Existence Result for a Superlinear Fractional Navier Boundary Value Problems

In this paper, we study the following fractional Navier boundary value problem

$$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$

where \(\alpha ,\beta \in (0,1]\) such that \(\alpha +\beta >1\), \(D^{\beta }\) and \(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that \(a+b>0\). The function g is a nonnegative continuous function in \([0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.

     

On the subordinate killed B.M in bounded domains and existence results for nonlinear fractional Dirichlet problems

We take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional Laplacian (-D_|D)^\fraca2{(-\Delta _{|D})^{\frac{\alpha }{2}}} associated to the subordinate killed Brownian motion process Z_a^D{Z_{\alpha }^{D}} in a bounded C ^1,1 domain D. Our arguments are based on potential theory tools on Z_a^D{Z_{\alpha }^{D}} and properties of an appropriate Kato class of functions K _α (D).     

Sur les Solutions D'un Opérateur Différentiel Singulier Semi-Linéaire

This paper deals with the following Dirichlet problem Lu = 1A ( Au – qu = – f ( , u ) on ] 0, [ , u, ( 0 ) = 0, u ( ) = 0, where ] 0, + ], q 0 is continuous on [ 0, [ × ] 0, + [ ] 0, + [ is continuous and A satisfies some appropriate conditions. The main result is the existence and the uniqueness of a strictly positive regular solution of the problem ( ). Moreover, we study the behaviour of this solution in a neighbourhood of . Our approach is based on the use of the Green's function of the homogeneous equation and Schauder's fixed point theorem.     

Perturbation des résolvantes et représentation des mesures excessives

Résumé Etant donnée une résolvante V=(V )_>0 sous-markovienne sur un espace mesurable (X, B) de noyau initial V propre; on étudie alors le balayage des mesures surmédianes au moyen de la résolvante perturbée V de V par une function mesurable positive bornée sur X.Dans le cas où (X, E _v) est un espace de balayage, on montre que toute mesure excessive vérifiant , s'écrit d'une manière unique sous la forme =V où est une mesure positive sur X.     

Perturbation semi-linéaire des résolvantes et des semi-groupes

Résumé La perturbation semi-linéaire des résolvantes et des semi-groupes linéaires, nous donne des résolvantes et des semi-groupes non linéaires. Nous étudions alors les propriétés de ces opérateurs non linéaires et en particulier les fonctions surmédianes et excessives associées.

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We are concerned with nonlinear resolvents and semi-groups. They are obtained by perturbing linear ones. Properties of these nonlinear operators are investigated, particularly supermedian and excessive functions.

Ce travail est soutenu par la fondation nationale pour la recherche scientifique. Projet MA4-89-FST.     

On the Solutions of a Singular Nonlinear Periodic Boundary Value Problem

In this paper, we consider the following singular nonlinear problem