creative_2019_28_2_163_182_001

Existence of countably many symmetric positive solutions for system of even order time scale boundary value problems in Banach spaces

Prsad, K. R. and Khuddush, Md.

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creative_2019_28_2_163_182

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This paper establishes the existence and uniqueness of the solutions to the system of even order differential equations on time scales,

    \[\begin{aligned} (-1)^n&u_1^{{(\Delta\nabla)}^n}(t)=\omega_1(t)f_1\big(u_1(t), u_2(t)\big), \; t \in [0, T]_{\T},\, n\in \Z^+,\\ (-1)^n&u_2^{{(\Delta\nabla)}^m}(t)=\omega_2(t)f_2\big(u_1(t), u_2(t)\big), \; t \in [0, T]_{\T},\, m\in \Z^+,\\ \end{aligned}\]

satisfying two-point Sturm–-Liouville integral boundary conditions

    \[\begin{aligned} \alpha_{i+1}u_1^{{(\Delta \nabla)}^i}(0)&-\beta_{i+1}u_1^{{(\Delta \nabla)}^i\Delta}(0)=\int_0^T a_{i+1}(s) u_1^{{(\Delta \nabla)}^i}(s)\nabla s, ~~0\leq i \leq n-1,\\ \alpha_{i+1}u_1^{{(\Delta \nabla)}^i}(T)&+\beta_{i+1}u_1^{{(\Delta \nabla)}^i\Delta}(T)=\int_0^T a_{i+1}(s) u_1^{{(\Delta \nabla)}^i}(s)\nabla s, ~~0\leq i \leq n-1,\\ \gamma_{j+1}u_2^{{(\Delta \nabla)}^j}(0)&-\delta_{j+1}u_2^{{(\Delta \nabla)}^j\Delta}(0)=\int_0^T b_{j+1}(s) u_2^{{(\Delta \nabla)}^j}(s)\nabla s, ~~0\leq j \leq m-1,\\ \gamma_{j+1}u_2^{{(\Delta \nabla)}^j}(T)&+\delta_{j+1}u_2^{{(\Delta \nabla)}^j\Delta}(T)=\int_0^T b_{j+1}(s) u_2^{{(\Delta \nabla)}^j}(s)\nabla s, ~~0\leq j \leq m-1, \end{aligned}\]

by utilizing Schauder fixed point theorem. We also establish the existence of countably many symmetric positive solutions for the above problem by applying Hölder’s inequality and Krasnoselskii’s fixed point theorem.

 

 

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