n=l ,., en 門l/p, where the supremum 1s taken over all orthonormal sequences {en). Set 令 (H) ={TE ... more n=l ,., en 門l/p, where the supremum 1s taken over all orthonormal sequences {en). Set 令 (H) ={TE L(H) : IITIIP < co}. The object of this paper is to define and study Cp(X, Y) where X and Y are sequences spaces
Sean X y Y espacios de Banach y L(X,Y) el espacio de operadores lineales acotados de X en Y. El s... more Sean X y Y espacios de Banach y L(X,Y) el espacio de operadores lineales acotados de X en Y. El subespacio de operadores compactos se denota K(X,Y). Se demuestra que bajo ciertas condiciones, si K(X,Y) es un M-ideal de L(X,Y) entonces Y es un M-ideal de Y**. Ademas, si X y Y son reflexivos y K(Y,Y) es un M-ideal de L(Y,Y), entonces K(X,Y)** es isometrico a L(X,Y). Esto generaliza resultados analogos de A. Lima y P. Harmand.
British Journal of Mathematics & Computer Science, 2015
LetX be a Banach space, andE X be a non-empty closed bounded subset ofX: The setE is called proxi... more LetX be a Banach space, andE X be a non-empty closed bounded subset ofX: The setE is called proximinal inX if for allx2 X there is somee2 E such thatkx ek = inffkx yk : y2 Eg: E is called remotal in X if for all x 2 X, there exists e 2 E such thatkx ek = supfkx yk : y2 Eg: The concept of strong proximinality is well known by now in the literature, and many results were obtained. In this paper we introduce the concept of strong remotality of sets. Many results are presented.
In this paper, we answer the question that many researchers did ask us about: "what is the geomet... more In this paper, we answer the question that many researchers did ask us about: "what is the geometrical meaning of the conformable derivative?". We answer the question using the concept of fractional cords. Fractional orthogonal trajectories are also introduced. Some examples illustrating the concepts of fractional cords and fractional orthogonal trajectories are given.
A (closed) subspace $ Y$ of a Banach space $ X$ is called proximinal if for every $ x\in X$ there... more A (closed) subspace $ Y$ of a Banach space $ X$ is called proximinal if for every $ x\in X$ there exists some $ y\in Y$ such that $ \|x-y\|\le\|x-z\|$ for $ z\in Y$. It is the object of this paper is to study the proximinality of $ L^\Phi(I,Y)$ in $ L^\Phi(I,X)$ for some class of…
Journal of Mathematical and Computational Science, 2018
In this paper we study the solution of the second order fractional differential equation of the f... more In this paper we study the solution of the second order fractional differential equation of the form F(x, y, y (α) , y (2α)) = 0, in case either x is missing or in case y is missing.
In this paper we introduce the concept of remotality in topological vector spaces. Some results a... more In this paper we introduce the concept of remotality in topological vector spaces. Some results are proved.
ABSTRACT In this article, we interpolate the well-known Young’s inequality for numbers and matric... more ABSTRACT In this article, we interpolate the well-known Young’s inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.
The object of this paper is to give a new definition of the infinitesimal generator for two param... more The object of this paper is to give a new definition of the infinitesimal generator for two parameter semigroups. We prove a result for such generators that gives known results in the literature as special cases.
European Journal of Pure and Applied Mathematics, 2021
In this paper, we find a solution of finite rank form of fractional Cauchy Problem. The fractiona... more In this paper, we find a solution of finite rank form of fractional Cauchy Problem. The fractional derivative used is the Conformable derivative. The main idea of the proofs are based on theory of tensor product of Banach spaces.
In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauch... more In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauchy problem with Caputo conformable derivative in Banach spaces. We establish a representation of a mild solution using a fractional Laplace transform. The existence of such solutions is proved under certain conditions, using the Mönch fixed point theorem and a general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non compactness. Applications illustrating our main abstract results and showing the applicability of the presented theory are also given.
Mathematical Proceedings of the Cambridge Philosophical Society, 1988
Let X, Y be Banach spaces and G a closed subspace of Y. Let L(X, Y) be the space of bounded linea... more Let X, Y be Banach spaces and G a closed subspace of Y. Let L(X, Y) be the space of bounded linear operators from X into Y. In this paper we investigate when L(X, G) is proximal in L(X, Y). Further, we discuss the related problem of proximinality of L∞(T, G) in L∞(T, Y). We improve results obtained by Light and Cheney in this direction.
Bulletin of the Australian Mathematical Society, 1986
For a modulus function φ, we define the Hardy-Orlicz space H (φ). Two main questions are discusse... more For a modulus function φ, we define the Hardy-Orlicz space H (φ). Two main questions are discussed in this paper. First, when is a linear map mg: H (φ) → H (φ), mg (f) = g.f an isometry? Second, when is H (φ) = H1?
Journal of Semigroup Theory and Applications, 2020
Some times it is not easy to find the exact solution of certain partial differential equations. I... more Some times it is not easy to find the exact solution of certain partial differential equations. In this paper we use tensor product technique of Banach spaces to find certain solutions of certain fractional differential equations.
If every point in a normed space X admits a unique farthest point in a given bounded subset E, th... more If every point in a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton?. This is known as the farthest point problem. In an attempt to solve this problem, we give our contribution toward solving it, in the positive direction, by proving that every such subset E in the sequence space ℓ 1 is a singleton.
Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X. B1(X) d... more Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X. B1(X) denotes the closed unit ball of X, and S1(X )i s unit sphere of X. An element T ∈ S1(L(X)) is called extreme operator if there is no A ∈ L(X) such thatT ± A �≤ 1. The set of extreme points of S1(X) will be denoted by ext(S1(X)) .T ∈ S1(L(X)) is called nice if T ∗ (ext(S1(L(X ∗ ))) ⊆ ext(S1(L(X ∗ ))). The object of this paper is to give simpler proofs of old results and present new results on extreme operators of S1(L(� p )). We introduce the concept of k-extreme points. Further, we characterize the nice operators on most of the classical function and sequence spaces. Nice compact operators onp −spaces are characterized. I. Introduction. Let X be a Banach space. The closed unit ball of X will be denoted by B1(X) , and the unit sphere by S1(X). An element x ∈ S1(X) is called an extreme point of B1(X) if whenever x = 1 (y + z), with y and z in S1(X), then x = y = z. The space of bounded line...
The object of this paper is to discuss the Abstract Cauchy Problem using tensor product technique... more The object of this paper is to discuss the Abstract Cauchy Problem using tensor product technique. We give two rank solution of the problem when the Banach space is a Hilbert space.
n=l ,., en 門l/p, where the supremum 1s taken over all orthonormal sequences {en). Set 令 (H) ={TE ... more n=l ,., en 門l/p, where the supremum 1s taken over all orthonormal sequences {en). Set 令 (H) ={TE L(H) : IITIIP < co}. The object of this paper is to define and study Cp(X, Y) where X and Y are sequences spaces
Sean X y Y espacios de Banach y L(X,Y) el espacio de operadores lineales acotados de X en Y. El s... more Sean X y Y espacios de Banach y L(X,Y) el espacio de operadores lineales acotados de X en Y. El subespacio de operadores compactos se denota K(X,Y). Se demuestra que bajo ciertas condiciones, si K(X,Y) es un M-ideal de L(X,Y) entonces Y es un M-ideal de Y**. Ademas, si X y Y son reflexivos y K(Y,Y) es un M-ideal de L(Y,Y), entonces K(X,Y)** es isometrico a L(X,Y). Esto generaliza resultados analogos de A. Lima y P. Harmand.
British Journal of Mathematics & Computer Science, 2015
LetX be a Banach space, andE X be a non-empty closed bounded subset ofX: The setE is called proxi... more LetX be a Banach space, andE X be a non-empty closed bounded subset ofX: The setE is called proximinal inX if for allx2 X there is somee2 E such thatkx ek = inffkx yk : y2 Eg: E is called remotal in X if for all x 2 X, there exists e 2 E such thatkx ek = supfkx yk : y2 Eg: The concept of strong proximinality is well known by now in the literature, and many results were obtained. In this paper we introduce the concept of strong remotality of sets. Many results are presented.
In this paper, we answer the question that many researchers did ask us about: "what is the geomet... more In this paper, we answer the question that many researchers did ask us about: "what is the geometrical meaning of the conformable derivative?". We answer the question using the concept of fractional cords. Fractional orthogonal trajectories are also introduced. Some examples illustrating the concepts of fractional cords and fractional orthogonal trajectories are given.
A (closed) subspace $ Y$ of a Banach space $ X$ is called proximinal if for every $ x\in X$ there... more A (closed) subspace $ Y$ of a Banach space $ X$ is called proximinal if for every $ x\in X$ there exists some $ y\in Y$ such that $ \|x-y\|\le\|x-z\|$ for $ z\in Y$. It is the object of this paper is to study the proximinality of $ L^\Phi(I,Y)$ in $ L^\Phi(I,X)$ for some class of…
Journal of Mathematical and Computational Science, 2018
In this paper we study the solution of the second order fractional differential equation of the f... more In this paper we study the solution of the second order fractional differential equation of the form F(x, y, y (α) , y (2α)) = 0, in case either x is missing or in case y is missing.
In this paper we introduce the concept of remotality in topological vector spaces. Some results a... more In this paper we introduce the concept of remotality in topological vector spaces. Some results are proved.
ABSTRACT In this article, we interpolate the well-known Young’s inequality for numbers and matric... more ABSTRACT In this article, we interpolate the well-known Young’s inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey.
The object of this paper is to give a new definition of the infinitesimal generator for two param... more The object of this paper is to give a new definition of the infinitesimal generator for two parameter semigroups. We prove a result for such generators that gives known results in the literature as special cases.
European Journal of Pure and Applied Mathematics, 2021
In this paper, we find a solution of finite rank form of fractional Cauchy Problem. The fractiona... more In this paper, we find a solution of finite rank form of fractional Cauchy Problem. The fractional derivative used is the Conformable derivative. The main idea of the proofs are based on theory of tensor product of Banach spaces.
In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauch... more In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauchy problem with Caputo conformable derivative in Banach spaces. We establish a representation of a mild solution using a fractional Laplace transform. The existence of such solutions is proved under certain conditions, using the Mönch fixed point theorem and a general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non compactness. Applications illustrating our main abstract results and showing the applicability of the presented theory are also given.
Mathematical Proceedings of the Cambridge Philosophical Society, 1988
Let X, Y be Banach spaces and G a closed subspace of Y. Let L(X, Y) be the space of bounded linea... more Let X, Y be Banach spaces and G a closed subspace of Y. Let L(X, Y) be the space of bounded linear operators from X into Y. In this paper we investigate when L(X, G) is proximal in L(X, Y). Further, we discuss the related problem of proximinality of L∞(T, G) in L∞(T, Y). We improve results obtained by Light and Cheney in this direction.
Bulletin of the Australian Mathematical Society, 1986
For a modulus function φ, we define the Hardy-Orlicz space H (φ). Two main questions are discusse... more For a modulus function φ, we define the Hardy-Orlicz space H (φ). Two main questions are discussed in this paper. First, when is a linear map mg: H (φ) → H (φ), mg (f) = g.f an isometry? Second, when is H (φ) = H1?
Journal of Semigroup Theory and Applications, 2020
Some times it is not easy to find the exact solution of certain partial differential equations. I... more Some times it is not easy to find the exact solution of certain partial differential equations. In this paper we use tensor product technique of Banach spaces to find certain solutions of certain fractional differential equations.
If every point in a normed space X admits a unique farthest point in a given bounded subset E, th... more If every point in a normed space X admits a unique farthest point in a given bounded subset E, then must E be a singleton?. This is known as the farthest point problem. In an attempt to solve this problem, we give our contribution toward solving it, in the positive direction, by proving that every such subset E in the sequence space ℓ 1 is a singleton.
Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X. B1(X) d... more Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X. B1(X) denotes the closed unit ball of X, and S1(X )i s unit sphere of X. An element T ∈ S1(L(X)) is called extreme operator if there is no A ∈ L(X) such thatT ± A �≤ 1. The set of extreme points of S1(X) will be denoted by ext(S1(X)) .T ∈ S1(L(X)) is called nice if T ∗ (ext(S1(L(X ∗ ))) ⊆ ext(S1(L(X ∗ ))). The object of this paper is to give simpler proofs of old results and present new results on extreme operators of S1(L(� p )). We introduce the concept of k-extreme points. Further, we characterize the nice operators on most of the classical function and sequence spaces. Nice compact operators onp −spaces are characterized. I. Introduction. Let X be a Banach space. The closed unit ball of X will be denoted by B1(X) , and the unit sphere by S1(X). An element x ∈ S1(X) is called an extreme point of B1(X) if whenever x = 1 (y + z), with y and z in S1(X), then x = y = z. The space of bounded line...
The object of this paper is to discuss the Abstract Cauchy Problem using tensor product technique... more The object of this paper is to discuss the Abstract Cauchy Problem using tensor product technique. We give two rank solution of the problem when the Banach space is a Hilbert space.
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