A note on the fractional calculus in Banach spaces

2005 ◽  
Vol 42 (2) ◽  
pp. 115-130 ◽  
Author(s):  
Hussein A. H. Salem ◽  
A. M. A. El-Sayed ◽  
O. L. Moustafa
Keyword(s):  
Integral Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Existence Result ◽  
Existence Of Weak Solutions ◽  
Weakly Continuous ◽  
The Cauchy Problem ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Fractional Order Integral

O'Regan fixed point theorem is used to establish an existence result for the fractional order integral equation x(t) = g(t)+ ?Ia f(.,x(.))(t), t?[0,1], a ? 0, where the vector-valued function f  is nonlinear weakly-weakly continuous. Moreover, existence of weak solutions to the Cauchy problem  dx/dt = f(t, x (t)), t ? [0,1], x(0) = x0, is obtained as a corollary.

scholarly journals Weak solutions of differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 407-418 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Continuous Vector ◽  
Weakly Continuous ◽  
Continuous Vector Field ◽  
The Cauchy Problem

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

scholarly journals Time decay for solutions to the Stokes equations with drift

2018 ◽  
Vol 20 (03) ◽  
pp. 1750046 ◽  
Author(s):  
M. Schonbek ◽  
G. Seregin
Keyword(s):  
Cauchy Problem ◽  
Stokes System ◽  
Stokes Equations ◽  
Time Decay ◽  
Smooth Vector ◽  
Scale Invariant ◽  
The Cauchy Problem ◽  
Divergence Free ◽  
Vector Valued Function ◽  
Vector Valued

In this note, we study the behavior of Lebesgue norms [Formula: see text] of solutions [Formula: see text] to the Cauchy problem for the Stokes system with drift [Formula: see text], which is supposed to be a divergence free smooth vector-valued function satisfying a scale invariant condition.

scholarly journals On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations

2015 ◽  
Vol 7 (1) ◽  
pp. 101-107 ◽  
Author(s):  
M.M. Osypchuk
Keyword(s):  
Cauchy Problem ◽  
Differential Operator ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Stable Process ◽  
Pseudo Differential Operator ◽  
The Cauchy Problem ◽  
Vector Valued Function ◽  
Vector Valued

A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric $\alpha$-stable process in multidimensional Euclidean space. Its generator $\mathbf{A}$ is a pseudo-differential operator whose symbol is given by $-c|\lambda|^\alpha$, were the constants $\alpha\in(1,2)$ and $c>0$ are fixed. The vector-valued operator $\mathbf{B}$ has the symbol $2ic|\lambda|^{\alpha-2}\lambda$. We construct a fundamental solution of the equation $u_t=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$ with a continuous bounded vector-valued function $a$.

scholarly journals A Study of a Nonlinear Ordinary Differential Equation in Modular Function Spaces Endowed with a Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jaauad Jeddi ◽  
Mustapha Kabil ◽  
Samih Lazaiz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Existence Result ◽  
Modular Function ◽  
Nonlinear Ordinary Differential Equation ◽  
Function Spaces ◽  
The Cauchy Problem ◽  
Modular Function Spaces

In this paper, we prove by means of a fixed-point theorem an existence result of the Cauchy problem associated to an ordinary differential equation in modular function spaces endowed with a reflexive convex digraph.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Regularity properties of Schrödinger equations in vector-valued spaces and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 149-166
Author(s):  
Veli Shakhmurov
Keyword(s):  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
The Cauchy Problem ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

Abstract In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A, we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems.

scholarly journals SOLVABILITY OF THE CAUCHY PROBLEM FOR EQUATIONS WITH RIEMANN–LIOUVILLE’S FRACTIONAL DERIVATIVES

2018 ◽  
Vol 62 (4) ◽  
pp. 391-397 ◽  
Author(s):  
Petr P. Zabreiko ◽  
Svetlana V. Ponomareva
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Unique Solution ◽  
Metric Space ◽  
Fractional Derivatives ◽  
Volterra Equation ◽  
Complete Metric Space ◽  
The Cauchy Problem ◽  
The Right

In this article we study the solvability of the analogue of the Cauchy problem for ordinary differential equations with Riemann–Liouville’s fractional derivatives with a nonlinear restriction on the right-hand side of functions in certain spaces. The conditions for solvability of the problem under consideration in given function spaces, as well as the conditions for existence of a unique solution are given. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.

Global well-posedness of the Cauchy problem for the 3D Jordan–Moore–Gibson–Thompson equation

2020 ◽  
pp. 2050069
Author(s):  
Reinhard Racke ◽  
Belkacem Said-Houari
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Small Data ◽  
Third Order ◽  
Bootstrap Argument ◽  
The Cauchy Problem ◽  
Linear Decay

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained.

scholarly journals Well-posedness of stochastic modified Kawahara equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
P. Agarwal ◽  
Abd-Allah Hyder ◽  
M. Zakarya
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Water Wave ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
Restriction Method ◽  
Fifth Order ◽  
Point Argument

AbstractIn this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in $H^{s}(\mathbb{R})$Hs(R), $s\geq -1/4$s≥−1/4. Moreover, we get the global existence for $L^{2}( \mathbb{R})$L2(R) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

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