Positive Solutions Of Equation Research Articles

Solutions and stability for p-Laplacian differential problems with mixed type fractional derivatives

Abstract In this note, the main emphasis is to study two kinds of high-order fractional p-Laplacian differential equations with mixed derivatives, which include Caputo type and Riemann–Liouville type fractional derivative. Based on fixed point theorems on the cone, the existence-uniqueness of positive solutions for equations and two iterative schemes to uniformly approximate the unique solutions are discussed theoretically. What’s more, the sufficient conditions for stability of the equations are given. Some exact examples are further provided to verify the analytical results at the end of the article.

Solutions for a Singular Hadamard-Type Fractional Differential Equation by the Spectral Construct Analysis

In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.

Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$

In this paper, we investigate the existence of positive solutions of the following equation \begin{eqnarray} (-\Delta)^s v +\lambda v=f(x) v^{p-1}+h(x)v^{q-1}, x\in R^N,\\ v\in H^s(R^N), \end{eqnarray} where $1\leq q 2s$ and $\lambda>0$ is a parameter. Since the concave and convex nonlinearities are involved, the variational functional of the equation has different properties. Via variational method, we show that the equation admits a positive ground state solution for all $\lambda>0$ strictly larger than a threshold value. Moreover, under certain conditions on $f$ and for sufficiently large $\lambda>0$, we also prove that there are at least $k+1$ ($k$ is a positive integer) positive solutions of the equation.

Nonexistence of Positive Solutions for an Integral Equation Related to the Hardy-Sobolev Inequality

Let ? and s be real numbers satisfying 0<s<?<n. We are concerned with the integral equation $$u(x)=\int_{R^n}\frac{u^p(y)}{|x-y|^{n-\alpha}|y|^s}dy, $$ where $\frac{n-s}{n-\alpha}< p< \alpha^{*}(s)-1$ with $\alpha^{*}(s)=\frac{2(n-s)}{n-\alpha}$ . We prove the nonexistence of positive solutions for the equation and establish the equivalence between the above integral equation and the following partial differential equation $$\begin{aligned} (-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{-s}u^p. \end{aligned}$$

Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

This paper is concerned with the higher order nonlinear neutral delay differential equation[a(t)(x(t)+b(t)x(t-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

Bounded Positive Solutions for a Third Order Discrete Equation

This paper studies the following third order neutral delay discrete equation , where τ, l ∈ ℕ, n0 ∈ ℕ ∪ {0}, , , are real sequences with an ≠ 0 for n ≥ n0, with lim n→∞(n − din) = +∞ for i ∈ {1,2, …, l} and . By using a nonlinear alternative theorem of Leray‐Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

Asymptotics for nonlinear heat equations

We study the Cauchy problem for the nonlinear heat equation { u t − Δ u + u 1 + σ = 0 , x ∈ R n , t > 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ R n , in the sub critical case of σ ∈ ( 0 , 2 n ) . In the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper Escobedo and Kavian (1987) [5] u ( t , x ) = t − 1 σ w 0 ( x t ) + o ( t − 1 σ ) as t → ∞ uniformly with respect to x ∈ R n , where w 0 ( ξ ) is a positive solution of equation − Δ w − ξ 2 ⋅ ∇ w + w 1 + σ = 1 σ w which decays rapidly at infinity: lim | ξ | → ± ∞ | ξ | 2 σ w 0 ( ξ ) = 0 .

Periodicities which preserve and periodicities which destroy boundedness

It is known that every positive solution of the difference equation with positive parameter β > 0 $$ x_{n + 1} = \beta + \frac{{x_{n - 2} }} {{x_n }}, n = 0,1, \ldots $$ is bounded. In this note we study the difference equation $$ x_{n + 1} = \beta _n + \frac{{x_{n - 2} }} {{x_n }}, n = 0,1, \ldots $$ . We show that every positive solution of this equation is bounded when {βn}n=0∞ is a period-2 sequence of positive real numbers, that is, “Period-2 Preserves Boundedness.” We also show that there exist prime period-3m, sequences {βn}n=0∞ of positive real numbers such that the equation has unbounded solutions. That is,“Period-3m Destroys Boundedness.”

Positive solutions for equations and systems with $p$-Laplace like operators

We prove the existence of positive solutions to boundary-value problems of the form \begin{align*} \begin{gathered} (\phi(u'))'+f(t,u)=0,\quad t\in(0,1)\\ \theta(u(0))=\beta \theta(u'(0)), \quad \theta(u(1))=-\delta \theta(u'(1)), \quad \beta,\delta\ge 0, \end{gathered} \end{align*} where $\phi$ and $\theta$ are odd increasing homeomorphisms of the real line. We also prove the existence of positive solutions to related systems. Our approach is via a priori estimates and Leray-Schauder degree.

Positive Solutions of Equations with Nonlinear Causal Mappings

We consider nonlinear scalar equations with causal mappings. These equations include differential, differential-delay, integral, integro-differential, difference and other traditional equations. Conditions that provide the existence of positive solutions are established. The suggested approach enables us to consider various classes of equations from the unified point of view.

Quasidifferentiability and positive solutions of equations for set-valued mappings

Quasidifferentiability and positive solutions of equations for set-valued mappings

On positive solutions of equations with two middle terms

On positive solutions of equations with two middle terms