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# Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps

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• In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem

$\left\{ \begin{array}{l} - u''(x) = \lambda f(u),\;\;\;\;0 < x < 1,\\u(0) = 0,\\\frac{{u(1)}}{{u(1) + 1}}u'(1) + \left[ {1 - \frac{{u(1)}}{{u(1) + 1}}} \right]u(1) = 0,\end{array} \right.$

where $λ>0$ is a bifurcation parameter, $f(u)>0$ for $u>0$. We give complete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for $f(u) = e^{u},\ f(u) = a^{u}(a>0),\ f(u) = u^{p}(p>0),\ f(u) = e^{u}-1,\ f(u) = a^{u}-1(a>1)$ and $f(u) = (1+u)^{p}(p>0)$. Our methods are based on a detailed analysis of time maps.

Mathematics Subject Classification: 34B18, 74G35.

 Citation: • • Figure 1.  Reversed $S$-shaped curve

Figure 2.  Broken reversed $S$-shaped curve

Figure 3.  Exactly $S$-shaped bifurcation curve $S$ of $(1.6)$

Figure 6.  Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}$

Figure 8.  Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = a^{u}$

Figure 10.  Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = u^p$

Figure 12.  Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}-1$

Figure 14.  Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p>1)$

Figure 16.  Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p\leq 1)$

Figure 4.  Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}$

Figure 5.  Graphs of $G(\lambda,\rho)$ and $H(\lambda,\rho)$ for $\lambda_{0},\lambda_{1},\lambda_{2}$ in the case $f(u) = e^{u}$

Figure 7.  Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = a^{u}$

Figure 9.  Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = u^p$

Figure 11.  Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}-1$

Figure 13.  Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = (1+u)^p(p>0)$

Figure 15.  Graphs of $\widetilde{H}(\rho,0).$

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