zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Non-local boundary value problems of arbitrary order. (English) Zbl 1165.34010
Many boundary value problems associated with $n$th-order differential equations may be formulated as fixed point problems for nonlinear operators involving kernels, say Green’s functions. In general, the boundary conditions at the end-points of some bounded interval $[0,1]$ may be linear, nonlinear, integral, local, nonlocal,… In this paper, the authors consider the existence of positive fixed points of the integral operator $$Tu(t)=Bu(t)+Fu(t)=:\sum_{i=1}^{i=N}\beta_i[u]\gamma_i(t)+\int_0^1k(t,s)g(s)f(s,u(s))ds,$$ under some conditions imposed on the kernel $k,$ the functions $g, \gamma_i$ and the nonlinearity $f.$ The integer $N$ lies between $0$ and the order of the underlying differential equation. The boundary operator involve linear functionals on $C[0,1]$, i.e. Stieltjes integrals $$\beta_i[u]=\int_0^1u(s)dB_j(s),$$ where $B_j$ are functions of bounded variation. These BCs contain as special cases multi-point BCs given by linear functionals $\beta_i[u]=\sum_{i=1}^{m-2}\beta_j^iu(\eta_i)$ with some $\eta_i\in(0,1).$ The $\beta_j^i$ may change sign and non-homogeneous conditions are also considered. The authors first prove cone invariance of some mappings in the space of continuous functions $C[0,1]$ and fixed point index results are obtained in theses cones. Some hypotheses on the nonlinearity $f$ including sublinear and upperlinear growths are assumed and an existence result is then derived. Also, a nonexistence result is proved. Finally, the authors illustrate their existence results by giving details for the weakly singular fourth-order equation $$u^{(4)}(t)=g(t)f(t,u(t)),\quad t\in(0,1)$$ with nonlocal BCs $$u(0)=\beta_1[u],\;u'(0)=\beta_2[u],\;u(1)=\beta_3[u],\;u''(1)=-\beta_4[u]$$ and local BCs $$ u(0)=0,\;u'(0)=0,\;u(1)=0,\;u''(1)=0$$ for which the Green’s function is easily determined. The paper ends with the case of non-homogeneous BCs.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
47H11Degree theory (nonlinear operators)
47H30Particular nonlinear operators
Full Text: DOI