Constant sign solution for a simply supported beam equation

Cabada, Alberto; Saavedra, Lorena

doi:10.14232/ejqtde.2017.1.59

Cited by 7 publications

(12 citation statements)

References 13 publications

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“…Proof From the proof of Cabada and Saavedra, , theorem 6.1 the following inequalities are fulfilled for M ∈(− λ 1 ,− λ 2 ].

\begin{matrix} false(+ \infty > false) g_{M} false(t, s false) & > 0, \forall false(t, s false) \in false(a, b false) \times false(a, b false), \\ false(+ \infty > false) {\frac{\partial g_{M} false(t, s false)}{\partial t}}_{false| t = a} & > 0, \forall s \in false(a, b false), \\ false(+ \infty > false) {\frac{\partial g_{M} false(t, s false)}{\partial s}}_{false| s = a} & > 0, \forall t \in false(a, b false), \\ false(- \infty < false) {\frac{\partial g_{M} false(t, s false)}{\partial t}}_{false| t = b} & < 0, \forall s \in false(a, b false), \\ false(- \infty < false) {\frac{\partial g_{M} false(t, s false)}{\partial s}}_{false| s = b} & < 0, \forall t \in false(a, b false) . \end{matrix}

Consider

normalΦ false(s false) = false(s - a false) false(b - s false) > 0, \forall s \in false(a, b false)

…”

Section: Fixed Point Formulation and Preliminary Resultsunclassified

“…Let us choose I =[0,1], and, in Cabada and Saavedra, there are obtained the different eigenvalues of operators

T_{4}^{0} false[0 false] = \frac{d^{4}}{d t^{4}}

. λ 1 = π 4 is the least positive eigenvalue of

T_{4}^{0} false[0 false]

in X .

λ_{2} = - m_{1}^{4} double-struck≊ - 5.5 5^{4}

, where m 1 is the least positive solution of

\tan (\frac{m}{\sqrt{2}}) = \tanh (\frac{m}{\sqrt{2}})

is the biggest negative eigenvalue of

T_{4}^{0} false[0 false]

in X 1 and X 3 .

λ_{3} = m_{2}^{4} double-struck≊ 3.92 7^{4}

, with m 2 the least positive solution of

\tan (m) = \tanh (m)

is the least positive eigenvalue of

T_{4}^{0} false[0 false]

in U 1 and U 2 . …”

Section: Existence Results For Particular Simply Supported Beam Problemsmentioning

confidence: 99%

“…Now, following the characterization of the constant sign Green's function for problem , given in Cabada and Saavedra, we obtain a characterization of the parameter set where g M satisfies either the condition ( P g 1 ) or ( N g 1 ).…”

Section: Fixed Point Formulation and Preliminary Resultsmentioning

confidence: 99%

“…From the constant sign Green's function characterization related to the operator T 4 [ M ], previously defined in , in the set

X = ({}, u \in C^{4} false(I) false| u false(a false) = u^{''} false(a false) = u false(b false) = u^{''} false(b false)),

obtained in Cabada and Saavedra by means of spectral theory, it can be characterized the parameter set for which the related Green's function of T 4 [ M ] in X verifies either the property ( P g 1 ) or the analogous for the negative case: ( N g 1 )There exist Φ, k 1 and k 2 continuous functions on I such that Φ( s )>0 for all s ∈( a , b ),

0 > k_{1} false(t false) \geq k_{2} false(t false)

for all t ∈( a , b ) and

Φ (s) k_{1} (t) \geq g_{M} (t, s) \geq Φ (s) k_{2} (t), \forall (t, s) \in I \times I .

…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Existence of solutions for a nonlinear simply supported beam equation

Saavedra

2018

Math Methods in App Sciences

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This paper is devotedto prove the existence of one or multiple solutions for a family of nonlinear fourth‐order boundary value problems. We use several fixed point theorems, previously developed by the author and her coauthor, to prove the existence of solutions of some simply supported beam problems. To finish the work, a particular case is studied, and the existence of multiple solutions is proved for 2 different particular nonlinear functions.

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“…Proof From the proof of Cabada and Saavedra, , theorem 6.1 the following inequalities are fulfilled for M ∈(− λ 1 ,− λ 2 ].

\begin{matrix} false(+ \infty > false) g_{M} false(t, s false) & > 0, \forall false(t, s false) \in false(a, b false) \times false(a, b false), \\ false(+ \infty > false) {\frac{\partial g_{M} false(t, s false)}{\partial t}}_{false| t = a} & > 0, \forall s \in false(a, b false), \\ false(+ \infty > false) {\frac{\partial g_{M} false(t, s false)}{\partial s}}_{false| s = a} & > 0, \forall t \in false(a, b false), \\ false(- \infty < false) {\frac{\partial g_{M} false(t, s false)}{\partial t}}_{false| t = b} & < 0, \forall s \in false(a, b false), \\ false(- \infty < false) {\frac{\partial g_{M} false(t, s false)}{\partial s}}_{false| s = b} & < 0, \forall t \in false(a, b false) . \end{matrix}

Consider

normalΦ false(s false) = false(s - a false) false(b - s false) > 0, \forall s \in false(a, b false)

…”

Section: Fixed Point Formulation and Preliminary Resultsunclassified

“…Let us choose I =[0,1], and, in Cabada and Saavedra, there are obtained the different eigenvalues of operators

T_{4}^{0} false[0 false] = \frac{d^{4}}{d t^{4}}

. λ 1 = π 4 is the least positive eigenvalue of

T_{4}^{0} false[0 false]

in X .

λ_{2} = - m_{1}^{4} double-struck≊ - 5.5 5^{4}

, where m 1 is the least positive solution of

\tan (\frac{m}{\sqrt{2}}) = \tanh (\frac{m}{\sqrt{2}})

is the biggest negative eigenvalue of

T_{4}^{0} false[0 false]

in X 1 and X 3 .

λ_{3} = m_{2}^{4} double-struck≊ 3.92 7^{4}

, with m 2 the least positive solution of

\tan (m) = \tanh (m)

is the least positive eigenvalue of

T_{4}^{0} false[0 false]

in U 1 and U 2 . …”

Section: Existence Results For Particular Simply Supported Beam Problemsmentioning

confidence: 99%

Section: Fixed Point Formulation and Preliminary Resultsmentioning

confidence: 99%

“…From the constant sign Green's function characterization related to the operator T 4 [ M ], previously defined in , in the set

X = ({}, u \in C^{4} false(I) false| u false(a false) = u^{''} false(a false) = u false(b false) = u^{''} false(b false)),

0 > k_{1} false(t false) \geq k_{2} false(t false)

for all t ∈( a , b ) and

Φ (s) k_{1} (t) \geq g_{M} (t, s) \geq Φ (s) k_{2} (t), \forall (t, s) \in I \times I .

…”

Section: Introductionmentioning

confidence: 99%

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Existence of solutions for a nonlinear simply supported beam equation

Saavedra

2018

Math Methods in App Sciences

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“…However, unlike Drábek and Holubová, we have to reflect the fact that our solution exhibits discontinuity of the first kind in its third derivative at the points where impulses are applied. The importance of the positivity and negativity of the Green function for the study of beam equations is also stressed in previous studies . Beam equations with singular data are studied in papers but different approach as well as different functional analytic framework is used there.…”

Section: Introductionmentioning

confidence: 99%

On fourth‐order boundary value problem with singular data

Drábek

Langerová

2020

Math Methods in App Sciences

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We consider a simply supported beam with restoring and external forces given as a sum of a continuous function and a Dirac delta distribution. We present sufficient conditions on these data in order to guarantee a unique positive or negative solution, respectively. KEYWORDS beam equation, boundary value problems, boundary value problems with impulses, impulsive problem, positive and negative solutions, singular data MSC CLASSIFICATION

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