| Summary: | Abstract In this paper, we study the existence and multiplicity of solutions of the quasilinear problems with minimum and maximum (ϕ(u′(t)))′=(Fu)(t),a.e. t∈(0,T),min{u(t)∣t∈[0,T]}=A,max{u(t)∣t∈[0,T]}=B, $$\begin{aligned}& \bigl(\phi \bigl(u'(t)\bigr)\bigr)'=(Fu) (t),\quad \mbox{a.e. }t\in (0,T), \\& \min \bigl\{ u(t) \mid t\in [0,T]\bigr\} =A, \qquad \max \bigl\{ u(t) \mid t\in [0,T]\bigr\} =B, \end{aligned}$$ where ϕ:(−a,a)→R $\phi :(-a,a)\rightarrow \mathbb{R}$ ( 0<a<∞ $0< a<\infty $) is an odd increasing homeomorphism, F:C1[0,T]→L1[0,T] $F:C^{1}[0,T]\rightarrow L^{1}[0,T]$ is an unbounded operator, T>1 $T>1$ is a constant and A,B∈R $A, B\in \mathbb{R}$ satisfy B>A $B>A$. By using the Leray–Schauder degree theory and the Brosuk theorem, we prove that the above problem has at least two different solutions.
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