Controllability of impulsive neutral stochastic integro-differential systems driven by a Rosenblatt process with unbounded delay

In this paper, the controllability of a class of impulsive neutral stochastic integro-differential systems with infinite delay driven by Rosenblatt process in a separable Hilbert space is studied. The controllability result is obtained by using stochastic analysis and a fixed-point strategy. A practical example is provided to illustrate the viability of the abstract result of this work.

Nonlinear Volterra delay evolution inclusions subjected to nonlocal initial conditions

This paper deals with a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition. The evolution inclusion involves an $m$-dissipative operator (possibly multivalued and/or nonlinear) and a noncompact interval. We first consider the evolution inclusion subjected to a local initial condition and prove an existence result for bounded $C^0$-solutions. Then, using a fixed point theorem for upper semicontinuous multifunctions with contractible values, we obtain a global solvability result for the original problem. Finally, we present an example to illustrate the abstract result.

Application of a fixed point theorem on infinite cartesian product to an infinite system of differential equations

"In the paper Operators on infinite dimensional cartesian product, (Analele Univ. Vest Timişoara, Mat. Inform., 48 (2010), 253–263), by I. A. Rus and M. A. Şerban, the authors give a generalization of the Fibre contraction theorem on infinite dimensional cartesian product. In this paper we give an application of this abstract result to an infinite system of differential equations. "

Functional differential equations with maxima, via step by step contraction principle

T. A. Burton presented in some examples of integral equations a notion of progressive contractions on C([a,\infty \lbrack). In 2019, I. A. Rus formalized this notion (I. A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3 (2019) No. 3, 111–120), put “step by step” instead of “progressive” in this notion, and give some variant of step by step contraction principle in the case of operators with Volterra property on C([a,b],\mathbb{B)} and C([a,\infty \lbrack,\mathbb{B}) where \mathbb{B} is a Banach space. In this paper we use the abstract result given by I. A. Rus, to study some classes of functional differential equations with maxima.

Quasilinear Dirichlet Problems with Degenerated p-Laplacian and Convection Term

The paper develops a sub-supersolution approach for quasilinear elliptic equations driven by degenerated p-Laplacian and containing a convection term. The presence of the degenerated operator forces a substantial change to the functional setting of previous works. The existence and location of solutions through a sub-supersolution is established. The abstract result is applied to find nontrivial, nonnegative and bounded solutions.

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

<p style='text-indent:20px;'>We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset <inline-formula><tex-math id="M1">\begin{document}$ \Lambda^* $\end{document}</tex-math></inline-formula> of the parameterized space such that the binary map <inline-formula><tex-math id="M2">\begin{document}$ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $\end{document}</tex-math></inline-formula> is continuous at all points of <inline-formula><tex-math id="M3">\begin{document}$ \Lambda^*\times \mathbb{R} $\end{document}</tex-math></inline-formula> with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space <inline-formula><tex-math id="M4">\begin{document}$ (0, \infty] $\end{document}</tex-math></inline-formula> and the infinity of noise means that the equation is deterministic.</p>

Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.

Monotone iterative method for fractional p-Laplacian differential equations with four-point boundary conditions

A four-point boundary problem for a fractional p-Laplacian differential equation is studied. The existence of two positive solutions is established by means of the monotone iterative method. An example supporting the abstract result is given.

Semiflow Selection to Models of General Compressible Viscous Fluids

We prove the existence of a semiflow selection with range the space of càglàd, i.e. left-continuous and having right-hand limits functions defined on $$[0,\infty )$$ [ 0 , ∞ ) and taking values in a Hilbert space. Afterwards, we apply this abstract result to the system arising from a compressible viscous fluid with a barotropic pressure of the type $$a\varrho ^{\gamma }, \gamma \ge 1$$ a ϱ γ , γ ≥ 1 , with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.

On the solvability of a nonlinear functional integral equations via measure of noncompactness in

Using the technique of a suitable measure of non-compactness and the Darbo fixed point theorem, we investigate the existence of a nonlinear functional integral equation of Urysohn type in the space of Lebesgue integrable functions Lp(RN). In this space, we show that our functional-integral equation has at least one solution. Finally, an example is also discussed to indicate the natural realizations of our abstract result.