Sequences of weak solutions for fractional equations

This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of nontrivial weak solutions for them exploiting the ${\mathbb{Z}}_2$-symmetric version of the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. As a particular case, we derive an existence theorem for the fractional Laplacian, finding nontrivial solutions of the equation \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & {\mbox{in}} \Omega\\ u=0 & {\mbox{in}} \erre^n\setminus \Omega. \end{array} \right. As far as we know, all these results are new and represent a fractional version of classical theorems obtained working with Laplacian equations

The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

The final open part of Strauss' conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang in 2006, or Zhou in 2007 independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou in 1995, the lifespan T(\e) of solutions of $u_{tt}-\Delta u=u^2$ in $\R^4\times[0,\infty)$ with the initial data u(x,0)=\e f(x),u_t(x,0)=\e g(x) of a small parameter \e>0, compactly supported smooth functions $f$ and $g$, has an estimate \exp(c\e^{-2})\le T(\e)\le\exp(C\e^{-2}), where $c$ and $C$ are positive constants depending only on $f$ and $g$. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations

Existence of solutions for fourth order three-point boundary value problems on a half-line

WOS: 000365262600001In this paper, we apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory to establish the existence of unbounded solutions for the following fourth order three-point boundary value problem on a half-line x''''(t) + q(t) f(t, x(t), x'(t), x ''(t), x'''(t)) = 0, t is an element of (0, +infinity), x ''(0) = A, x(eta) = B-1, x'(eta) = B-2, x'''(+infinity) = C, where eta is an element of (0, +infinity), but fixed, and f : [0, +infinity) x R-4 -> R satisfies Nagumo's condition. We present easily verifiable sufficient conditions for the existence of at least one solution, and at least three solutions of this problem. We also give two examples to illustrate the importance of our results.Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK)This work was done when the first author was on academic leave, visiting Texas A&M University-Kingsville, Department of Mathematics. He gratefully acknowledges the financial support of The Scientific and Technological Research Council of Turkey (TUBITAK)

Three positive solutions to initial-boundary value problems of nonlinear delay differential equations

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation $$\left\{ \begin{array}{lll} (\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t<1,\\ x_{0}=0,\\ x(1)=0, \end{array}\right.$$ where $\phi: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\mathbb{R})$ defined by $x_{t}(\sigma)=x(t+\sigma)$ for $-\tau \leq \sigma\leq 0.$ By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results

On the structure of attainable sets for generalized differential equations and control systems

AbstractSection 1 deals with the study of properties of the set of solutions of (1) /.x ϵ R(t), x(0) = 0, where the set valued map R is measurable with nonempty compact subsets (of a ball of finite radius in En) as values. This is equivalent to the study of solutions of a linear control system. If Mr ⊂ (L∞[0, T] denotes the set of all measurable selections of R, and for r ϵ MR, (Ir)(t) = ∫0tr(τ)dτ, then I(MR)⊂C[0, T] is the space of all solutions of (1). One type of typical “cost functional” for an associated optimization problem is a continuous map F: C[0, T] → E1. An extension of Aumanns theorem is used, together with the Stone-Weierstrass theorem, to show that the set of F: C[0, T] → E1 such that F(I(MR) is compact is dense in the space of all continuous maps from C[0, T] → E1 with the uniform topology. The implications to optimal control problems are evident.Section 2 deals with the nonlinear problem (2) /.x ϵ R(x), x(0) = 0, where R has values as in Section 1. Using the machinery of Section 1, the existence of solutions of Eq. (2), when R is Lipschitzian, a result of Filippov 1966, is shown to be a trivial consequence of a fixed point theorem for contracting set valued mappings. If R is continuous and convex valued, the fact that the set of solutions in C[0, T] is compact and has compact fixed time cross section (Filippov-Roxin theorem) is also an immediate consequence of, now, the Bohnenblust-Karlin fixed point theorem for set valued maps. The remainder of Section 2 gives an example in which the set of points attainable by solutions of an equation of the form (2), at some time T > 0, is actually open! In fact, in this example R has the control representation R(x) = {ƒ (x, u) : u ϵ U} with U compact and ƒ smooth. To construct this, every point of the boundary of the attainable set of the convexified problem is attainable only as a limit of “chattering solutions” of the original system. This is quite difficult to accomplish (in fact many people conjectured it was not possible)