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Start Over Subject upper solution

1. Boundary value problem with an inner point for the singularly perturbed semilinear differential equations

Creator:
Vrábeľ, Róbert
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
singular perturbation, boundary value problem, upper solution, and lower solution
Language:
English
Description:
In this paper we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem εy ′′ + ky = f(t, y), t ∈ ha,bi, k < 0, 0 < ε ≪ 1 satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and delicate estimations.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

Creator:
Mawhin, Jean and Szymańska-Dębowska, Katarzyna
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
nonlocal boundary value problem, lower solution, upper solution, lower slope, upper slope, and Leray-Schauder degree
Language:
English
Description:
A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem x ′′ = f(t, x, x′ ), x(0) = 0, x ′ (1) = ∫ 1 0 x ′ (s) dg(s), with g increasing on [0, 1] such that ∫ 1 0 dg = 1, is a couple of functions σ, τ ∈ C 1 ([0, 1]) such that σ(t) ≤ τ (t) for all t ∈ [0, 1], σ ′ (t) ≥ f(t, x, σ(t)), σ(1) ≤ ∫ 1 0 σ(s) dg(s), τ ′ (t) ≤ f(t, x, τ (t)), τ (1) ≥ ∫ 1 0 τ (s) dg(s), in the stripe ∫ t 0 σ(s) ds ≤ x ≤ ∫ t 0 τ (s) ds and t ∈ [0, 1]. It is proved that the existence of such a couple (σ, τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Nonlinear elliptic differential equations with multivalued nonlinearities

Creator:
Fiacca, Antonella, Matzakos, Nikolas, Papgeorgiou, Nikolas S., and Servadei, Raffaella
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
upper solution, lower solution, order interval, truncation function, pseudomonotone operator, coercive operator, extremal solution, Yosida approximation, nonsmooth Palais-Smale condition, critical point, and eigenvalue problem
Language:
English
Description:
In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).
Rights:
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4. Periodic problems and problems with discontinuities for nonlinear parabolic equations

Creator:
Cardinali, Tiziana and Papageorgiou, Nikolaos S.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
pseudomonotone operator, $L$-pseudomonotonicity, operator of type $(S)_{+}$, operator of type $L$-$(S)_{+}$, coercive operator, surjective operator, evolution triple, compact embedding, multifunction, upper solution, lower solution, extremal solution, and $R_{\delta }$-set
Language:
English
Description:
In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact $R_{\delta }$-set in $(CT,L^2(Z))$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

5. The method of upper and lower solutions for a Lidstone boundary value problem

Creator:
Guo, Yanping and Gao, Ying
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
$n$-parameter eigenvalue problem, Lidstone boundary value problem, lower solution, and upper solution
Language:
English
Description:
In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 0<t<1, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] where $f\:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public