Coupled fixed point theorems in partially ordered metric spaces and application

In this paper, we prove some coupled fixed point theorems for contractive mappings in partially ordered complete metric spaces under certain conditions to extend and complement the recent fixed point theorems according to Lakshmikantham and Ćirić [V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70(2009), 4341-4349 and Luong and Thuan [N. V. Luong, N. X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011), 983-992]. As an application, we give a result of existence and uniqueness for the solutions of a class of nonlinear integral equations

Common fixed point of generalized weak contractive mappings in partially ordered Gb-metric spaces

In this work, using the concepts of G-metric and b-metric we define a new type of metric which we call Gb-metric. We study some basic properties of such metric. We also prove a common fixed point theorem for six mappings satisfying weakly compatible condition in complete partially ordered Gb-metric spaces. A nontrivial example is presented to verify the e ectiveness and applicability of our main result.http://www.pmf.ni.ac.rs/filomathb201

Multiplicity of positive solutions for quasilinear elliptic p-Laplacian systems

We study the existence and multiplicity of solutions to the elliptic system $$displaylines{ -hbox{div}(|abla u|^{p-2} abla u)+m_1(x)|u|^{p-2}u =lambda g(x,u) quad xin Omega,cr -hbox{div}(|abla v|^{p-2} abla v)+m_2(x)|v|^{p-2}v=mu h(x,v) quad xin Omega,cr |abla u|^{p-2}frac{partial u}{partial n}=f_u(x,u,v),quad |abla v|^{p-2}frac{partial v}{partial n}=f_v(x,u,v), }$$ where $Omegasubset mathbb{{R}}^N$ is a bounded and smooth domain. Using fibering maps and extracting Palais-Smale sequences in the Nehari manifold, we prove the existence of at least two distinct nontrivial nonnegative solutions

Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian

We derive a priori bounds for positive supersolutions of $-\Delta_p u = \rho(x) f(u)$, where p >1 and $\Delta_p$ is the p-Laplace operator, in a smooth bounded domain of $\mathbb{R}^N$ with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem $-\Delta_p u = \lambda f(u)$, with Dirichlet boundary condition, where $f$ is a nondecreasing continuous differentiable function on such that f(0)>0, $f(t) ^{1/(p-1)}$ is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter $\lambda_p^*$. In particular, we consider the nonlinearities $f(u) = e^u$ and $f(u)=(1+u) ^m$ ($m > p-1$) and give explicit estimates on $\lambda_p^*$. As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p-Laplacian that improves obtained results in the recent literature for some range of p and N

Existence of solutions for mixed Volterra-Fredholm integral equations

In this article, we give some results concerning the continuity of the nonlinear Volterra and Fredholm integral operators on the space $L^{1}[0,infty)$. Then by using the concept of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results extend some previous works

Oscillation of solutions to second-order nonlinear differential equations of generalized Euler type

We are concerned with the oscillatory behavior of the solutions of a generalized Euler differential equation where the nonlinearities satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. Some implicit necessary and sufficient conditions and some explicit sufficient conditions are given for all nontrivial solutions of this equation to be oscillatory or nonoscillatory. Also, it is proved that solutions of the equation are all oscillatory or all nonoscillatory and cannot be both