Dynamic equations on time scales, dynamic inequalities on time scales, Applications of opial and Wirtinger inequalities on the zeros of Riemann Zeta Function.

Volterra Integro-differential Systems

These have their roots in biological growth problems, whose origins can be traced from the Malthusian model through the logistic equation, the predator-prey system of Lotka and Volterra and Volterra's own formulation of integral equations regarding age distribution in population.

Artificial Neural Networks

ANNs are crude mathematical models of biological neural systems. They have to be designed in such a way that their synaptic weights, which are the strengths of signals or communications between neurons, could effectively store and

retrieve memories.

Planning Algorithms

In robotics, motion planning is an important component. The focus is on designing algorithms that generate useful motions by processing complicated geometric models.

Swarm Intelligence

Swarming, or aggregations of organisms in groups, can be found in nature in many organisms ranging from simple bacteria to mammals. A relatively new area of research looks into the behavior of swarms, in particular to how a swarm's collective behavior could be mimicked to solve challenging engineering problems.

Internet Congestion Control (new area of interest)

One of the most recent and exciting areas of research in the stability analysis of systems deals with the need to control traffic in the ever-growing Internet in a more systematic, rigorous and efficient manner.

Partial functional differential equations

Evolution equations

Delay and ordinary differential equations

Functional Differential Equations: Oscillation, Positive solutions.

Lyapunov's type Inequalities.

Difference Equations.

Fractional differential equations

Impulsive differential equations

Biomedical applications

Major research interests are in the theory of functions, functional analysis and the theory of differential equations:

- Linear and nonlinear integral and matrix operators;

- Weighted inequalities;

- Weighted embedding theorem;

- Spectral theory of operators;

- The qualitative properties of quasilinear differential and difference equations.

integral inequalities, evolution equations

Mathematical Biology,

Epidemiology

integro-differential systems): stability theory of integral equations. Ordinary

differential equations: boundary value problems on infinite intervals,

boundedness, asymptotic properties, dichotomy, trichotomy.

Bound value problem

Dynamical Systems

Control Theory

Ordinary Differential Equations

- Techniques for approximating solutions of delay differential equations: numerical methods using successive approximations sequences and trapezoidal quadrature formula and spline functions method, the step method.

dynamical systems

Differential Equations with Impulses,

Functional Analysis, Topology

Differential Equations and Dynamical Systems

Boundary value problems of impulsive differential inclusions.

boundary value problem,

Partial Differential Equation.

viscoplastic, electro-elastic, ... ect (with or

without friction, damage, adhesion, Normal

compliance, ... )

Applications to the Life Sciences;

Functional Differential Equations;

Almost Periodic Differential Equations;

Delay Differential Equations with Applications in

Population Dynamics

Nonlinear elliptic equations: local and Nonlocal

Hamiltonian systems

boundary value problems; nonlocal boundary value problems for system of hyperbolic

equations with mixed derivatives

Inverse problems for Partial Differential Equations,

Asymptotic Behavior of Ordinary Differential Equations

Functional Differential Equations, Difference Equations, Time Scale Calculus.

Dynamic Equations on Time Scales

Qualitative and stability theory of functional differential equations

$u_i'(t)=f_i\big(t,u_1(\tau_{i1}(t)),u_2(\tau_{i2}(t))\big) (i=1,2)$

with the boundary conditions (1.2)

$\varphi\big(u_1(0),u_2(0)\big)=0, u_1(t)=u_1(a), u_2(t)=0 for t\geq a,$

where $f_i: [0,a]\times \Bbb{R}^2\to \Bbb{R}$ $(i=1,2)$ satisfy the local Carathéodory conditions, while $\varphi: \Bbb{R}^2\to \Bbb{R}$ and $\tau_{ik}: [0,a]\to [0,+\infty[$ $(i,k=1,2)$ are continuous functions. The optimal, in a certain sense, sufficient conditions are obtained for the existence and uniqueness of a nonnegative solution of the problem (1.1), (1.2).

\begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*}

where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$.

$$\partial_t u-\sum_{i,j=1}^N \partial_{x_i}(a_{i,j}(x,t)\partial_{x_j}u)+\sum_{i=1}^N b_i(x,t)\partial_{x_i}u=f(x,t,u)\hbox{ in }\Omega\times R$$

$$u(x,t)=0\hbox{ on }\partial\Omega\times R$$

$$u(x,t)=u(x,t+T)\hbox{ in }\Omega\times R$$

assuming certain conditions on the ratio $2\int_0^s f(x,t,\sigma) d\sigma/s^2$ with respect to the principal eigenvalue of the associated linear problem. The method of proof, whcih is based on the construction of upper and lower solutions, also yields information on the localization and the stability of the solution.

\begin{equation*}\tag{1}

x'_i(t)=\sum\limits_{k=1}^n\ell_{ik}(x_k)(t)+q_i(t)\qquad (i=1,\dots,n)

\end{equation*}

and its particular case

\begin{equation*}x'_i(t)=\sum\limits_{k=1}^n p_{ik}(t)x_k(\tau_{ik}(t))+q_i(t)\qquad (i=1,\dots,n)\tag{1'}\end{equation*}

with the boundary conditions

\begin{equation}\tag{2}

\int_a^b x_i(t)d\varphi_i(t)=c_i\qquad (i=1,\dots,n).

\end{equation}

Here $\ell_{ik}:C(I;\Bbb R)\to L(I;\Bbb R)$ are linear bounded operators, $p_{ik}$ and $q_i\in L(I;\Bbb R)$, $c_i\in\Bbb R$ $(i,k=1,\dots,n)$, $\varphi_i:I\to\Bbb R$ $(i=1,\dots,n)$ are the functions with bounded variations, and $\tau_{ik}:I\to I$ $(i,k=1,\dots,n)$ are measurable functions. The optimal, in some sense, conditions of unique solvability of the problems $(1)$, $(2)$ and $(1')$, $(2)$ are established.

$$u_tt-\Delta_p u+(-\Delta)^\alpha u_t+g(u)=f$$

where $0<\alpha\leq 1$ and $g$ does not satisfy the sign condition $g(u)u \geq 0$.

$$x'=F(t,x,\int_0^t C(at-s) x(s)\,ds)$$

where $a$ is a constant satisfying $0<a<\infty$. Thus, the integral represents the memory of past positions of the solution $x$. We make the assumption that $\int_0^\infty |C(t)|\, dt<\infty$ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of $a$. Very different properties of solutions emerge as we vary $a$ and we are interested in developing an approach which handles them in a unified way.

$$[a(t)\psi(x(t))|x'(t)|^{\alpha-1}x'(t)]'+q(t)f(x(t))=0, \alpha>0$$

where $a,q:[t0,\infty)\rightarrow R, \psi,f:R\rightarrow R$ are continuous, $a(t)>0$ and $\psi(x)>0$, $xf(x)>0$ for $x\not=0$. These criteria involve the use of averaging functions.

$$\frac{d^n}{dt^n} [ x(t) + \lambda x(t-\tau) ] + f(t,x(g(t))) = 0\tag{1.1}$$

is considered under the following conditions: $n\ge 2$ is even; $\lambda>0$; $\tau>0$; $g \in C[t_0,\infty)$, $\lim_{t\to\infty} g(t) = \infty$; $f \in C([t_0,\infty) \times {\bf R})$, $u f(t,u) \ge 0$ for $(t,u) \in [t_0,\infty) \times {\bf R}$, and $f(t,u)$ is nondecreasing in $u \in {\bf R}$ for each fixed $t\ge t_0$. It is shown that equation (1.1) is oscillatory if and only if the non-neutral differential equation

$$x^{(n)}(t) + \frac{1}{1+\lambda} f(t,x(g(t))) = 0$$

is oscillatory.

$$ax''(t)+bx'(t)+cx(t)+g(x(t-\tau_1), \ x'(t-\tau_2), x''(t-\tau_3))=p(t)=p(t+2\pi).$$

and establish a sufficient coudition for the existence of $2\pi$-periodic solution of above equations.

$$\left|

\begin{array}{lcr}

x_1^{\prime}(t)={{A(t)}\over {1+x_2^n(t)}}-x_1(t)\\

x_2^{\prime}(t)={{B(t)}\over {1+x_1^n(t)}}-x_2(t),

\end{array}

\right.\tag{1}$$

where $A$ and $B$ belong to ${\cal C}_+$ and ${\cal C}_+$ is the set of continuous functions $g:{\cal R}\longrightarrow {\cal R}$, which are bounded above and below by positive constants. $n$ is fixed natural number. The system (1) describes cell differentiation, more precisely - its passes from one regime of work to other without loss of genetic information. The variables $x_1$ and $x_2$ make sense of concentration of specific metabolits. The parameters $A$ and $B$ reflect degree of development of base metabolism. The parameter $n$ reflects the highest row of the repression's reactions.

\begin{equation*}\label{eq3.1}\tag{1}

\ddot{\xi}+\xi=u,

\end{equation*}

where an external force (the control function) $u$ depends on the coordinate $\xi$, only. It can be shown that no ordinary (even nonlinear) feedback controls of the form $u=f(\xi(t))$ can asymptotically stabilize the solutions of the system \eqref{eq3.1}. However, one is able to make the system \eqref{eq3.1} asymptotically stable if one designs a special feedback control $u$ depending on $\xi(\cdot)$ which is called a hybrid feedback control. We demonstrate in the paper that the dynamics of a typical linear system of ordinary differential equations equipped with a linear hybrid feedback control possesses some irregular properties that dynamical systems without delay do not have. For example, solutions with different initial conditions may cross or even partly coincide. This proves that the hybrid dynamics cannot, in general, be described by a system of ordinary differential equations, neither linear, nor nonlinear, so that time-delays have to be incorporated into the system.

$$\eqalign{- u''(x) &= \lambda f(u(x)) \ \text{for} \ x \in (-1, 1), \lambda > 0, \cr

u(-1)&= 0\ = u(1) ,}$$

where $f : [0, \infty) \to \Bbb R$ is semipositone ($f(0)<0$) and superlinear ($\lim_{t \to \infty} f(t) /t = \infty)$. We consider the case when the nonlinearity $f$ is of concave-convex type having exactly one inflection point. We establish that $f$ should be appropriately concave (by establishing conditions on $f$) to allow multiple positive solutions. For any $\lambda > 0$, we obtain the exact number of positive solutions as a function of $f(t)/t$ and establish how the positive solution curves to the above problem change. Also, we give examples where our results apply. This work extends the work in [1] by giving a complete classification of positive solutions for concave-convex type nonlinearities.

$$ u''(t)+\lambda a(t)f(u(t)) = 0, \;\;\;t\in(0,1)$$

$$u(0)=0,\;\;\;\; \alpha u(\eta)=u(1),$$

where $0<\eta<1$ and $0<\alpha <\frac{1}{\eta}.$

$$\Delta_{p_i} u_i=H_i(|x|)u_{i+1}^{\alpha_i}, x\in \mathbb{R}^N, i=1,2,...,m$$

with nonnegative continuous function $H_i$. Sufficient conditions are given to have nonnegative nontrivial radial entire solutions. When $H_i$, $i = 1, 2, ..., m$, behave like constant multiples of $|x|^\lambda$, $\lambda\in \mathbb{R}$, we can completely characterize the existence property of nonnegative nontrivial radial entire solutions.

$$-(|u'|^{p-2}u')'=\lambda m|u|^{p-2}u\qquad u \in I=]a,b[,\quad u(a)=u(b)=0,$$

where $p>1$, $\lambda$ is a real parameter, $m$ is an indefinite weight, and $a$, $b$ are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight $m$ and the domain $I$, the k-th eigenfunction, corresponding to the $k$-th eigenvalue, has exactly $k-1$ zeros in $(a,b)$. At the end, we give a simple variational formulation of eigenvalues.

$$\Delta u(x)+f(|x|,u(x),|\nabla u(x)|)=0\qquad x\in B,\ u|_\Gamma=a\in{\mathbb{R}}\ (\Gamma:=\partial B)$$

is proved, where $B$ is the unit ball in ${\mathbb{R}}^n$ centered at the origin $(n\ge2)$, $a$ is arbitrary $(a>a_0\ge-\infty);f$ is positive, continuous and bounded. It is shown that these solutions belong to $C^2(\overline{B})$. Moreover, in the case $f\in C^1$ a sufficient condition (near necessary) for the smoothness property $u(x;a)\in C^3(\overline{B})\quad\forall a>a_0$ is also obtained.

$${d\over dt} D(x_t) = f (x_t).$$

A new proof based on local integral manifold theory and the implicit function theorem is given for the classical result that a simple periodic orbit of the equation above is asymptotically orbitally stable with asymptotic phase. The technique used overcomes the difficulty that the solution operator of a NFDE does not smooth as $t$ increases.

The goal of the present paper is to detect those mathematical constructions that are related to the existence of alternative fields dictated by differential equations. With this in mind we investigate differential and integral calculus based on the commutative algebra that is generated by a given differential equation. It turns out that along with the standard differential and integral calculus there always exists an isomorphic alternative calculus. Moreover, every system of differential equations generates its own calculus that is isomorphic (or homomorphic) to the standard one. The given system written in its own calculus appears to be linear.

It is also shown that there always exist two alternative to each other geometries, and matrix algebra has its alternative isomorphic to the classical one.

\[

\left\{

\begin{array}{c}

-\left( \varphi \left( u^{\prime }\right) \right) ^{\prime }=f\left(

u\right) \text{ in }\left( 0,1\right) \\

u\left( 0\right) =u\left( 1\right) =0

\end{array}

\right.

\]

where $\varphi $ is an odd increasing homeomorphism of $\Bbb{R}$ concave on $\Bbb{R}^{+}$ and $f$ $\in C\left( \Bbb{R}\text{, }\Bbb{R}\right) $is odd and superlinear.

$$\left\{\begin{array}{lll}

-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,+ f(x,u,v,\lambda)&

\mbox{in} \ \Omega\\

-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \, + g(x,u,v,\lambda) &

\mbox{in} \ \Omega\\

(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). & \

\end{array}

\right.

$$

We prove that the principal eigenvalue $\lambda_1$ of the following eigenvalue problem

$$\left\{\begin{array}{lll}

-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,& \mbox{in} \ \Omega\\

-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \,& \mbox{in} \ \Omega\\

(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)& \

\end{array}

\right.$$

is simple and isolated and we prove that $(\lambda_1,0,0)$ is a bifurcation point of the system mentioned above.

$u_{\varepsilon}=(u_{\varepsilon 1},u_{\varepsilon 2},u_{\varepsilon 3})$ of a p-energy functional as $\varepsilon \to 0$, and the zeros of $u_{\varepsilon 1}^2+u_{\varepsilon 2}^2$ are located roughly. In addition,the estimates of the convergent rate of $u_{\varepsilon 3}^2$ (to $0$) are presented. At last, based on researching the Euler-Lagrange equation of symmetric solutions and establishing its $C^{1,\alpha}$ estimate, the author obtains the $C^{1,\alpha}$ convergence of some symmetric minimizer.

\begin{gather*}

u^{\Delta\nabla}(t) + \lambda a(t)f(u(t)) = 0, \quad t \in (0,T) \cap \mathbb{T},\\

u(0) = 0, \quad \alpha u(\eta) = u(T),

\end{gather*}

where $\eta \in (0, \rho(T)) \cap \mathbb{T}$, and $0 < \alpha <T/\eta$, has a positive solution.

\begin{eqnarray*}

\left| \left( Ax\right) \left( t\right) -\left( Ay\right) \left( t\right)

\right| &\leq &\beta \left| x\left( \nu \left( t\right) \right) -y\left( \nu

\left( t\right) \right) \right| + \\

&&+\frac{k}{t^{\alpha }}\int_{0}^{t}\left| x\left( \sigma \left( s\right)

\right) -y\left( \sigma \left( s\right) \right) \right| ds,

\end{eqnarray*}

(where $\alpha ,$ $\beta \in \lbrack 0,1)$, $k\geq 0$, and $\nu ,$ $\sigma :\left[ 0,T\right] \rightarrow \left[ 0,T\right] $ are continuous functions such that $\nu \left( t\right) \leq t,$ $\sigma \left( t\right)\leq t,$ $\forall t\in \left[ 0,T\right] $) has exactly one fixed point in $M $. Then the result is extended in $C\left( \mathbb{R}_{+},X\right) ,$ where $\mathbb{R}_{+}:=[0,\infty ).$

\begin{equation}

\left\{\begin{array}{lll}

-\Delta_p u=& \lambda m(x)|u|^{p-2}u + f(\lambda,x,u)& \mbox{in} \ \Omega\\

\frac{\partial u}{\partial \nu}\hspace{0.55cm}= & 0 & \mbox{on}

\ \partial\Omega.

\end{array}

\right.

\end{equation}

We prove that the principal eigenvalue $\lambda_1$ of the corresponding eigenvalue problem with $f\equiv 0,$ is a bifurcation point by using a generalized degree type of Rabinowitz.

We study such equations with more general delays by means of two successive applications of contraction mappings. Given the initial function, we explicitly locate the constant to which the solution converges, show that the solution is stable, and show that its limit function is a type of "selective global attractor." In the last section we examine a problem of Minorsky in the guidance of a large ship. Knowledge of that constant to which solutions converge is critical for guidance and control.

$$\displaylines{ u^{(4)}(t) = a(t)f(u(t)), \quad 0 < t < 1, \cr u(0) = u(1) = u'(0) = u'(1) = 0, }$$

where $a(t)$ is $L^p$-integrable and $f$ satisfies certain growth conditions.

\begin{equation*}

\operatorname{div}(\|\nabla u\|^{p-2}\nabla u)+ \left\langle \vec b(x), \|\nabla u\|^{p-2}\nabla u\right\rangle + c(x)|u|^{q-2}u=0,

\end{equation*}

via the Riccati technique and prove an integral sufficient condition on the potential function $c(x)$ and the damping $\vec b(x)$ which ensures that no positive solution of the equation satisfies a lower (if $p>q$) or upper (if $q>p$) bound eventually.

$$

\sum_{i,\,j=1}^{N}D_i[\,a_{ij}(x)D_jy\,]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0,

\tag{E}

$$

which are different from most known ones in the sense that they are based on a new weighted function $H(r,s,l)$ defined in the sequel. Both the cases when $D_ib_i(x)$ exists for all $i$ and when it does not exist for some $i$ are considered.

\[

u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.e.}\ t \in [0,T],

\]

\[

u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m,

\]

\[

g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0,

\]

where $f \in Car([0,T]\times\mathbb{R}^{2})$, $g_1$, $g_2 \in C(\mathbb{R}^2)$, $J_j$, $M_j \in C(\mathbb{R})$. An existence theorem is proved for non-ordered lower and upper functions. Proofs are based on the Leray–Schauder degree and on the method of a priori estimates.

$$

\sum_{k=0}^{n}

(-1)^k\nu_k\left(\frac{y^{(k)}}{t^{2n-2k-\alpha}}\right)^{(k)}

=(-1)^m\left(q_m(t)y^{(m)}\right)^{(m)},\; \nu_n:=1,

$$

where $m \in \{0, 1\}$, $\alpha \not\in \{1, 3, \dots , 2n-1\}$ and $\nu_0, \dots, \nu_{n-1},$ are real constants satisfying certain conditions, are investigated. In particular, the case when $q_m(t)=\frac{\beta}{t^{2n-2m-\alpha}\ln^2 t }$ is studied.

Then $u\equiv 0$ and $v\equiv 0.$

$\left (\phi_p(u''(t)) \right )'' = \lambda h(t) u(t), \, u'(0) = 0, \,

\beta_0 u(\eta_0) = u(1), \, \phi_p'(u''(0)) = 0, \,

\beta_1\phi_p(u''(\eta_1)) = \phi_p(u''(1))$, $p > 2$,

$0 < \eta_1,\eta_0 < 1, 0 < \beta_1, \beta_0 < 1$,

using the theory of u$_0$-positive operators with respect to a cone in a Banach space. We then obtain a comparison theorem for the smallest positive eigenvalues, $\lambda_1$ and $\lambda_2$, for the differential equations

$\left ( \phi_p(u''(t)) \right )'' = \lambda_1 f(t) u(t)$ and

$\left ( \phi_p(u''(t)) \right )'' =\lambda_2 g(t) u(t)$ where $0 \leq f(t) \leq g(t), t \in [0,1]$.

$$ -M(\|u\|^2)\Delta u \geq f(x,u) $$

Making use of the penalized method and Galerkin approximations, we establish existence theorems for both cases when $M$ is continuous and when $M$ is discontinuous.

$ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \,

u(\cdot , 0) = \phi (\cdot)$ and

$ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x + \mathcal{H} v_{xxx}) = 0, \,

v(\cdot , 0) = \psi (\cdot)$

has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When $\eta>0$ and the initial data is complex-valued we prove local well-posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis.

$$x^{(3)}(t) \in F(x(t),x'(t),x''(t)),\quad x(0)=x_{0}, \quad x'(0)=y_{0}, \quad x''(0)=z_{0}.$$

\begin{equation*}

\left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t),

\end{equation*}

on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions which ensure that every solution oscillates or satisfies $\lim \inf_{t\rightarrow \infty }\left\vert x(t)\right\vert =0.$ Our oscillation results when $r(t)=0$ improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc \ 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491--507] since our results do not require $\int_{t_{0}}^{\infty }q(t)\Delta t>0$ and $\int_{\pm t_{0}}^{\pm \infty } \frac{du}{f(u)}<\infty .$ Also, as a special case when ${\mathbb{T=R}}$, and $r(t)=0$ our results improve some oscillation results for differential equations. Some examples are given to illustrate the main results.

$$x\left( t\right) =q\left( t\right) +\int_{0}^{t}K\left( t,s,x\left( s\right) \right) ds

+\int_{0}^{\infty }G\left( t,s,x\left( s\right) \right) ds$$

is presented.

\begin{equation} x(t+1)=a(t)x(t)+c(t)\Delta x(t-g(t))+q(x(t),x(t-g(t))\big).\nonumber \end{equation}

In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis.

$$ x(t)=a(t)-\int^t_0 C(t,s) x(s)ds,$$ a resolvent $$ R(t,s),$$ and a variation-of-parameters formula

$$ x(t)=a(t)-\int^t_0 R(t,s) a(s)ds $$ with special accent on the case in which $a(t)$ is unbounded. We use contraction mappings to establish close relations between $a(t)$ and $\int^t_0R(t,s) a(s)ds$.

\[ x'(t) = A(t)x(t) + \int_s^t B(t,u)x(u)\,du \]

in the same way that it is defined for $x' = A(t)x$ and prove that it is the unique matrix solution of

\[ \frac{\partial}{\partial{t}}Z(t,s) = A(t)Z(t,s) + \int_{s}^t B(t,u)Z(u,s)\,du, \quad Z(s,s) = I. \]

Furthermore, we prove that the solution of

\[ x'(t) = A(t)x(t) + \int_{\tau}^t B(t,u)x(u)\,du + f(t), \quad x(\tau) = x_0\]

is unique and given by the variation of parameters formula

\[ x(t) = Z(t,\tau)x_0 + \int_{\tau}^t Z(t,s)f(s)\,ds.\]

We also define the principal matrix solution $R(t,s)$ of the adjoint equation

\[ r'(s) = -r(s)A(s) - \int_s^t r(u)B(u,s)\,du \]

and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of

\[ \frac{\partial}{\partial{s}}R(t,s) = -R(t,s)A(s) - \int_{s}^t R(t,u)B(u,s)\,du, \quad R(t,t) = I. \]

Finally, we prove that despite the difference in their definitions $R(t,s)$ and $Z(t,s)$ are in fact identical.

$$-{\rm div}a(x, u, \nabla u) + g(x, u, \nabla u) = f-{\rm div}F $$

where $a(.)$ is a Carathéodory function satisfying the classical condition of type Leray-Lions hypothesis, while $g(x, s, \xi)$ is a non-linear term which has a growth condition with respect to $\xi$ and no growth with respect to $s$, but it satisfies a sign condition on $s$.

\begin{equation}

x(n+1)=a(n)x(n)+c(n)\Delta x(n-g(n))+\sum^{n-1}_{s=n-g(n)}k(n,s)h(x(s)).\nonumber

\end{equation}

A Krasnoselskii fixed point theorem is used in the analysis.

\begin{eqnarray*}

&&u_{tt}+\Delta^2 u-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla

v||^2_{L^2(\Omega_t)})\Delta u\\

&&+\int^{t}_{0}g_1(t-s)\Delta u(s)ds

+\alpha u_{t}+h(u-v)=0 \quad \mbox{in} \quad \hat{Q},\\

&&v_{tt}+\Delta^2 v-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla

v||^2_{L^2(\Omega_t)})\Delta v \\

&&+\int^{t}_{0}g_2(t-s)\Delta v(s)ds + \alpha v_{t}-h(u-v)=0 \quad

\mbox{in} \quad \hat{Q}

\end{eqnarray*}

in a non cylindrical domain of $\mathbb{R}^{n+1}$ $(n\ge1)$ under suitable hypothesis on the scalar functions $M$, $h$, $g_1$ and $g_2$, and where $\alpha$ is a positive constant. We show that such dissipation is strong enough to produce uniform rate of decay. Besides, the coupling is nonlinear which brings up some additional difficulties, which plays the problem interesting. We establish existence and uniqueness of regular solutions for any $n\ge 1$.

$$

(u_{tt}-\Delta u)_{g_s}=f(u)+g(|x|),\quad t\in [0, 1], x\in {\cal R}^3,

\tag{1}

$$

$$

u(1, x)=u_0\in {\dot H}^1({\cal R}^3),\quad

u_t(1, x)=u_1\in L^2({\cal R}^3),

\tag{2}

$$

where $g_s$ is the Reissner-Nordström metric (see [2]); $f\in {\cal C}^1({\cal R}^1)$, $f(0)=0$, $a|u|\leq f'(u)\leq b|u|$, $g\in {\cal C}({\cal R}^+)$, $g(|x|)\geq 0$, $g(|x|)=0$ for $|x|\geq r_1$, $a$ and $b$ are positive constants, $r_1>0$ is suitable chosen. When $g(r)\equiv 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous. When $g(r)\ne 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous.

\begin{equation}

\frac{d}{dt}[x(t) - ax(t-\tau)]= r(t)x(t)- f(t, x(t-\tau))

\end{equation}

has a positive periodic solution. An example will be provided as an application to our theorems.

\begin{equation*}

\left( r(t)\psi (x)f(\dot{x})\right) ^{\cdot }+p(t)\varphi \left(g(x),r(t)\psi(x)f(\dot{x})\right)+q(t)g(x)=0,

\end{equation*}

where $p$, $q$, $r:[t_{o},\infty )\rightarrow \mathbf{R}$ and\ $\psi $, $g $, $f:\mathbf{R}\rightarrow \mathbf{R}$ are continuous, $r(t)>0$,\ $p(t)\geq 0$ and $\psi (x)>0$, $xg(x)>0$ for $x\neq 0$, $uf(u)>0$ for $u\neq0 $. Our results generalize and extend some known oscillation criteria in the literature. The relevance of our results is illustrated with a number of examples.

$$

\left\{ \begin{array}{ll}

(-1)^nu^{(2n)}(x)=\lambda p(x)f(u(x)),\quad 0<x<1 \\

u^{(2i)}(0)=\sum_{j=1}^{m}a_ju^{(2i)}(\eta _j), \quad

u^{(2i+1)}(1)=\sum_{j=1}^{m}b_ju^{(2i+1)}(\eta _j), \quad i=0, 1,

\ldots , n-1

\end{array} \right.

$$

where $a_j,b_j\in[0,\infty), \ j=1, 2, \ldots, m,$ with $0<\sum_{j=1}^{m}a_j<1, \ 0<\sum_{j=1}^{m}b_j<1,$ and $ \eta_j \in(0,1)$ with $0<\eta_1<\eta_2<\ldots <\eta_m<1,$ under certain conditions on $f$ and $p$ using the Krasnosel'skii fixed point theorem for certain values of $\lambda$. We use the positivity of the Green's function and cone theory to prove our results.

$$ \frac{\partial u}{\partial t}-\triangle_{p}u+\alpha(u)=f \quad \text{in } ]0,\ T[\times\Omega, $$ with Neumann-type boundary conditions and initial data in $L^1$. Our approach is based essentially on the time discretization technique by Euler forward scheme.

$$

-\Delta u+a_uu=u^3-\beta uv^2, \quad u=u(x),

$$

$$

-\Delta v+a_vv=v^3-\beta u^2v, \quad v=v(x), \ x\in \mathbb {R}^3,

$$

$$

u\big| _{|x|\to \infty }=v\big| _{|x|\to \infty }=0,

$$

where $a_u,a_v$ and $\beta $ are positive constants. We prove the existence of a component-wise positive smooth radially symmetric solution of this system. This result is a part of the results presented in the recent paper by Sirakov [1]; in our opinion, our method allows one to treat the problem simpler and shorter.

\begin{eqnarray*}

u^{(n)}(t)+\lambda a(t)f(u(t))=0,\,\,\, 0<t<1,

\end{eqnarray*}

satisfying three kinds of different boundary value conditions. Our analysis relies on Krasnoselskii's fixed point theorem of cone. An example is also given to illustrate the main results.

The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t)$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient.

The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t)$ as the sum of a bounded, but badly behaved function, and a large well behaved function.

\begin{eqnarray*}

&&D^{\alpha}u + a(t) f(u) = 0, \quad 0<t<1, 1<\alpha\leq2,\\

&&u(0) = 0 ,u'(1)= 0,

\end{eqnarray*}

where $ D^{\alpha}$ is the Riemann-Liouville differential operator of order $\alpha $, $f: [0,\infty)\rightarrow [0,\infty)$ is a given continuous function and $a$ is a positive and continuous function on $[0,1]$.

\begin{equation*}\left\{ \begin{aligned}

& x^{^{\prime \prime \prime }}(t)=\alpha \left( t\right) f(t,x(t),x^{\prime}\left( t\right) ,x^{\prime \prime }\left( t\right) ),\;\;\;0<t<1, \\

& x\left( 0\right) =x^{\prime }\left( \eta \right) =x^{\prime \prime }\left(1\right) =0,

\end{aligned}\right.\end{equation*}

under positivity of the nonlinearity. Existence results for a positive and concave solution $x\left( t\right) ,\ 0\leq t\leq 1$ are given, for any $1/2<\eta <1.\ $ In addition, without any monotonicity assumption on the nonlinearity, we prove the existence of a sequence of such solutions with \begin{equation*} \lim_{n\rightarrow \infty }||x_{n}||=0. \end{equation*} Our principal tool is a very simple applications on a new cone of the plane of the well-known Krasnosel’skiĭ’s fixed point theorem. The main feature of this approach is that, we do not use at all the associated Green's function, the necessary positivity of which yields the restriction $\eta \in \left( 1/2,1\right) $. Our method still guarantees that the solution we obtain is positive.

\begin{equation*}

\frac{\partial u}{\partial t}=(\frac{\partial ^{m}}{\partial x^{m}}u)^{p},

t\geq 0, x\in \mathbb{R}, m, p\in \mathbb{N}, p>1,

\end{equation*}

by a new method that we call the travelling profiles method. This method allows us to find several forms of exact solutions including the classical forms such as travelling-wave and self-similar solutions.

$$

\big(y(t)- \sum_{i=1}^n p_i(t) y(\delta_i(t))\big)'+\sum_{i=1}^m q_i(t) y(\sigma_i(t)) = f(t)

$$

oscillates or tends to zero as $t \to \infty$. Here the coefficients $p_i(t), q_i(t)$ and the forcing term $f(t)$ are allowed to oscillate; such oscillation condition in all coefficients is very rare in the literature. Furthermore, this paper provides an answer to the open problem 2.8.3 in [7, p. 57]. Suitable examples are included to illustrate our results.

$D_{0+}^{\alpha} u(t) + a(t)f(t,u(t), u^{\prime \prime}(t))=0, \quad 0 < t < 1, \quad 3 < \alpha \leq 4,$

$u(0) = u^{\prime}(0) = u^{\prime \prime}(0)= u^{\prime \prime}(1)=0 $,

where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in the fact that the nonlinear term is allowed to depend on the second order derivative $u^{\prime \prime}$.

\[\left\{\begin{array}{l}

\left[ \Phi _p(u^{\bigtriangleup }(t))\right] ^{\bigtriangledown}+a(t)f(u(t),u(\mu (t)))=0,t\in \left(0,T\right),\\

u_0(t)=\varphi (t), t\in \left[ -r,0\right], u(0)-B_0(u^{\bigtriangleup }(\eta ))=0, u^{\bigtriangleup }(T)=0,

\end{array}\right.\]

are established by using the well-known Five Functionals Fixed Point Theorem.

$$

\begin{array}{rlll}

y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash

\{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $J=[0,b]$ and $F: J \times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m.$). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion

$$

\begin{array}{rlll}

y'(t) &\in& \varphi(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash

\{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $\varphi: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincar\'e operator.

$$

\bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime}

+q(t)|x|^{\beta-1}x=0\,,\tag{E}

$$

where $\alpha>0$, $\beta>0$ are constants and $p,q:[a,\infty)\to(0,\infty)$ are continuous functions satisfying conditions

$$

\int_a^{\infty}\left( \frac{t}{p(t)}\right)^{\frac{1}{\alpha}}\,dt<\infty,

\int_a^{\infty}\frac{t}{\left(p(t)\right)^{\frac{1}{\alpha}}}\,dt<\infty .

$$

We will establish necessary and sufficient condition for oscillation of all solutions of the sub-half-linear equation (E) (for $\beta<\alpha$) as well as of the super-half-linear equation (E) (for $\beta>\alpha$).

$$

\left\{\begin{aligned} &u''=-f(t,v), \ \ 0< t< 1,\\&v''=-g(t,u), \ \ 0< t< 1\\&u(0)=v(0)=0,\varsigma u(\zeta)=u(1),\varsigma v(\zeta)=v(1),\end{aligned}\right.

$$

where $f:(0,1)\times [0,+\infty)\to [0,+\infty),g:[0,1]\times [0,+\infty)\to [0,+\infty),0<\zeta<1, \varsigma>0,$ and $\varsigma\zeta< 1,f$ may be singular at $t = 0$ and/or $t = 1.$ Under some rather simple conditions, by means of monotone iterative technique, a necessary and sufficient condition for the existence of positive solutions is established, a result on the existence and uniqueness of the positive solution and the iterative sequence of solution is given.

\begin{equation}

\Delta x(n)=-a(n)x(n-\tau(n))

\end{equation}

and its generalization

\begin{equation}

\Delta x(n)=-\sum^{N}_{j=1}a_j(n)x(n-\tau_j(n)).

\end{equation}

Fixed point theorems are used in the analysis.

${ \left \{\begin{array} {l}

u''(t) + f(t,u(t),u'(t))=0, \quad t \not= t_k, \ t \in [0, T], \\

\triangle u(t_k) = I_k(u(t_k)), \quad k = 1, \cdots , m, \\

\triangle u'(t_k) = I_k^*(u(t_k)), \quad k = 1, \cdots , m, \\

u(0) + u(T) = 0, \ u'(0) + u'(T) = 0.

\end{array}\right.} $

New criteria are established based on Schaefer's fixed-point theorem.

\[\left\{\begin{array}{l}

\frac{dx}{dt}=-r(t)x(t)+F(t,x_t,u(t-\delta(t))),\\

\frac{du}{dt}=-h(t)u(t)+g(t)x(t-\sigma(t)).\\

\end{array}\right.\]

We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on $F$.

$$

\left\{ \begin{array}{lll}

(\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t<1,\\

x_{0}=0,\\

x(1)=0,

\end{array}\right.

$$

where $\phi: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\mathbb{R})$ defined by $x_{t}(\sigma)=x(t+\sigma)$ for $ -\tau \leq \sigma\leq 0.$ By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.

\[

\left\{\begin{array}{lll} - (\varphi( u')) ' = \psi(u), \quad -T< x < T; \\

\quad u(-T)=0, \quad u(T)=0 \\

\end{array} \right.\tag{*}

\]

where $\varphi (s) = \alpha s_+^{p-1} -\beta s_-^{p-1}, \psi (s) = \lambda s_+^{p-1} -\mu s_-^{p-1}, p >1.$ We obtain a explicit characterization of Fucik spectrum $(\alpha, \beta, \lambda, \mu),$ i.e., for which the (*) has a nontrivial solution.

$$\left\{\begin{array}{llll}

(c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\

x(0)=\sum\limits_{i=1}^{n}\mu_ix(\xi_{i}),

~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0

\end{array}\right.$$

and

$$\left\{\begin{array}{llll}

(c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\

x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow

+\infty}c(t)\phi_{p}(x'(t))=0

\end{array}\right.

$$

with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results.

\begin{equation*}

f^{^{\prime \prime }}+Af^{^{\prime }}+Bf=F,

\end{equation*}

where $A,$ $B,$ $F\not\equiv 0$ are finite order meromorphic functions having only finitely many poles.

$$

\begin{array}{rlll}

y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in

J\backslash \{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $J=[0,b]$ and $F: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved.

\begin{equation}

\big[x(t)+A(t)x(\alpha(t))\big]^{\Delta^{n}}+B(t)F(x(\beta(t)))=\varphi(t)\qquad\text{for}\ t\in[t_{0},\infty)_{\mathbb{T}},\label{asbeq1}\tag{$\star$}

\end{equation}

where $n\in[2,\infty)_{\mathbb{Z}}$, $t_{0}\in\mathbb{T}$, $\sup\{\mathbb{T}\}=\infty$, $A\in\rm{C_{rd}}([t_{0},\infty)_{\mathbb{T}},\mathbb{R})$ is allowed to alternate in sign infinitely many times, $B\in\rm{C_{rd}}([t_{0},\infty)_{\mathbb{T}},\mathbb{R}^{+})$, $F\in\rm{C_{rd}}(\mathbb{R},\mathbb{R})$ is nondecreasing, and $\alpha,\beta\in\rm{C_{rd}}([t_{0},\infty)_{\mathbb{T}},\mathbb{T})$ are unbounded increasing functions satisfying $\alpha(t),\beta(t)\leq t$ for all sufficiently large $t$. We give change of order formula for double(iterated) integrals to prove our main result. Some simple examples are given to illustrate the applicability of our results too. In the literature, almost all of the results for \eqref{asbeq1} with $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$ hold for bounded solutions. Our results are new and not stated in the literature even for the particular cases $\mathbb{T}=\mathbb{R}$ and/or $\mathbb{T}=\mathbb{Z}$.

\begin{equation*}

\dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\tag{A(t)}

\end{equation*}

is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem

\begin{equation*}

\left\{\begin{split}

\dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\

y(0) &= 0

\end{split}\right.

\end{equation*}

is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries.

$$

\left\{\begin{aligned}

&(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\

&u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0,

\end{aligned}\right.

$$

where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem.

$$

\left\{\begin{array}{ll}

x''(t)+a(t)f(t,x(t),x'(t))=0,0<t<1,\\

x'(0)=0,\ \ x(1)=\sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}),

\end{array}\right.

$$

where $0<\xi_{1}<\xi_{2}<\cdots<\xi_{m-2}<1, \alpha_{i}\in [0,1)$, $i = 1, 2, \cdots m-2$ , with $0<\sum_{i=1}^{m-2}\alpha_{i}<1$ and $f$ may change sign and may be singular at $x=0$ and $x'=0$.

\begin{equation}

\left\vert\begin{array}{l}

-\dfrac{\partial ^{2}u}{\partial x^{2}}-\sum\limits_{s=1}^{n}\dfrac{\partial}{\partial y_{s}}a_{s}(y,\frac{\partial u}{\partial y_{s}})+f(y,u)=0\text{in }\Omega =\mathbb{R}\times D, \\ \\

u+\varepsilon \dfrac{\partial u}{\partial n}=0\text{ on }\mathbb{R}\times \partial D.

\end{array}\right. \tag*{$\left( P\right) $}

\end{equation}

where $a_{s}:D\times \mathbb{R}\rightarrow \mathbb{R}$ are $H^{1}$-functions with constant sign such that

\begin{equation}\begin{array}{c}

2\int\limits_{0}^{\xi _{s}}a_{s}(y,t_{s})dt_{s}-\xi _{s}a_{s}(y,\xi_{s})\leq 0,s=1,...,n

\end{array}\tag*{$\left( H_{1}\right) $}\end{equation}

and $f:D\times \mathbb{R}\rightarrow \mathbb{R}$ is a real continuous locally Liptschitz function such that

\begin{equation} 2F(y,u)-uf(y,u)\leq 0. \tag*{$\left( H_{2}\right) $} \end{equation}

We show that the function \begin{equation*} E(x)=\int\limits_{D}\left\vert u(x,y)\right\vert ^{2}dy \end{equation*} is convex on $\mathbb{R}$ . Our proof is based on energy (integral) identities.