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1
TWIN POSITIVE SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS FOR THE ONE-DIMENSIONAL ρ-LAPLACIAN

Bai, Chuan-Zhi,Fang, Jin-Xuan Korean Mathematical Society 2003 대한수학회보 Vol.40 No.2
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For the boundary value problem (BVP) of second order functional differential equations for the one-dimensional $\rho$-Laplaclan: ($\Phi$$_{\rho}$(y'))'(t)+m(t)f(t, $y^{t}$ )=0 for t$\in$[0,1], y(t)=η(t) for t$\in$[-$\sigma$,0], y'(t)=ξ(t) for t$\in$[1,d], suitable conditions are imposed on f(t, $y^{t}$ ) which yield the existence of at least two positive solutions. Our result generalizes the main result of Avery, Chyan and Henderson.
2
Twin positive solutions of functional differential equations for the one-dimensional $p$-laplacian

Chuan-Zhi Bai,Jin-Xuan Fang 대한수학회 2003 대한수학회보 Vol.40 No.2
For the boundary value problem (BVP) of second order functional differential equations for the one-dimensional $p$-Laplac-ian: \begin{eqnarray*} &\mbox{}& (\Phi_p (y^{\prime}))^{\prime}(t) + m(t) f(t, y^t) = 0 \mbox{} \hspace{0.5cm} {\rm for} \ t \in [0, 1],\\ &\mbox{}& \hspace{12mm} y(t) = \eta (t) \mbox{} \hspace{ 0.5cm} {\rm for} \ t \in [- \sigma, 0],\\ &\mbox{}& \hspace{12mm} y^{\prime}(t) = \xi (t) \mbox{} \hspace{ 0.5cm} {\rm for} \ t \in [1, d], \end{eqnarray*} suitable conditions are imposed on $f(t, y^t)$ which yield the existence of at least two positive solutions. Our result generalizes the main result of Avery, Chyan and Henderson.
3
A system of nonlinear variational inclusions in real Banach spaces

Chuan-Zhi Bai,Jin-Xuan Fang 대한수학회 2003 대한수학회보 Vol.40 No.3
In this paper, we introduce and study a system of nonlinear implicit variational inclusions (SNIVI) in real Banach spaces: determine elements $x^{\ast}, \ y^{\ast}, \ z^{\ast} \in E$ such that $$\theta \in \alpha T(y^{\ast}) + g(x^{\ast}) - g(y^{\ast}) + A(g(x^{\ast})) \hspace{5mm} {\rm for} \ \alpha > 0, $$ $$\theta \in \beta T(z^{\ast}) + g(y^{\ast}) - g(z^{\ast}) + A(g(y^{\ast})) \hspace{5mm} {\rm for} \ \beta > 0,$$ $$\theta \in \gamma T(x^{\ast}) + g(z^{\ast}) - g(x^{\ast}) + A(g(z^{\ast})) \hspace{5mm} {\rm for} \ \gamma > 0, $$ where $T, g : E \to E$, $\theta$ is zero element in Banach space $E$, and $A : E \to 2^E$ be $m$-accretive mapping. By using resolvent operator technique for $m$-accretive mapping in real Banach spaces, we construct some new iterative algorithms for solving this system of nonlinear implicit variational inclusions. The convergence of iterative algorithms be proved in $q$-uniformly smooth Banach spaces and in real Banach spaces, respectively.
4
A SYSTEM OF NONLINEAR VARIATIONAL INCLUSIONS IN REAL BANACH SPACES

Bai, Chuan-Zhi,Fang, Jin-Xuan Korean Mathematical Society 2003 대한수학회보 Vol.40 No.3
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In this paper, we introduce and study a system of nonlinear implicit variational inclusions (SNIVI) in real Banach spaces: determine elements $x^{*},\;y^{*},\;z^{*}\;\in\;E$ such that ${\theta}\;{\in}\;{\alpha}T(y^{*})\;+\;g(x^{*})\;-\;g(y^{*})\;+\;A(g(x^{*}))\;\;\;for\;{\alpha}\;>\;0,\;{\theta}\;{\in}\;{\beta}T(z^{*})\;+\;g(y^{*})\;-\;g(z^{*})\;+\;A(g(y^{*}))\;\;\;for\;{\beta}\;>\;0,\;{\theta}\;{\in}\;{\gamma}T(x^{*})\;+\;g(z^{*})\;-\;g(x^{*})\;+\;A(g(z^{*}))\;\;\;for\;{\gamma}\;>\;0,$ where T, g : $E\;{\rightarrow}\;E,\;{\theta}$ is zero element in Banach space E, and A : $E\;{\rightarrow}\;{2^E}$ be m-accretive mapping. By using resolvent operator technique for n-secretive mapping in real Banach spaces, we construct some new iterative algorithms for solving this system of nonlinear implicit variational inclusions. The convergence of iterative algorithms be proved in q-uniformly smooth Banach spaces and in real Banach spaces, respectively.

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