| Summary: | Abstract In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions {u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0, $$\begin{aligned}& \left \{ \textstyle\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\quad t\in(0,1),\\ u(0)=u(1)=\beta_{1}[u],\qquad u''(0)+\beta_{2}[u]=0,\qquad u''(1)+\beta_{3}[u]=0, \end{array}\displaystyle \right . \end{aligned}$$ where f:[0,1]×R+×R×R−→R+ $f: [0,1]\times\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{-}\to \mathbb{R}_{+}$ is continuous, h∈L1(0,1) $h\in L^{1}(0,1)$ and βi[u] $\beta_{i}[u]$ is Stieltjes integral ( i=1,2,3 $i=1,2,3$). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C2[0,1] $C^{2}[0,1]$. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.
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