Ercan Tunç

On the asymptotic behavior of solutions of certain second-order differential equations

Abstract In this paper, the second order non-linear differential equation x ¨ + a ( t ) f ( x , x ˙ ) x ˙ + b ( t ) g ( x ) = p ( t , x , x ˙ ) is considered, and Lyapunovs second method is used to show that uniform boundedness and convergence to zero of all solutions of this equation together with their derivatives of the first order.

Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions

In this study we consider one boundary-value problem type Sturm-Liouville with eigenparameter dependent boundary conditions and with transmission conditions at one inner point of considered finite interval. We give operator-theoretical formulation, construct fundamental solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions of the considered problem.

Oscillation of third-order nonlinear damped delay differential equations

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout.

Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives

Abstract In this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the form C , H D a r x ( t ) = e ( t ) + f ( t , x ( t ) ) , a > 1 ,

New Results on the Nonoscillation of Solutions of Some Nonlinear Differential Equations of Third Order

\begin{equation*}^{C,H}\mathcal{D}_{a}^{r}x(t)=e(t)+f(t,x(t)), \quad a\gt1, \end{equation*}

On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations

where r = n+α–1, α ∊ (0,1), and n ∊ℤ+. They also apply their technique to investigate the oscillatory and asymptotic behavior of solutions of the related integral equation x ( t ) = e ( t ) + ∫ a t ln ⁡ t s r − 1 k ( t , s ) f ( s , x ( s ) ) d s s , a > 1 , r  is as above .

New Ultimate Boundedness and Periodicity Results for Certain Third-order Nonlinear Vector Differential Equations

\begin{equation*}x(t)=e(t)+\int\limits_{a}^{t}\left( \ln \frac{t}{s}\right) ^{r-1}k(t,s)f(s,x(s))\frac{ds}{s}, \quad a\gt1, \quad r\textrm{ is as above}. \end{equation*}