Non-well-ordered lower and upper solutions for semilinear systems of PDEs

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

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Infinite Boundary Value Problems for Second-Order Nonlinear Impulsive Differential Equations with Supremum

Discrete Dynamics in Nature and Society ◽

10.1155/2010/393757 ◽

2010 ◽

Vol 2010 ◽

pp. 1-15

Author(s):

Piao-Piao Shi ◽

Wen-Xia Wang

Keyword(s):

Differential Equations ◽

Boundary Value Problems ◽

Solution Method ◽

Boundary Value ◽

Impulsive Differential Equations ◽

Existence Results ◽

Second Order ◽

Comparison Result ◽

Upper Solution ◽

Lower And Upper Solution

We investigate the infinite boundary value problems for second-order impulsive differential equations with supremum by establishing a new comparison result and using the lower and upper solution method, and obtain the existence results for their maximal and minimal solutions.

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Existence results for second-order boundary-value problems with variable exponents

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2018.04.003 ◽

2018 ◽

Vol 44 ◽

pp. 40-53 ◽

Cited By ~ 1

Author(s):

Shapour Heidarkhani ◽

Shahin Moradi ◽

David Barilla

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Existence Results ◽

Second Order ◽

Variable Exponents

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Boundary value problems on noncompact intervals

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500030341 ◽

1995 ◽

Vol 125 (4) ◽

pp. 777-799 ◽

Cited By ~ 3

Author(s):

Donal O'Regan

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Boundary Value Problems ◽

Boundary Value ◽

Existence Results ◽

Second Order

Existence results are established for second-order boundary value problems for ordinary differential equations on non-compact intervals.

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Difference equations in abstract spaces

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700001762 ◽

1998 ◽

Vol 64 (2) ◽

pp. 277-284 ◽

Cited By ~ 17

Author(s):

Ravi P. Agarwal ◽

Donal O'Regan

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Value Problems ◽

Point Theorem ◽

Difference Equations ◽

Boundary Value ◽

Existence Results ◽

Second Order ◽

Abstract Spaces

AbstractExistence results are presented for second order discrete boundary value problems in abstract spaces. Our analysis uses only Sadovskii's fixed point theorem.

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Existence results for second order boundary value problems using the barrier strip technique

Aequationes Mathematicae ◽

10.1007/s00010-010-0019-0 ◽

2010 ◽

Vol 79 (3) ◽

pp. 281-291

Author(s):

P. S. Kelevedjiev ◽

D. O’Regan ◽

N. Popivanov ◽

R. P. Agarwal

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Existence Results ◽

Second Order

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Nonmonotone impulse effects in second-order periodic boundary value problems

Abstract and Applied Analysis ◽

10.1155/s1085337504306299 ◽

2004 ◽

Vol 2004 (7) ◽

pp. 577-590 ◽

Cited By ~ 3

Author(s):

Irena Rachůnková ◽

Milan Tvrdý

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Periodic Boundary ◽

Boundary Value ◽

Existence Results ◽

Second Order ◽

Periodic Boundary Value Problem ◽

Lower Upper ◽

Periodic Boundary Value Problems

We deal with the nonlinear impulsive periodic boundary value problemu″=f(t,u,u′),u(ti+)=Ji(u(ti)),u′(ti+)=Mi(u′(ti)),i=1,2,…,m,u(0)=u(T),u′(0)=u′(T). We establish the existence results which rely on the presence of a well-ordered pair(σ1,σ2)of lower/upper functions(σ1≤σ2 on [0,T])associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functionsJi,Miis not required here.

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Existence results for boundary value problems of second order impulsive differential equations

Journal of Mathematical Analysis and Applications ◽

10.1016/0022-247x(90)90285-n ◽

1990 ◽

Vol 149 (1) ◽

pp. 56-69 ◽

Cited By ~ 36

Author(s):

L.H. Erbe ◽

Xinzhi Liu

Keyword(s):

Differential Equations ◽

Boundary Value Problems ◽

Boundary Value ◽

Impulsive Differential Equations ◽

Existence Results ◽

Second Order

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Examples of the nonexistence of a solution in the presence of upper and lower solutions

The ANZIAM Journal ◽

10.1017/s1446181100012955 ◽

2003 ◽

Vol 44 (4) ◽

pp. 591-594 ◽

Cited By ~ 13

Author(s):

Patrick Habets ◽

Rodrigo L. Pouso

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Boundary Value Problems ◽

Boundary Value ◽

Upper And Lower Solutions ◽

Second Order ◽

Lower And Upper Solutions ◽

Existence Of A Solution ◽

Nagumo Condition

AbstractStandard results for boundary value problems involving second-order ordinary differential equations ensure that the existence of a well-ordered pair of lower and upper solutions together with a Nagumo condition imply existence of a solution. In this note we introduce some examples which show that existence is not guaranteed if no Nagumo condition is satisfied.

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