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Existence result of second-order differential equations with integral boundary conditions at resonance. (English) Zbl 1180.34016

The paper concerns the boundary value problem

${x}^{\text{'}\text{'}}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0
${x}^{\text{'}}\left(0\right)={\int }_{0}^{1}h\left(t\right){x}^{\text{'}}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt,\phantom{\rule{2.em}{0ex}}{x}^{\text{'}}\left(1\right)={\int }_{0}^{1}g\left(t\right){x}^{\text{'}}\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt·$

It is assumed that the functions $h$ and $g$ are continuous and nonnegative and such that the conditions ${\int }_{0}^{1}h\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=1$ and ${\int }_{0}^{1}g\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt=1$ hold. The linear operator $L,$ $Lx:={x}^{\text{'}\text{'}}$, defined on the subspace of functions that belong to the Sobolev space ${W}^{2,1}\left(0,1\right)$ and satisfy the boundary conditions is a Fredholm operator with index zero when the functions $h$ and $g$ satisfy a condition formulated in the preliminary part of the paper. Moreover, its kernel is two-dimensional.

Next, a fixed point theorem due to J. Mawhin [Topological degree methods in nonlinear boundary value problems. Regional Conference Series in Mathematics. No. 40. R.I.: The American Mathematical Society (1979; Zbl 0414.34025)] is recalled. The existence of at least one solution of the boundary value problem is proved when certain conditions are satisfied. They are formulated in the main result which proof is based on Mawhin’s fixed point theorem. An example is presented.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations