Evolution equations for abstract differential operators

We study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a Banach space. The results can be applied to many partial differential equations on different function spaces.     

Mild solutions for abstract fractional differential equations

We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a,?k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C ₀-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

On a class of fractional non-selfadjoint operators associated with differential equations

We present a research method for non-selfadjoint integral operators associated with fractional differential equations. With the help of this method we, in particular, estimate eigen-functions and eigenvalues of the boundary-value problem for a fractional oscillatory equation.     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

Global attractivity for nonlinear fractional differential equations

We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

Positive solutions for nonlinear fractional differential equations

Positivity - We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation $$\begin{aligned} \left\{ \begin{array}{l} ^{C}D^{\alpha }x\left( t\right)...     

Extremal solutions of Cauchy problems for abstract fractional differential equations

In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L ¹-Lipschitz-like, L ¹-Carathéodory-like and L ¹-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.     

Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations

In this paper we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results of mixed monotone operators on cones.     

On abstract degenerate neutral differential equations

We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.     

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula AE, B(x)0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) H₁ according to the formula: L₀y=–y + Q(x)y (–0 is a self-adjoint operator in H₁ under the given assumptions.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 65–72, July, 1969.     

On bounded solutions of abstract differential equations

Summary We construct nontrivial bounded solutions of an abstract evolution equationu'(t)=Au(t) (—∞<t<∞) whereA generates a (C ₀) semigroup of operators {T _t;t≥0} such thatT _t converges strongly to zero ast→∞.

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Riassunto Si costruiscono soluzioni limitate non triviali di una equazioneu'(t)=Au(t) (—∞<t<∞) doveA è il generatore di un semigruppo di operatori {T _t;t≥0} tale cheT _t converge fortemente verso 0 pert→∞.

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Supported by National Science Foundation grant GP-12722.