Existence Of Unbounded Solutions Research Articles

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯γ,uγ) of the equationF(D2uγ)+λ¯γuγrγ=0inB(0,1)∖{0},uγ=0on∂B(0,1) where uγ>0 in B(0,1)∖{0} and γ>0. We prove existence of radial solutions which are continuous on B(0,1)‾ in the case γ<2, existence of unbounded solutions in the case γ=2 and a non existence result for γ>2. We also give, in the case of Pucci's operators, the explicit value of λ¯2, which generalizes the Hardy–Sobolev constant for the Laplacian.

Closed-form formulas for solutions to a difference equation and existence of unbounded solutions

Closed-form formulas for solutions to a difference equation and existence of unbounded solutions

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapping region around the zero equilibrium.

Periodic pattern repetition for unbounded solutions to finite difference equations

ABSTRACT We study a large class of finite difference equations that exhibit a type of periodic pattern repetition in their solutions for certain choices of initial conditions and prove the existence of unbounded solutions.

Bounded or unbounded solutions of a functional equation with nonautonomous iteration

Motivated by nonautonomous difference equations, we study a functional equation with nonautonomous iteration of order n for bounded solutions and unbounded solutions. We give a condition for the existence of bounded solutions in the case that a given function is bounded and a condition for the existence of unbounded solutions in the case that a given function is unbounded. We also discuss the existence of unbounded solutions in the case that a given function is bounded.

On the asymptotic character of a generalized rational difference equation

We investigate the global asymptotic stability of the solutions of \begin{document}$X_{n+1}=\frac{β X_{n-l} + γ X_{n-k}}{A + X_{n-k}} $\end{document} for \begin{document}$n=1,2, ...$\end{document} , where \begin{document}$l$\end{document} and \begin{document}$k$\end{document} are positive integers such that \begin{document}$l≠ k$\end{document} . The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms ( \begin{document}$l$\end{document} , \begin{document}$k$\end{document} ) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.

On unbounded solutions of singular IVPs withϕ-Laplacian

The paper deals with a singular nonlinear initial value problem with a φ-Laplacian (p(t)φ(u′(t)))′ + p(t)f(φ(u(t))) = 0, t > 0, u(0) = u0 ∈ [L0, L], u′(0) = 0. Here, f is a continuous function with three roots φ(L0) < 0 < φ(L), φ: R → R is an increasing homeomorphism and function p is positive and increasing on (0, ∞). The problem is singular in the sense that p(0) = 0 and 1/p may not be integrable in a neighbourhood of the origin. The goal of this paper is to prove the existence of unbounded solutions. The investigation is held in two different ways according to the Lipschitz continuity of functions φ−1 and f. The case when those functions are not Lipschitz continuous is more involved that the opposite case and it is managed by means of the lower and upper functions method. In both cases, existence criteria for unbounded solutions are derived.

Periodic and unbounded motions in asymmetric oscillators at resonance

We consider the existence of periodic and unbounded solutions for the asymmetric oscillatorx″+ax+−bx−+g(x)=p(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, a and b are two positive constants, p(t) is a 2π-periodic smooth function and g(x) satisfies lim|x|→+∞⁡x−1g(x)=0. We have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions Φp(θ) and Λ(h). In this paper, we consider some critical cases and obtain some new sufficient conditions for the existence of 2π-periodic and unbounded solutions. In particular, the obtained periodic solutions are new and cannot be deduced from Landesman–Lazer's and Fabry–Fonda's existence conditions. In contrast with many existing results, the function g(x) may be unbounded or oscillatory without asymptotic limits.

A Permutation Related to Non-compact Global Attractors for Slowly Non-dissipative Systems

We consider scalar reaction-diffusion equations with non-dissipative nonlinearities generating global semiflows which exhibit blow-up in infinite time. This type of equations was only recently approached and the corresponding dynamical systems are known as slowly non-dissipative systems. The existence of unbounded solutions, referred to as grow-up solutions, requires the introduction of some objects interpreted as equilibria at infinity. By extending known results, we are able to obtain a complete decomposition of the associated non-compact global attractor. The connecting orbit structure is determined based on the Sturm permutation method, which yields a simple criterion for the existence of heteroclinic connections.

Existence of unbounded solutions of a linear homogenous system of differential equations with two delays

Behavior of solutions of a linear homogeneous system of differential equations with deviating arguments in the form \begin{equation*} \dot y(t)=\beta(t)\left[y(t-\delta)-y(t-\tau)\right] \end{equation*} is discussed for $t\to\infty$. It is assumed that $y$ is an $n$-dimensional column vector, $n\geq 1$ is an integer, $\delta,\tau\in{\mathbb{R}}$, $\tau>\delta>0$, and $\beta(t)$ is an $n\times n$ matrix defined for $t\geq t_{0}$, $t_{0}\in\mathbb{R}$, and such that its elements are nonnegative, continuous functions and in every row of this matrix at least one element is nonzero. The existence of solutions in an exponential form under certain assumptions is proved. Sufficient conditions for the existence of unbounded solutions and the estimations for a solution are derived. A comparison with the known results and an illustrative example are given.

Nonlinear Functional Analysis of Boundary Value Problems 2013

and Applied Analysis 3 solve a class of boundary value problems for second-order singular differential equations. (iii) In the paper titled “Existence and uniqueness of solution to nonlinear boundary value problems with sign-changing Green’s function,” by using the cone theory and the Banach contraction mapping principle, some existence and uniqueness results are established for a nonlinear higher-order differential equation boundary value problem with signchanging Green’s function. (iv) In the paper titled “Onboundedness and attractiveness of nonlinear switched delay systems,” the authors study the boundedness and attractiveness problems for a class of nonlinear switched delay systems. Some sufficient conditions are established to guarantee the system’s boundedness. The authors also work out the region where the solution will remain and the relationship between the initial function and the bounded region. (v) In the paper titled “Unbounded solutions of asymmetric oscillator,” the authors establish sufficient conditions for the existence of unbounded solutions of a nonlinear differential equation with a continuous, bounded, and periodic source function. (6) Stochastic Differential Equation Models (i) In the paper titled “A stochastic string with a compound poisson process,” the authors introduce a compound Poisson process with a constant jump intensity and random jump-size to capture information burst and the resulting discontinuous path and derive the no-arbitrage condition on the drift of instantaneous forward rates in the compound model. With the proposedmodel, the impact of random jump on the price of the zero coupon bond is shown in the paper. (ii) In the paper titled “Equilibrium asset and option pricing under jump diffusion model with stochastic volatility,” the authors study the equity premium and option pricing under a jump-diffusion model with stochastic volatility. The pricing kernel is established and the exact expression for the option price is derived by using the Fourier transformation method. (7) Fixed-Point Theory (i) In the paper titled “Generalizations of fixed-point theorems of Altman and Rothe types,” the authors present some extensions of the Altman and Rothe type fixed-point theorems. Some new fixed-point theorems for continuous operators are established by use of the theory of topological degree. (ii) In the paper titled “Commonfixed-points for weak contractive mappings in ordered metric space with applications,” the authors establish some new common fixed-point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Bounded and Unbounded Motions in Asymmetric Oscillators at Resonance

We consider the boundedness and unboundedness of solutions for the asymmetric oscillator $$\begin{aligned} x''+ax^+-bx^-+g(x)=p(t), \end{aligned}$$ where $$x^+=\max \{x,0\},x^-=\max \{-x,0\}, a$$ and $$b$$ are two positive constants, $$ p(t)$$ is a $$2\pi $$ -periodic smooth function and $$g(x)$$ satisfies $$\lim _{|x|\rightarrow +\infty }x^{-1}g(x)=0$$ . We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. It turns out that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions $$\Phi _p(\theta )$$ and $$\Lambda (h)$$ . Some explicit conditions are given for the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function $$g(x)$$ is required to be a bounded function with asymptotic limits, here we allow $$g(x)$$ be unbounded or oscillatory without asymptotic limits.

Upper and Lower Solution Method for Fractional Boundary Value Problems on the Half-Line

We establish the existence of unbounded solutions for nonlinear fractional boundary value problems on the half-line. By the upper and lower solution method technique, sufficient conditions for the existence of solutions for the fractional boundary value problems are established. An example is presented to illustrate our main result.

Unbounded Solutions of Asymmetric Oscillator

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation(φp(x′))+(p−1)[αφp(x+)−βφp(x−)]=(p−1)f(t,x,x′), whereφp(u)=|u|p−2u, p&gt;1, x+=max{x,0}, x−=max{−x,0},α,βare positive constants, andfis continuous, bounded, andT-periodic intfor someT&gt;0.

Unboundedness results for fourth order rational difference equations

We present some new results regarding the boundedness character of the kth order rational difference equation When applied to the general fourth order rational difference equation, these results prove the existence of unbounded solutions for 49 special cases of the fourth order rational difference equation, where the boundedness character has not been established yet. This resolves 49 conjectures posed by E. Camouzis and G. Ladas.

On the nature of solutions of the difference equation $mathbf{x_{n+1}=x_{n}x_{n-3}-1}$

We investigate the long-term behavior of solutions of the difference equation[ x_{n+1}=x_{n}x_{n-3}-1 ,, n=0 ,, 1 ,, ldots ,, ]noindent where the initial conditions $x_{-3} ,, x_{-2} ,, x_{-1} ,, x_{0}$ are real numbers. In particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

Solutions of the Difference Equation xn+1 = xnxn−1 − 1

Our goal in this paper is to investigate the long‐term behavior of solutions of the following difference equation: xn+1 = xnxn−1 − 1, n = 0, 1, 2, …, where the initial conditions x−1 and x0 are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.

Dynamics for Nonlinear Difference Equation xn+1=(αxn−k)/(β+γxn−lp)

We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.

Global Behavior of a Higher Order Difference Equation

In this work we study the asymptotic stability of the nonnegative equilibrium points of the difference equation. We discuss the conditions under which there exist prime period two solutions and semicycles. Finally we investigate the oscillation and the existence of unbounded solutions.

Solutions in weighted spaces of singular boundary value problems on the half-line

In this paper, we establish the existence of an unbounded solution to a second order boundary value problem on the half line. Also the existence of multiple unbounded positive solutions is discussed using the theory of fixed point index.