepsilon ),u(1) = u(1 - \varepsilon ) $$ , where ε ∈ (0, 1/2), M ∈ (0,∞) is a constant and r > 0 is a parameter, g ∈ C([0, 1], (0,+∞)), f ∈ C(?,?) with sf(s) > 0 for s ≠ 0. The proof of the main results is based upon bifurcation techniques.     

The Jump Conditions for Second Order Quasilinear Degenerate Parabolic Equations

ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0     

Higher-order energy expansions and spike locations

We consider the following singularly perturbed semilinear elliptic problem: where is a bounded domain in R ^N with smooth boundary , is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional defined by where . Ni and Takagi ([29, 30]) proved that for a single boundary spike solution , the following asymptotic expansion holds: where c ₁ > 0 is a generic constant, is the unique local maximum point of and is the boundary mean curvature function at . In this paper, we obtain a higher-order expansion of where c ₂, c ₃ are generic constants and is the scalar curvature at . In particular c ₃ > 0. Some applications of this expansion are given.Received: 14 January 2003, Accepted: 28 July 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): Primary 35B40, 35B45; Secondary 35J25     

Multiple symmetric positive solutions of fourth-order two point boundary value problem

In this paper, by using the Krasnoselskii fixed point theorem, we study the existence of 2m or 2m + 1 symmetric positive solutions of fourth-order two point boundary value problem $y^{(4)} (t) - f(t,y(t),y''(t)) = 0,y(0) = y(1) = y'(0) = y''(1) = 0$ .     

Existence of positive solutions of second-order periodic boundary value problems with sign-changing Green’s function

In this paper, we consider the existence of positive solutions of second-order periodic boundary value problem

$$u'' + {\left( {\frac{1}{2} + \varepsilon } \right)^2}u = \lambda g\left( t \right)f\left( u \right),t \in \left[ {0,2\pi } \right],u\left( 0 \right) = u\left( {2\pi } \right),u'\left( 0 \right) = u'\left( {2\pi } \right)$$

, where 0 < ε < 1/2, g: [0, 2π] → ? is continuous, f: [0,∞) → ? is continuous and λ > 0 is a parameter.

     

Existence to singular boundary value problems with sign changing nonlinearities using an approximation method approach

This paper studies the existence of solutions to the singular boundary value problem

Isomorphic pairs of homogeneous functions and their morphisms

Summary. In this paper we determine all iseomorphic pairs (isomorphic pairs with monotonic, thus continuous isomorphisms) of continuous, strictly increasing, linearly homogeneous functions defined on cartesian squares I ² and J ² of intervals of positive numbers or on their restrictions or and or We prove that, if the iseomorphy is nontrivial, then each homogeneous function is a (weighted) geometric or power mean or a joint pair of such means. In functional equations terminology this means that all nontrivial continuous strictly increasing linearly homogeneous solutions G, H (with the continuous strictly monotonic F also unknown) of the equation on D _< or D _> are weighted geometric or power means, while on I ² they are joint pairs of weighted geometric means or of weighted power means.     

On the oscillation of the second order neutral delay differential equations

In this paper, we consider the oscillation of the second order neutral delay differential equations[x(t) cx(t-τ)]" p(t)x(t-σ)=0 (1)and obtain some sufficient conditions of the oscillation of (1) for the case c≥0, -1≤c<0 and c<-1.     

Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces

In this paper, we study the well-posedness of the third-order differential equation with finite delay(P_3): αu'"(t) + u"(t) = Au(t) + Bu'(t) + Fut +f(t)(t ∈ T := [0,2π]) with periodic boundary conditions u(0) = u(2π), u'(0) = u"(2π),u"(0)=u"(2π) in periodic Lebesgue-Bochner spaces Lp(T;X) and periodic Besov spaces B_(p,q)~s(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from Lp([-2π, 0]; X)(resp. Bp,qs([-2π, 0]; X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [-2π,0]. Necessary and sufficient conditions for the Lp-well-posedness(resp. B_(p,q)~s-well-posedness)of(P_3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.     

On the Solvability of Impulsive Differential-Algebraic Equations

We establish theorems on the existence and uniqueness of a solution of the impulsive differential-algebraic equation

Asymptotic behavior of the solutions of an integrodifferential system

Summary We consider the system(L): , t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each A_i, B(t) are n × n matrices. System(L) generates a semigroup by means of T_tf(s)=y (t+s, f), f(s) ∈ BC_l(− ∞, 0]. Under some hypotheses concerning the roots ofdet where is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L₁[0, ∞), thendet forRe λ>0 iff for every ɛ>0 there is an M_ɛ>0 such that ∥T_tf∥_l ⩽ ⩽ M_ɛ exp [ɛt]∥f∥_l for t ⩾ 0. Corollary 3.1.1: suppose exp [at]B(t) ∈ ∈ L₁[0, ∞) for some a>0 anddet forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable. Entrata in Redazione il 21 marzo 1975. The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper.     

Non-existence of one-signed solutions for variational equations with locally amplifying potentials

In questo lavoro si considerano equazioni differenziali ordinarie del secondo ordine di tipo singolare della forma

Types and criteria of nonoscillatory solutions for some second order

In this paper we discuss the following NFDE $$[r(t)[x(t) - cx(t - \tau )']' + \smallint _a^b p(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0$$ where τ>0, 1>c≥0, 0≤g(t, ε)≤t,r(t)>0,p(t, ε)>0, and some sufficient and necessary conditions are given, under which there are three types of nonoscillatory solutions for the above equation.     

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.