Suchergebnisse

In this paper, we establish some (p,q)-Opial type inequalities and generalization of (p,q)-Opial type inequalities.

For any $\mu _{j}\ (\mu _{j}\in \mathbb{C},\left\vert \mu _{j}\right\vert =1,j=1,2)$, we consider the rotations $f_{\mu _{1}}$ and $F_{\mu _{2}}$ of right half-plane harmonic mappings $f,F\in S_{\mathcal{H}}$ which are CHD with the prescribed dilatations $\omega _{f}(z)=\left( a-z\right) /\left(1-az\right) $ for some $a$ $\left( -1<a<1\right) $ and $\omega _{F}(z)=$ $e^{i\theta }z^{n}$ $\left( n\in \mathbb{N},\theta \in \mathbb{R}\right) $, $\omega _{F}(z)=$ $\left( b-z\right) /\left( 1-bz\right) $, $\omega_{F}(z)=\left( b-ze^{i\phi }\right) /\left( 1-bze^{i\phi }\right) $ $(-1<b<1,\phi \in \mathbb{R})$, respectively. It is proved that the convolution $f_{\mu _{1}}\ast F_{\mu _{2}}\in S_{\mathcal{H}}$ and is convex in the direction of $\overline{\mu _{1}\mu _{2}}$ under certain conditions on the parameters involved.

In this article, we have interested the study of the existence and uniqueness of positive solutions of the first-order nonlinear Hilfer fractional differential equation $$D_{0^{+}}^{\alpha ,\beta }y(t)=f(t,y(t)),\text{ }0<t\leq 1,$$ with the integral boundary condition $$I_{0^{+}}^{1-\gamma }y(0)=\lambda \int_{0}^{1}y(s)ds+d,$$ where $0<\alpha \leq 1,$ $0\leq \beta \leq 1,$ $\lambda \geq 0,$ $d\in \mathbb{R}^{+},$ and $D_{0^{+}}^{\alpha ,\beta }$, $I_{0^{+}}^{1-\gamma }$ are fractional ope\-rators in the Hilfer, Riemann-Liouville concepts, respectively. In this approach, we transform the given fractional differential equation into an equivalent integral equation. Then we establish sufficient conditions and employ the Schauder fixed point theorem and the method of upper and lower solutions to obtain the existence of a positive solution of a given problem. We also use the Banach contraction principle theorem to show the existence of a unique positive solution. The result of existence obtained by structure the upper and lower control functions of the nonlinear term is without any monotonous conditions. Finally, an example is presented to show the effectiveness of our main results.

To obtain the main result of the present paper we use the technique of differential subordination. As special cases of our main result, we obtain sufficient conditions for $f\in\mathcal A$ to be $\phi-$like, starlike and close-to-convex in a parabolic region.

"In this paper, we show the existence of continuous positive solutions of a class of nonlinear parabolic reaction di usion systems with initial conditions using techniques of functional analysis and potential analysis."

The notions of strong differential subordination and superordination have been studied recently by many authors. In the present paper, using these concepts, we obtain some preserving properties of certain nonlinear integral operator defined on the space of normalized analytic functions in $\mathbb{D}\times\overline{\mathbb{D}}$. The sandwich-type theorems and consequences of the main results are also considered.