The fourth order periodic boundary value problem $u^{\left(4\right)}-m⁴u+F(t,u)=0$, 0 < t < 2π, with $u^{\left(i\right)}\left(0\right)=u^{\left(i\right)}\left(2\pi \right)$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $\pm {10}^{-7}$.

Let $a₁,...,a_k$ be arbitrary nonzero real numbers. An $(a₁,...,a_k)$-decomposition of a function f:ℝ → ℝ is a sum $f₁+\cdots +f_k=f$ where $f_i:ℝ\to ℝ$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h₁+\cdots +h_k=0$ with $h_i:ℝ\to ℝa_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a₁,...,a_k)$-decomposition is essentially unique. We characterize those periods for which essential...

The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2n+1)}=f(t,u,u^{\text{'}},...,u^{\left(2n\right)})$, where $f:ℝ\times {ℝ}^{2n+1}\to ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f(t,x₀,x₁,...,x_{2n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].

Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu’)(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu’)(0) = (Mu’)(2π), in Lebesgue-Bochner spaces $L^p(,X)$, periodic Besov spaces $B_{p,q}^s(,X)$ and periodic Triebel-Lizorkin spaces $F_{p,q}^s(,X)$, where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M)....

The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay $$x^{\text{'}}\left(t\right)=-{\int }_{t-\tau \left(t\right)}^{t}a(t,s)g\left(x\left(s\right)\right)ds+\frac{d}{dt}Q(t,x(t-\tau \left(t\right)))+G(t,x\left(t\right),x(t-\tau \left(t\right))).$$ We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $\tau$, $g$, $a$, $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of...

We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation $\left(\right|u^{\text{'}}{|}^{p-2}{u}^{\text{'}}{\left)\right)}^{\text{'}}+f\left(u\right)u^{\text{'}}+g(x,u)=t$, when $f$ is arbitrary and $g(x,u)\to +\infty$ or $g(x,u)\to -\infty$ when $\left|u\right|\to \infty$. The proof uses upper and lower solutions and the Leray–Schauder degree.

By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t\ne t_j$, j ∈ ℤ, ⎨ ⎩$y\left(t{⁺}_j\right)=y\left(t{¯}_j\right)+I_j\left(y\left(t_j\right)\right)$, where $A\left(t\right)={\left(a_{ij}\left(t\right)\right)}_{n\times n}$ is a nonsingular matrix with continuous real-valued entries.

In this paper we study existence and uniqueness of solutions for the Hammerstein equation $$u\left(x\right)=v\left(x\right)+\lambda {\int }_{I_a^b}K(x,y)f(y,u\left(y\right))dy$$ in the space of function of bounded total $\varphi$-variation in the sense of Hardy-Vitali-Tonelli, where $\lambda \in ℝ$, $K:I_a^b\times I_a^b⟶ℝ$ and $f:I_a^b\times ℝ⟶ℝ$ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family $F^t:t\ge 0$ of continuous linear set-valued functions $F^t:K\to cc\left(K\right)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $\Phi (t,x)=F^t\left(x\right)$ is a solution of the problem $D_t\Phi (t,x)=\Phi (t,G\left(x\right)):=\bigcup \Phi (t,y):y\in G\left(x\right)$, Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_t\Phi (t,x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G\left(x\right):=li{m}_{s\to 0+}(F^s\left(x\right)-x)/s$ for x ∈ K.

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_j$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,\nu }$ be the space of piecewise linear continuous functions on the torus with knots $t_j:0\le j\le N-1$. Finally, let $P_{n,\nu }$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,\nu }$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical...

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain...

Let ${}_{\infty }\left(B_X\right)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let ${}_u\left(B_X\right)$ be the subspace of ${}_{\infty }\left(B_X\right)$ of those functions which are uniformly continuous on $B_X$. A subset $B\subset B_X$ is a boundary for ${}_{\infty }\left(B_X\right)$ if $\parallel f\parallel =su{p}_{x\in B}\left|f\left(x\right)\right|$ for every $f{\in }_{\infty }\left(B_X\right)$. We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for ${}_{\infty }\left(B_X\right)$. On the other hand, for X = , the Schreier...

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $\cdots \&lt;{\lambda }_k^{-}\le {\lambda }_k^{+}\&lt;{\lambda }_{k+1}^{-}\le \cdots$ of a Zakharov-Shabat operator is symmetric,. ${\lambda }_k^{\pm }=-{\lambda }_{-k}^{\mp }$ for all $k$, if and only if the sequence ${\left({\gamma }_k\right)}_{k\in \mathbb{Z}}$ of gap lengths, ${\gamma }_k:={\lambda }_k^{+}-{\lambda }_k^{-}$, is symmetric with respect to $k=0$.

We consider functionals of a potential energy ${\mathbb{P}}_{\psi }\left(u\right)$ corresponding to $\mathrm{𝑎𝑛}\mathrm{𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}-\mathrm{𝑣𝑎𝑙𝑢𝑒}\mathrm{𝑝𝑟𝑜𝑏𝑙𝑒𝑚}$. We are dealing with $𝑎\mathrm{𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛}\mathrm{𝑜𝑓}𝑎\mathrm{𝑡ℎ𝑖𝑛}\mathrm{𝑎𝑛𝑛𝑢𝑙𝑎𝑟}\mathrm{𝑝𝑙𝑎𝑡𝑒}$ with $\mathrm{𝑁𝑒𝑢𝑚𝑎𝑛𝑛}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}\mathrm{𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠}$. Various types of the subsoil of the plate are described by various types of the $\mathrm{𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒}$ nonlinear term $\psi \left(u\right)$. The aim of the paper is to find a suitable computational algorithm.

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...