scholarly journals On Second-Order Differential Equations with Nonsmooth Second Member

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
M. Milla Miranda ◽  
A. T. Lourêdo ◽  
L. A. Medeiros
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Initial Value Problem ◽  
Positive Function ◽  
Linear Operators ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Open Question ◽  
Abstract Framework

In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), u(0)=u0,u′(0)=u1, where μ is a positive function and f a nonsmooth function. Given u0, u1, and f we determine Fu in order to have a solution u of the previous equation. We analyze two cases of Fu. In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).

scholarly journals Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

2021 ◽  
Vol 8 (1) ◽  
pp. 87-100
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Jamal Eddine Lazreg ◽  
Juan J. Nieto ◽  
Yong Zhou
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Banach Algebras ◽  
Existence Results ◽  
Initial Value ◽  
Hilfer Fractional Derivative ◽  
Implicit Differential Equations

Abstract In this paper, we prove some existence results of solutions for a class of nonlocal initial value problem for nonlinear fractional hybrid implicit differential equations under generalized Hilfer fractional derivative. The result is based on a fixed point theorem on Banach algebras. Further, examples are provided to illustrate our results.

scholarly journals Linear Hyperbolic Functional-Differential Equations with Essentially Bounded Right-Hand Side

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Alexander Domoshnitsky ◽  
Alexander Lomtatidze ◽  
Abraham Maghakyan ◽  
Jiří Šremr
Keyword(s):  
Differential Equations ◽  
Unique Solvability ◽  
Initial Value Problem ◽  
Hyperbolic Equations ◽  
Functional Differential Equations ◽  
Linear Operators ◽  
Initial Value ◽  
Bounded Functions ◽  
Functional Differential ◽  
Characteristic Initial Value Problem

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problemu1,1(t,x)=l0(u)(t,x)+l1(u1,0)(t,x)+l2(u0,1)(t,x)+q(t,x),  u(t,c)=α(t)fort∈[a,b], u(a,x)=β(x)  for  x∈[c,d]given on the rectangle[a,b]×[c,d]are established, where the linear operatorsl0,l1,l2map suitable function spaces into the space of essentially bounded functions. General results are applied to the hyperbolic equations with essentially bounded coefficients and argument deviations.

On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ashwini D. Mali ◽  
Kishor D. Kucche ◽  
José Vanterler da Costa Sousa
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Initial Value ◽  
Fractional Differential

Abstract This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results.

Initial Value Problem of Fractional Integro-Differential Equations in Banach Space

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Adel Jawahdou
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Initial Value Problem ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Initial Value

AbstractThis paper is devoted to study the existence of solutions of nonlinear fractional integro-differential equation, via the techniques of measure of noncompactness. The investigation is based on a Schauder's fixed point theorem. The main result is less restrictive than those given in the literature. An illustrative example is given.

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