Non-local boundary value problems of arbitrary order.

*(English)*Zbl 1165.34010Many 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.

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.

Reviewer: Smail Djebali (Algiers)

##### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

47H11 | Degree theory for nonlinear operators |

47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |