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This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results

Abstract In this paper, we consider the oscillation theory for fractional differential equations. We obtain oscillation criteria for three classes of fractional differential equations of the forms Tαt0x(t)+∑i=1mpi(t)x(τi(t))=0,t⩾t0,Tαt0(r(t)(Tαt0(x(t)+p(t)x(τ(t))))β)+q(t)xβ(σ(t))=0,t⩾t0, $$\begin{aligned}& T_{\alpha}^{t_{0}} x(t)+\sum_{i=1}^{m}p_{i}(t)x \bigl(\tau_{i}(t)\bigr)=0,\quad t\geqslant t_{0}, \\& T_{\alpha}^{t_{0}} \bigl(r(t) \bigl(T_{\alpha}^{t_{0}} \bigl(x(t)+p(t)x\bigl(\tau(t)\bigr)\bigr)\bigr)^{\beta}\bigr)+q(t)x^{\beta}\bigl(\sigma(t)\bigr)=0, \quad t\geqslant t_{0}, \end{aligned}$$ and Tαt0(r2Tαt0(r1(Tαt0y)β))(t)+p(t)(Tαt0y(t))β+q(t)f(y(g(t)))=0,t⩾t0, $$T_{\alpha}^{t_{0}}\bigl(r_{2}T_{\alpha}^{t_{0}} \bigl(r_{1}\bigl(T_{\alpha}^{t_{0}} y \bigr)^{\beta}\bigr)\bigr) (t)+p(t) \bigl(T_{\alpha}^{t_{0}} y(t)\bigr)^{\beta}+q(t)f\bigl(y\bigl(g(t)\bigr)\bigr)=0, \quad t \geqslant t_{0}, $$ where Tα $T_{\alpha}$ denotes the conformable differential operator of order α, 0

We investigated remediation of phenol from water using microbe–plant partnerships. Co-introduction of maize seedlings, Pseudomonas fluorescens rifampicin-resistant P13 and P. stutzeri P7 carrying self-transmissible TOL-like plasmids reduced phenol content in water at lower phenol concentrations (25, 50, and 75 mg/L), similar to individual introduction of the bacteria. Co-introduction of plants and bacteria significantly reduced phenol content in water at higher phenol concentrations (100, 125, and 150 mg/L) compared to using individual introduction of the bacteria. Moreover, TOL-like plasmids were transferred from P7 to P13. Addition of plants promoted the growth of both strains, leading to increased plasmid transfer. At higher phenol concentrations, addition of plants resulted in increases of catechol 2, 3-dioxygenase (C23O) activity and reduction in level of reactive oxygen species (ROS) of bacteria in the degradation experiments. Increased plasmid transfer and C23O activity and reduction in ROS level might be the major reasons why plants promote bacterial degradation of phenol at higher phenol concentrations. Furthermore, root exudate of maize seedlings and artificial root exudate (ARE) constructed using major components of the root exudate had the same effects on bacterial activities. Unlike the ARE, deletion of glucose, arabinose, or fructose or all the monosaccharides from ARE resulted in no increase in numbers of both strains and in plasmid transfer. At the higher phenol concentrations, deletion of glutamic acid, aspartic acid, alanine, or glycine or all the amino acids did not stimulate bacterial C23O activity. Deletion of fumaric, oxaloacetic or citric acids still reduced bacterial ROS level as ARE did, but, deletion of all the organic acids or DIMBOA, a hydroxamic acid, did not reduce bacterial ROS level as ARE did. The data showed that each monosaccharide might be important for sufficient numbers of plant-associated bacteria and increased plasmid transfer while each amino acid might be important for maintaining bacterial C23O activity and that DIMBOA might be responsible for the decrease in ROS levels. These results are the basis for efficient remediation of phenol from water by microbe–plant partnerships and further studies on the mechanism of rhizobacterium-plant interaction. Keywords: Phenol, Microbe–plant remediation, Root exudate, C23O activity, ROS level

Abstract In this paper, we investigate the existence and uniqueness of solutions for mixed fractional q-difference boundary value problems involving the Riemann–Liouville and the Caputo fractional derivative. By using the Guo–Krasnoselskii fixed point theorem and Banach contraction mapping principle as well as Schaefer’s fixed point theorem, we obtain the main results. In addition, several examples are given to illustrate the main results.

Abstract In this paper, we establish some Lyapunov-type inequalities for a class of linear and nonlinear fractional q-difference boundary value problems under Cauchy boundary conditions. As applications, we use the inequality to obtain an interval, where Mittag-Leffler function has no real zeros. In addition, we also derive nonexistence results for fractional q-difference boundary value problem.

Abstract In this paper, we consider the boundary value problem of a class of nonlinear fractional q-difference equations involving the Riemann–Liouville fractional q-derivative on the half-line. By means of Schauder fixed point theorem and Leggett–Williams fixed point theorem, some results on existence and multiplicity of solutions are obtained.

Abstract The objective of this paper is to offer sufficient conditions for the oscillation of all solutions of the third order nonlinear damped dynamic equation with mixed arguments of the form (r2(r1(yΔ)α)Δ)Δ(t)+p(t)ψ(t,yΔ(a(t)))+q(t)f(t,y(g(t)))=0 $$\bigl(r_{2}\bigl(r_{1}\bigl(y^{\Delta}\bigr)^{\alpha}\bigr)^{\Delta}\bigr)^{\Delta}(t)+p(t)\psi \bigl(t,y^{\Delta}\bigl(a(t)\bigr)\bigr)+q(t)f\bigl(t,y\bigl(g(t)\bigr) \bigr)=0 $$ on time scales, where a(t)≥t $a(t)\geq t$ and g(t)≤t $g(t)\leq t$. Using Riccati transformation, integral averaging technique, and comparison theorem, we give some new criteria for the oscillation of the studied equation. Our results essentially improve and complement the earlier ones.

Abstract In this paper, we investigate the existence of solutions for a coupled system of fractional compartmental models as differential inclusions with coupled nonlocal and integral boundary conditions, whose multivalued terms depend on lower-order fractional derivatives. By means of nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems together with Leray-Schauder degree theory, some sufficient conditions for the existence of solutions are presented, which extend some known results. Several examples are given to demonstrate the application of our main results.

In this article, we consider the following boundary-value problem of nonlinear fractional differential equation with $p$-Laplacian operator $$displaylines{ D_{0+}^eta(phi_p(D_{0+}^alpha u(t)))+a(t)f(u)=0, quad 01$, $phi_p^{-1}=phi_q$, $1/p+1/q=1$, $0leqslantgamma0$ is a parameter, $a:(0,1)o [0,+infty)$ and $f:[0,+infty)o[0,+infty)$ are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameter $lambda$ are obtained. The uniqueness of positive solution on the parameter $lambda$ is also studied. Some examples are presented to illustrate the main results.

In this paper, we consider the oscillation criteria for even order nonlinear neutral differential equations of the form $$ \left(r(t)z^{(n-1)}(t)\right)'+q(t)f(x(\sigma(t)))=0, $$ where $z(t)=x(t)+p(t)x(\tau(t)),$ $n\geq2$ is a even integer. The results are obtained both for the case $\int^\infty r^{-1}(t){\rm d}t=\infty,$ and in case $\int^\infty r^{-1}(t){\rm d}t

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term $$ x^{\Delta\Delta}(t)+a(t)f(x(q(t)))=e(t) $$ on a time scale $\mathbb{T}.$ Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.

We study boundary value problems for the following nonlinear fractional Sturm-Liouville functional differential equations involving the Caputo fractional derivative: CDβ(p(t)CDαu(t)) + f(t,u(t-τ),u(t+θ))=0, t∈(0,1), CDαu(0)= CDαu(1)=( CDαu(0))=0, au(t)-bu′(t)=η(t), t∈[-τ,0], cu(t)+du′(t)=ξ(t), t∈[1,1+θ], where CDα, CDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([-τ,1+θ],ℝ), 10, and η∈C([-τ,0],[0,∞)), ξ∈C([1,1+θ],[0,∞)). By means of the Guo-Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form (r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)(xσ(t))α=F(t,xσ(t)) on a time scale 𝕋 which is unbounded, where α is a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

This paper studies the eigenvalue comparisons for second-order linear equations with boundary conditions on time scales. Using results from matrix algebras, the existence and comparison results concerning eigenvalues are obtained.

We discuss the initial value problem for the nonlinear fractional differential equation L(D)u=f(t,u), t∈(0,1], u(0)=0, where L(D)=Dsn-an-1Dsn-1-⋯-a1Ds1, 0

New oscillation criteria are established for the second-order nonlinear neutral functional differential equations of the form (r(t)|z′(t)|α−1z′(t))’+f(t,x[σ(t)])=0, t≥t0, where z(t)=x(t)+p(t)x(τ(t)), p∈C1([t0,∞),[0,∞)), and α≥1. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results.

We study the existence of solutions for the boundary value problem -Δνy1(t)=f(y1(t+ν-1),y2(t+μ-1)), -Δμy2(t)=g(y1(t+ν-1),y2(t+μ-1)), y1(ν-2)=Δy1(ν+b)=0, y2(μ-2)=Δy2(μ+b)=0, where 1

We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations D0+αu(t)+λf(u(t))=0, 0

We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type, [a(t)(x(t)+r(t)x(τ(t)))Δ]Δ+p(t)xγ(δ(t))=0, on a time scale unbounded above. Here γ>0 is a quotient of odd positive integers with a and p being real-valued positive functions defined on 𝕋. Our results in this paper not only extend and improve the results in the literature but also correct an error in one of the references.

We study the following second-order neutral functional differential equation with mixed nonlinearities (r(t)|(u(t)+p(t)u(t-σ))'|α-1(u(t)+p(t)u(t-σ))′)′+q0(t)|u(τ0(t))|α-1u(τ0(t))+q1(t)|u(τ1(t))|β-1u(τ1(t))+q2(t)|u(τ2(t))|γ-1u(τ2(t))=0, where γ>α>β>0, ∫t0∞(1/r1/α(t))dt

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale 𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for 𝕋=ℝ and 𝕋=ℤ within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) M(λ) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

We consider coupled boundary value problems for second-order symmetric equations on time scales. Existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only unify the existing ones of coupled boundary value problems for second-order symmetric differential equations but also contain more complicated time scales.

Some new oscillation criteria for the second-order neutral delay differential equation (r(t)z′(t))′+q(t)x(σ(t))=0, t≥t0 are established, where ∫t0∞(1/r(t))dt=∞, z(t)=x(t)+p(t)x(τ(t)), 0≤p(t)≤p00. These oscillation criteria extend and improve some known results. An example is considered to illustrate the main results.

By using Riccati transformation technique, we will establish some new oscillation criteria for the even order neutral delay differential equations [z(t)](n)+q(t)f(x(σ(t)))=0, t≥t0, where n is even, z(t)=x(t)+p(t)x(τ(t)), 0≤p(t)≤p0

By topological degree theory and some analysis skills, we consider a class of generalized Liénard type p-Laplacian equations. Upon some suitable assumptions, the existence and uniqueness of periodic solutions for the generalized Liénard type p-Laplacian differential equations are obtained. It is significant that the nonlinear term contains two variables.

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden-Fowler delay dynamic equations (rxΔ)Δ(t)+p(t)xγ(τ(t))=0 on a time scale 𝕋; here γ is a quotient of odd positive integers with r and p as real-valued positive rd-continuous functions defined on 𝕋. Our results in this paper not only extend the results given in Agarwal et al. (2005), Akin-Bohner et al. (2007) and Han et al. (2007) but also unify the results about oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation.

We will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic equations (r(t)(x(t)−a(t)x(τ(t)))ΔΔ)Δ+p(t)xγ(δ(t))=0 on a time scale 𝕋, where γ>0 is a quotient of odd positive integers with r, a, and p real-valued positive rd-continuous functions defined on 𝕋. To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. Some examples are considered to illustrate the main results.

By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.

We systematically explore the periodicity of Liénard type p-Laplacian equations on time scales. Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scale is chosen as the set of the real numbers. The main method is based on the Mawhin's continuation theorem.

This paper studies Sturm-Picone comparison theorem of second-order linear equations on time scales. We first establish Picone identity on time scales and obtain our main result by using it. Also, our result unifies the existing ones of second-order differential and difference equations.

We consider the existence of positive solutions for a class of second-order multi-point boundary value problem with p-Laplacian on time scales. By using the well-known Krasnosel'ski's fixed-point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, an example is given to illustrate the main results.

We employ Kranoselskii's fixed point theorem to establish the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation [x(t)+p(t)x(τ0(t))]ΔΔ+q1(t)x(τ1(t))−q2(t)x(τ2(t))=e(t) on a time scale 𝕋. To dwell upon the importance of our results, one interesting example is also included.

This paper studies general coupled boundary value problems for second-order difference equations. Existence of eigenvalues is proved, numbers of their eigenvalues are calculated, and their relationships between the eigenvalues of second-order difference equation with three different coupled boundary conditions are established.

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations (r(t)(xΔ(t))γ)Δ+p(t)f(x(τ(t)))=0 on a time scale 𝕋; here γ>0 is a quotient of odd positive integers with r and p real-valued positive rd-continuous functions defined on 𝕋. Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations (p(t)(xΔ(t))γ)Δ+q(t)f(x(Ä(t)))=0 on a time scale T, here γ≥1 is a quotient of odd positive integers with p and q real-valued positive rd-continuous functions defined on T.

The study of fractional Langevin equation has obtained abundant results in recent years. However, there are few studies on resonant fractional Langevin equation. In this paper, we investigate boundary value problems for fractional Langevin equation at resonance. By virtue of Banach contraction mapping principle and Leray-Schauder fixed point theorem, we obtain the uniqueness and existence of solutions. In addition, we get different stability results, including Ulam-Hyres stability and generalized Ulam-Hyres stability. Finally, give relevant examples to demonstrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the positive solutions for Hilfer frac-tional differential equation boundary value problems at resonance. First, we give the expression of the solution with Mittag-Leer function. Next, we ob-tain the existence of the positive solutions by using fixed point index theorem. Finally, we give relevant examples to prove our main results. [ABSTRACT FROM AUTHOR]