Solvability of a generalized third-order left focal problem at resonance in Banach spaces

Youwei Zhang

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 4, page 361-382
  • ISSN: 0862-7959

Abstract

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This paper deals with the generalized nonlinear third-order left focal problem at resonance ( p ( t ) u ' ' ( t ) ) ' - q ( t ) u ( t ) = f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , t ∈ ] t 0 , T [ , m ( u ( t 0 ) , u ' ' ( t 0 ) ) = 0 , n ( u ( T ) , u ' ( T ) ) = 0 , l ( u ( ξ ) , u ' ( ξ ) , u ' ' ( ξ ) ) = 0 , where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the topological degree technique as well as some recent generalizations of this technique. Our results are generalizations and extensions of the results of several authors. An application is included to illustrate the results obtained.

How to cite

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Zhang, Youwei. "Solvability of a generalized third-order left focal problem at resonance in Banach spaces." Mathematica Bohemica 138.4 (2013): 361-382. <http://eudml.org/doc/260695>.

@article{Zhang2013,
abstract = {This paper deals with the generalized nonlinear third-order left focal problem at resonance \[ \{\left\lbrace \begin\{array\}\{ll\} (p(t)u^\{\prime \prime \}(t))^\{\prime \}-q(t)u(t)=f(t, u(t), u^\{\prime \}(t), u^\{\prime \prime \}(t)), \quad t\in \mathopen ]t\_0, T[, m(u(t\_0), u^\{\prime \prime \}(t\_0))=0, n(u(T), u^\{\prime \}(T))=0, l(u(\xi ), u^\{\prime \}(\xi ), u^\{\prime \prime \}(\xi ))=0, \end\{array\}\right.\} \] where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the topological degree technique as well as some recent generalizations of this technique. Our results are generalizations and extensions of the results of several authors. An application is included to illustrate the results obtained.},
author = {Zhang, Youwei},
journal = {Mathematica Bohemica},
keywords = {Fredholm operator; coincidence degree; left focal problem; nontrivial solution; resonance; Fredholm operator; coincidence degree; left focal problem; resonance},
language = {eng},
number = {4},
pages = {361-382},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solvability of a generalized third-order left focal problem at resonance in Banach spaces},
url = {http://eudml.org/doc/260695},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Zhang, Youwei
TI - Solvability of a generalized third-order left focal problem at resonance in Banach spaces
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 4
SP - 361
EP - 382
AB - This paper deals with the generalized nonlinear third-order left focal problem at resonance \[ {\left\lbrace \begin{array}{ll} (p(t)u^{\prime \prime }(t))^{\prime }-q(t)u(t)=f(t, u(t), u^{\prime }(t), u^{\prime \prime }(t)), \quad t\in \mathopen ]t_0, T[, m(u(t_0), u^{\prime \prime }(t_0))=0, n(u(T), u^{\prime }(T))=0, l(u(\xi ), u^{\prime }(\xi ), u^{\prime \prime }(\xi ))=0, \end{array}\right.} \] where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions by using the topological degree technique as well as some recent generalizations of this technique. Our results are generalizations and extensions of the results of several authors. An application is included to illustrate the results obtained.
LA - eng
KW - Fredholm operator; coincidence degree; left focal problem; nontrivial solution; resonance; Fredholm operator; coincidence degree; left focal problem; resonance
UR - http://eudml.org/doc/260695
ER -

References

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