scholarly journals Existence of solutions to quasilinear differential equations in a Banach space

1976 ◽  
Vol 15 (3) ◽  
pp. 421-430
Author(s):  
James R. Ward
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Linear Systems ◽  
Function Space ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Reflexive Banach Space ◽  
Initial Value ◽  
Suitable Function

Initial value problems of the form x′ + A(t, x)x = f(t, x), x(0) = a, t ≥ 0, are considered in a real, separable, reflexive Banach space. Results concerning the existence of solutions on (0, ∞) are given by considering linear systems of the form x′ + A(t, u(t))x = f(t, u(t)). Here u(t) belongs to a suitable function space.

Solvability of initial value problems with fractional order differential equations in banach spaces by α-dense curves

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Gonzalo García
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Initial Value ◽  
Measures Of Noncompactness ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

AbstractIn this paper we study the existence of solutions for an initial value problem, posed in a given Banach space, with a fractional differential equation via densifiability techniques. For our goal, we will prove a new fixed point result (not based on measures of noncompactness) which is, in forms, a generalization of the well-known Darbo’s fixed point theorem but essentially different. Some illustrative examples are given.

scholarly journals On the correct formulation of a nonlinear differential equations in Banach space

2001 ◽  
Vol 26 (7) ◽  
pp. 437-444
Author(s):  
Mahmoud M. El-Borai ◽  
Osama L. Moustafa ◽  
Fayez H. Michael
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Nonlinear Differential Equations ◽  
Initial Value ◽  
Correct Formulation

We study, the existence and uniqueness of the initial value problems in a Banach spaceEfor the abstract nonlinear differential equation(dn−1/dtn−1)(du/dt+Au)=B(t)u+f(t,W(t)), and consider the correct solution of this problem. We also give an application of the theory of partial differential equations.

Existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Tiberiu Trif
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Growth Conditions ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equations ◽  
Global Existence Of Solutions ◽  
Fractional Differential

AbstractThe purpose of the paper is to investigate the global existence of solutions to initial value problems for nonlinear fractional differential equations on the semi-axis. More precisely, it deals with the initial value problem (*)$\left\{ \begin{gathered} D_{0 + }^\alpha x(t) = f(t,x(t)),t \in [0,\infty ], \hfill \\ \lim _{t \to 0 + } t^{1 - \alpha } x(t) = x_0 , \hfill \\ \end{gathered} \right. $ where 0 < α < 1, D 0+α denotes the Riemann-Liouville fractional derivative of order α, and f: (0,∞) × ℝ → ℝ is a continuous function. Unlike all the previous papers dealing with the problem of existence of solutions to (*), this problem is solved here by constructing a special locally convex space which is metrizable and complete. Then Schauder’s fixed point theorem enables to provide sufficient conditions on f, ensuring that (*) possesses at least one solution. The growth conditions imposed to f are weaker than other similar conditions already used in the literature.

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