Mostrando recursos 1 - 20 de 25

  1. On the Oscillation of Solutions of First-Order Difference Equations with Delay

    Shoukaku, Y.
    Consider the first order delay difference equation $$\Delta x_{n} + p_{n} x_{\sigma(n)} = 0, \quad n \in {\mathbb N}_0,$$ where $\{p_{n}\}_{n \in {\mathbb N}_0}$ is a sequence of nonnegative real numbers, and $\{\sigma(n)\}_{n \in {\mathbb N}_0}$ is a sequence of integers such that $\sigma(n) \le n-1$, and $\displaystyle \lim_{n \to \infty}\sigma(n) = +\infty$. We obtain similar oscillation criteria of delay differential equations. This criterion is used by more simple method until now.

  2. Weak Solutions for Implicit Hilfer Fractional Differential Equations With Not Instantaneous Impulses

    Abbas, S.; Benchohra, M.; Henderson, J.
    In this paper, we present results concerning the existence of weak solutions for some functional implicit Hilfer fractional differential equations with not instantaneous impulses in Banach spaces. The main results are proved by applying Mönch's fixed point theorem associated with the technique of measure of weak non compactness, and we present an illustrative example.

  3. General Adjoint on a Banach Space

    Gill, Tepper L.
    In this paper, we show that the continuous dense embedding of a separable Banach space $\mathcal B$ into a Hilbert space $\mathcal H$ offers a new tool for studying the structure of operators on a Banach space. We use this embedding to demonstrate that the dual of a Banach space is not unique. As a application, we consider this non-uniqueness within the $\mathbb C[0,1] \subset L^2[0,1]$ setting. We then extend our theory every separable Banach space $\mathcal B$. In particular, we show that every closed densely defined linear operator $A$ on $\mathcal B$ has a unique adjoint $A^*$ defined on...

  4. General Adjoint on a Banach Space

    Gill, Tepper L.
    In this paper, we show that the continuous dense embedding of a separable Banach space $\mathcal B$ into a Hilbert space $\mathcal H$ offers a new tool for studying the structure of operators on a Banach space. We use this embedding to demonstrate that the dual of a Banach space is not unique. As a application, we consider this non-uniqueness within the $\mathbb C[0,1] \subset L^2[0,1]$ setting. We then extend our theory every separable Banach space $\mathcal B$. In particular, we show that every closed densely defined linear operator $A$ on $\mathcal B$ has a unique adjoint $A^*$ defined on...

  5. General Adjoint on a Banach Space

    Gill, Tepper L.
    In this paper, we show that the continuous dense embedding of a separable Banach space $\mathcal B$ into a Hilbert space $\mathcal H$ offers a new tool for studying the structure of operators on a Banach space. We use this embedding to demonstrate that the dual of a Banach space is not unique. As a application, we consider this non-uniqueness within the $\mathbb C[0,1] \subset L^2[0,1]$ setting. We then extend our theory every separable Banach space $\mathcal B$. In particular, we show that every closed densely defined linear operator $A$ on $\mathcal B$ has a unique adjoint $A^*$ defined on...

  6. Vector Inequalities For Two Projections in Hilbert Spaces and Applications

    Dragomir, Silvestru Sever
    In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$ ¶ Applications for norm and numerical radius inequalities of two bounded...

  7. Vector Inequalities For Two Projections in Hilbert Spaces and Applications

    Dragomir, Silvestru Sever
    In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$ ¶ Applications for norm and numerical radius inequalities of two bounded...

  8. Vector Inequalities For Two Projections in Hilbert Spaces and Applications

    Dragomir, Silvestru Sever
    In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$ ¶ Applications for norm and numerical radius inequalities of two bounded...

  9. Vector Inequalities For Two Projections in Hilbert Spaces and Applications

    Dragomir, Silvestru Sever
    In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$ ¶ Applications for norm and numerical radius inequalities of two bounded...

  10. Vector Inequalities For Two Projections in Hilbert Spaces and Applications

    Dragomir, Silvestru Sever
    In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$ ¶ Applications for norm and numerical radius inequalities of two bounded...

  11. Norm Estimates for Powers of Products of Operators in a Banach Space

    Gil’, Michael
    Let $A$ and $B$ be bounded linear operators in a Banach space. We consider the following problem: if $\Sigma_{k=0}^{\infty} || A^{k} |||| B^{k} || \lt\infty$, under what conditions $\Sigma_{k=0}^{\infty} || (AB)^{k} || \lt \infty$?

  12. Norm Estimates for Powers of Products of Operators in a Banach Space

    Gil’, Michael
    Let $A$ and $B$ be bounded linear operators in a Banach space. We consider the following problem: if $\Sigma_{k=0}^{\infty} || A^{k} |||| B^{k} || \lt\infty$, under what conditions $\Sigma_{k=0}^{\infty} || (AB)^{k} || \lt \infty$?

  13. Norm Estimates for Powers of Products of Operators in a Banach Space

    Gil’, Michael
    Let $A$ and $B$ be bounded linear operators in a Banach space. We consider the following problem: if $\Sigma_{k=0}^{\infty} || A^{k} |||| B^{k} || \lt\infty$, under what conditions $\Sigma_{k=0}^{\infty} || (AB)^{k} || \lt \infty$?

  14. Norm Estimates for Powers of Products of Operators in a Banach Space

    Gil’, Michael
    Let $A$ and $B$ be bounded linear operators in a Banach space. We consider the following problem: if $\Sigma_{k=0}^{\infty} || A^{k} |||| B^{k} || \lt\infty$, under what conditions $\Sigma_{k=0}^{\infty} || (AB)^{k} || \lt \infty$?

  15. Norm Estimates for Powers of Products of Operators in a Banach Space

    Gil’, Michael
    Let $A$ and $B$ be bounded linear operators in a Banach space. We consider the following problem: if $\Sigma_{k=0}^{\infty} || A^{k} |||| B^{k} || \lt\infty$, under what conditions $\Sigma_{k=0}^{\infty} || (AB)^{k} || \lt \infty$?

  16. A Poincaré Inequality for Functions with Locally Bounded Variation in $\mathbb{R}^{d}$

    Savadogo, Bacary; Fofana, Ibrahim
    We prove a weighted Poincaré inequality in a subspace of $BV_\text{loc}$ whose elements have variation measure in a Wiener amalgam space of Radon measures.

  17. A Poincaré Inequality for Functions with Locally Bounded Variation in $\mathbb{R}^{d}$

    Savadogo, Bacary; Fofana, Ibrahim
    We prove a weighted Poincaré inequality in a subspace of $BV_\text{loc}$ whose elements have variation measure in a Wiener amalgam space of Radon measures.

  18. Nonlinear Eigenvalue Problem for the p-Laplacian

    Tsouli, Najib; Chakrone, Omar; Darhouche, Omar; Rahmani, Mostafa
    This article is devoted to the study of the nonlinear eigenvalue problem $$-\Delta_{p} u \quad=\quad \lambda |u|^{p-2}u \;\mbox{in}\; \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\quad+\quad\beta |u|^{p-2}u=\lambda |u|^{p-2}u \;\mbox{on}\quad\partial\Omega,$$ where $ν$ denotes the unit exterior normal, $1 \lt p \lt ∞ \,\mathrm {and} ∆_{p}u = div(|∇u|^{p−2}∇u)$ denotes the p-laplacian. $Ω ⊂ \mathbb{R}^{N}$ is a bounded domain with smooth boundary where $N ≥ 2$ and $β \in L^{∞}(∂Ω) \,\mathrm{with}\, β^{−} := \mathrm{inf}_{x∈∂Ω}β(x) > 0$. Using Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of positive eigenvalues and the first eigenvalue is simple and isolated. Moreover, we will prove that the second eigenvalue...

  19. Nonlinear Eigenvalue Problem for the p-Laplacian

    Tsouli, Najib; Chakrone, Omar; Darhouche, Omar; Rahmani, Mostafa
    This article is devoted to the study of the nonlinear eigenvalue problem $$-\Delta_{p} u \quad=\quad \lambda |u|^{p-2}u \;\mbox{in}\; \Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial \nu}\quad+\quad\beta |u|^{p-2}u=\lambda |u|^{p-2}u \;\mbox{on}\quad\partial\Omega,$$ where $ν$ denotes the unit exterior normal, $1 \lt p \lt ∞ \,\mathrm {and} ∆_{p}u = div(|∇u|^{p−2}∇u)$ denotes the p-laplacian. $Ω ⊂ \mathbb{R}^{N}$ is a bounded domain with smooth boundary where $N ≥ 2$ and $β \in L^{∞}(∂Ω) \,\mathrm{with}\, β^{−} := \mathrm{inf}_{x∈∂Ω}β(x) > 0$. Using Ljusternik-Schnirelman theory, we prove the existence of a nondecreasing sequence of positive eigenvalues and the first eigenvalue is simple and isolated. Moreover, we will prove that the second eigenvalue...

  20. Boundary Value Problems for Degenerate Coupled Systems with Variable Time Delay

    Bokalo, Mykola; Ilnytska, Olga
    The boundary value problems for coupled systems of parabolic and ordinary differential equations, where all equations contain time depended delay and degenerate at initial moment, are considered. Existence and uniqueness of classical solutions of these problems are proved. A priori estimates are obtained.

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