scholarly journals Positive Solutions for Nonlinear Differential Equations with Periodic Boundary Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Shengjun Li ◽  
Li Liang ◽  
Zonghu Xiu
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Positive Solutions ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Nonlinear Differential Equations ◽  
Nonseparated Boundary Conditions ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Alternative Principle

We study the existence of positive solutions for second-order nonlinear differential equations with nonseparated boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved.

scholarly journals The Existence of Positive Solutions for Singular Impulse Periodic Boundary Value Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Zhaocai Hao ◽  
Tanggui Chen
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Value Problem ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Alternative Principle

We obtain new result of the existence of positive solutions of a class of singular impulse periodic boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require the monotonicity of functions in paper (Zhang and Wang, 2003). An example is also given to illustrate our result.

scholarly journals Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Shengjun Li ◽  
Fang Zhang
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Multiple Positive Solutions ◽  
Difference Systems ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Alternative Principle

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

scholarly journals Existence and Multiplicity Results for Nonlinear Differential Equations Depending on a Parameter in Semipositone Case

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Hailong Zhu ◽  
Shengjun Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Multiplicity Results ◽  
Second Order Differential Equations ◽  
Existence And Multiplicity ◽  
Nonlinear Alternative ◽  
Krasnosel’Skii’S Fixed Point Theorem ◽  
Alternative Principle

The existence and multiplicity of solutions for second-order differential equations with a parameter are discussed in this paper. We are mainly concerned with the semipositone case. The analysis relies on the nonlinear alternative principle of Leray-Schauder and Krasnosel'skii's fixed point theorem in cones.

Optimality of Hyperbolic Partial Differential Equations With Dynamically Constrained Periodic Boundary Control—A Flow Control Application

2006 ◽  
Vol 128 (4) ◽  
pp. 946-959 ◽  
Author(s):  
Nhan Nguyen ◽  
Mark Ardema
Keyword(s):  
Boundary Condition ◽  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Control Problem ◽  
Boundary Control ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential

This paper is concerned with optimal control of a class of distributed-parameter systems governed by first-order, quasilinear hyperbolic partial differential equations that arise in optimal control problems of many physical systems such as fluids dynamics and elastodynamics. The distributed system is controlled via a forced nonlinear periodic boundary condition that describes a boundary control action. Further, the periodic boundary control is subject to a dynamic constraint imposed by a lumped-parameter system governed by ordinary differential equations that model actuator dynamics. The partial differential equations are thus coupled with the ordinary differential equations via the periodic boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to solve a feedback control problem of the Mach number in a wind tunnel.

Existence of Positive Solutions to the Singular Initial Value Problems of Fractional Differential Equations

2011 ◽  
Vol 403-408 ◽  
pp. 563-569
Author(s):  
Qiu Ping Li ◽  
Shu Rong Sun ◽  
Zhen Lai Han ◽  
Yi Ge Zhao
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

In this paper, we consider the existence of positive solutions for the initial value problem of nonlinear fractional differential equations where and is the Riemann–Liouville fractional derivative. By using the Nonlinear Alternative of Leray and Schauder theorem, some sufficient conditions for the existence of at least one positive solution for the initial value problem are established.

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