Hyers-Ulam stability for coupled random fixed point theorems and applications to periodic boundary value random problems

In this paper, we prove some existence, uniqueness and Hyers-Ulam stability results for the coupled random fixed point of a pair of contractive type random operators on separable complete metric spaces. The approach is based on a new version of the Perov type fixed point theorem for contractions. Some applications to integral equations and to boundary value problems are also given.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Junta de Andalucí

Global existence and Asymptotic behavior for a system of wave equation in presence of distributed delay term

In this paper, we consider the following viscoelastic coupled wave equation with a delay term: $$\begin{gathered} u_{tt}(x,t)-Lu(x,t)-\int_0^t g_1(t-\sigma)L u(x,\sigma)d\sigma + \mu_{1}u_{t}(x,t) + \int_{\tau_1}^{\tau_2} \mu_2(s)u_{t}(x,t-s)ds + f_1(u,\upsilon)=0, \\ \upsilon_{tt}(x,t) - L\upsilon(x,t) - \int_0^t g_{2}(t-\sigma)L \upsilon(x,\sigma)d\sigma + \mu_3\upsilon_t(x,t) + \int_{\tau_1}^{\tau_2} \mu_4(s)\upsilon_{t}(x,t-s)ds + f_{2}(u,\upsilon)=0, \end{gathered}$$ in a bounded domain. Under appropriate conditions on $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$, we prove global existence result by combining the energy method with the Faedo-Galerkin's procedure. In addition , we focus on asymptotic behavior by using an appropriate Lyapunov functional

Topological method for coupled systems of impulsive stochastic semilinear differential inclusions with fractional Brownian motion

In this paper we prove the existence of mild solutions for a first-order impulsive semilinear stochastic differential inclusion with an infinite-dimensional fractional Brownian motion. We consider the cases in which the right hand side can be either convex or nonconvex-valued. The results are obtained by using two different fixed point theorems for multivalued mappings, more precisely, the technique is based on a multivalued version of Perov’s fixed point theorem and a new version of a nonlinear alternative of Leray–Schauder’s fixed point theorem in generalized Banach spaces.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Transportation inequalities for coupled systems of stochastic delay evolution equations with a fractional Brownian motion

We prove an existence and uniqueness result of mild solution for a system of stochastic semilinear differential equations with fractional Brownian motions and Hurst parameter H < 1/2. Our approach is based on Perov’s fixed point theorem, and we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution

Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion

Some results on the existence and uniqueness of mild solution for a system of semilinear impulsive differential equations with infinite fractional Brownian motions are proved. The approach is based on Perov’s fixed point theorem and a new version of Schaefer’s fixed point theorem in generalized Banach spaces. The relationship between mild and weak solutions and the exponential stability of mild solutions are investigated as well. The abstract theory is illustrated with an example.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Existence and uniquenes results for systems of impulsive functional stochastic differential equations driven by fractional Brownian motion with multiple delay

We present some existence and uniqueness results on impulsive functional differential equations with multiple delay with fractional Brownian motion. Our approach is based on the Perov fixed point theorem and a new version of Schaefer's fixed point in generalized metric and Banach spaces

Global existence and Asymptotic behavior for a system of wave equation in presence of distributed delay term

In this paper, we consider the following viscoelastic coupled wave equation with a delay term: u tt (x, t) − Lu(x, t) − t 0 g 1 (t − σ)Lu(x, σ)dσ + µ 1 u t (x, t) + τ 2 τ 1 µ 2 (s)u t (x, t − s)ds + f 1 (u, υ) = 0, υ tt (x, t) − Lυ(x, t) − t 0 g 2 (t − σ)Lυ(x, σ)dσ + µ 3 υ t (x, t) + τ 2 τ 1 µ 4 (s)υ t (x, t − s)ds + f 2 (u, υ) = 0, in a bounded domain. Under appropriate conditions on µ 1 , µ 2 , µ 3 and µ 4 , we prove global existence result by combining the energy method with the Faedo-Galerkin's procedure. In addition , we focus on asymptotic behavior by using an appropriate Lyapunov functional

Corrigendum to the paper: Existence and stability results for semilinear systems of impulsivestochastic differential equations with fractionalBrownian motion. Stoch. Anal. Appl. 34 (2016), no. 5, 792–834

In this paper we correct an error made in our paper [Blouhi, T.; Caraballo, T.; Ouahab, A. Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion. Stoch. Anal. Appl. 34 (2016), no. 5, 792-834]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown