Solvability of Hammerstein integral equations with applications to boundary value problems. (English) Zbl 1384.45005
The paper contains several somewhat independent results.
One is related to nonlinear eigenvalue problems for the Hammerstein operator \[ \lambda x(t)=\int_\Omega k(t,s)f(s,x(s))\,ds. \] Under some nonnegativity condition for \(k\) on a subset \(\Omega_0\) it had previously been shown by the first and last author that a positive eigenvalue exists with a continuous eigenfunction which is nonnegative on \(\Omega_0\) and which has a given \(L_p\)-norm. This result is extended: In hypothesis and conclusion the nonnegativitity is relaxed to a mean-nonnegativity on \(\Omega_0\), i.e., on the nonnegativity of a corresponding integral over \(\Omega_0\).
In the second part, the existence of a solution of the problem \[ x(t)=\alpha[x]v(t)+\beta[x]w(t)+\lambda\int_0^1k(t,s)f(s,x(s))\,ds \] with given functions \(v,w\) and bounded linear functionals \(\alpha,\beta\) are considered in the space of continuous functions of bounded variations on \([0,1]\). To this end, a variant of Krasnoselskiĭ’s fixed point theorem and recent compactness result in the space of bounded variations by the second author are applied.
Finally, the results are applied to the second-order equation \(x''(t)=-\lambda f(t,x(t))\) with nonlocal boundary conditions of Riemann-Stieltjes type \[ x(0)=\int_0^1A(s)\,dx(s),\quad x(1)=\int_0^1B(s)\,dx(s). \]
One is related to nonlinear eigenvalue problems for the Hammerstein operator \[ \lambda x(t)=\int_\Omega k(t,s)f(s,x(s))\,ds. \] Under some nonnegativity condition for \(k\) on a subset \(\Omega_0\) it had previously been shown by the first and last author that a positive eigenvalue exists with a continuous eigenfunction which is nonnegative on \(\Omega_0\) and which has a given \(L_p\)-norm. This result is extended: In hypothesis and conclusion the nonnegativitity is relaxed to a mean-nonnegativity on \(\Omega_0\), i.e., on the nonnegativity of a corresponding integral over \(\Omega_0\).
In the second part, the existence of a solution of the problem \[ x(t)=\alpha[x]v(t)+\beta[x]w(t)+\lambda\int_0^1k(t,s)f(s,x(s))\,ds \] with given functions \(v,w\) and bounded linear functionals \(\alpha,\beta\) are considered in the space of continuous functions of bounded variations on \([0,1]\). To this end, a variant of Krasnoselskiĭ’s fixed point theorem and recent compactness result in the space of bounded variations by the second author are applied.
Finally, the results are applied to the second-order equation \(x''(t)=-\lambda f(t,x(t))\) with nonlocal boundary conditions of Riemann-Stieltjes type \[ x(0)=\int_0^1A(s)\,dx(s),\quad x(1)=\int_0^1B(s)\,dx(s). \]
Reviewer: Martin Väth (Prague)
MSC:
45G10 | Other nonlinear integral equations |
26A45 | Functions of bounded variation, generalizations |
45C05 | Eigenvalue problems for integral equations |
45M20 | Positive solutions of integral equations |
47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
34B15 | Nonlinear boundary value problems for ordinary differential equations |