Uniqueness Implies Existence and Uniqueness Conditions for a Class of (k + j)-Point Boundary Value Problems for n-th Order Differential Equations

2012 ◽  
Vol 55 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Paul W. Eloe ◽  
Johnny Henderson ◽  
Rahmat Ali Khan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Differential Equation ◽  
Unique Solvability ◽  
Existence And Uniqueness ◽  
Existence Results ◽  
Point Boundary ◽  
Uniqueness And Existence ◽  
Uniqueness And Existence Results

AbstractFor the n-th order nonlinear differential equation, y(n) = f (x, y, y′, … , y(n–1)), we consider uniqueness implies uniqueness and existence results for solutions satisfying certain (k + j)-point boundary conditions for 1 ≤ j ≤ n – 1 and 1 ≤ k ≤ n – j. We define (k; j)-point unique solvability in analogy to k-point disconjugacy and we show that (n – j0; j0)-point unique solvability implies (k; j)-point unique solvability for 1 ≤ j ≤ j0, and 1 ≤ k ≤ n – j. This result is analogous to n-point disconjugacy implies k-point disconjugacy for 2 ≤ k ≤ n – 1.

Unique Solution for Multi-point Fractional Integro-Differential Equations

2020 ◽  
Vol 21 (2) ◽  
pp. 219-226
Author(s):  
Chengbo Zhai ◽  
Lifang Wei
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Iterative Sequence ◽  
Point Boundary ◽  
Uniqueness Of Solutions

AbstractWe study a fractional integro-differential equation subject to multi-point boundary conditions: $$\left\{\begin{array}{l} D^\alpha_{0^+} u(t)+f(t,u(t),Tu(t),Su(t))=b,\ t\in(0,1),\\u(0)=u^\prime(0)=\cdots=u^{(n-2)}(0)=0,\\ D^p_{0^+}u(t)|_{t=1}=\sum\limits_{i=1}^m a_iD^q_{0^+}u(t)|_{t=\xi_i},\end{array}\right.$$where $\alpha\in (n-1,n],\ n\in \textbf{N},\ n\geq 3,\ a_i\geq 0,\ 0<\xi_1<\cdots<\xi_m\leq 1,\ p\in [1,n-2],\ q\in[0,p],b>0$. By utilizing a new fixed point theorem of increasing $\psi-(h,r)-$ concave operators defined on special sets in ordered spaces, we demonstrate existence and uniqueness of solutions for this problem. Besides, it is shown that an iterative sequence can be constructed to approximate the unique solution. Finally, the main result is illustrated with the aid of an example.

scholarly journals Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

2018 ◽  
Vol 23 (5) ◽  
pp. 771-801 ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Existence Results ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Existence And Nonexistence ◽  
Fractional Differential

>We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

scholarly journals Existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions

2018 ◽  
Vol 1 (1) ◽  
pp. 21-36 ◽  
Author(s):  
Mısır J. Mardanov ◽  
Yagub A. Sharifov ◽  
Kamala E. Ismayilova
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Point Boundary ◽  
Uniqueness Of Solutions

AbstractThis paper is devoted to a system of nonlinear impulsive differential equations with three-point boundary conditions. The Green function is constructed and considered original problem is reduced to the equivalent impulsive integral equations. Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the problem and Schaefer’s fixed point theorem is used to prove the existence of a solution of the problem under consideration. We illustrate the application of the main results by two examples.

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.

scholarly journals Monotone iterative method for fractional p-Laplacian differential equations with four-point boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoping Li ◽  
Minyuan He
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Iterative Method ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Positive Solutions ◽  
Monotone Iterative Method ◽  
Point Boundary ◽  
Abstract Result ◽  
Monotone Iterative

AbstractA four-point boundary problem for a fractional p-Laplacian differential equation is studied. The existence of two positive solutions is established by means of the monotone iterative method. An example supporting the abstract result is given.

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