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Zbl 1165.34010
Webb, J.R.L.; Infante, Gennaro
Non-local boundary value problems of arbitrary order.
(English)
[J] J. Lond. Math. Soc., II. Ser. 79, No. 1, 238-258 (2009). ISSN 0024-6107; ISSN 1469-7750/e

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,\dots 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.
[Smail Djebali (Algiers)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B10 Multipoint boundary value problems
47H11 Degree theory
47H30 Particular nonlinear operators

Keywords: Nonlocal boundary value problems; Stieltjes integrals; Green's function; fixed point index

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