Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus

Author:

Samei Mohammad EsmaelORCID

Abstract

AbstractIn this study, we discuss the existence of positive solutions for the system of m-singular sum fractional q-differential equations $$ \begin{gathered} D_{q}^{\alpha_{i}} x_{i} + g_{i} \bigl(t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma _{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr) \\ \quad{} +h_{i} \bigl(t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr)=0 \end{gathered} $$Dqαixi+gi(t,x1,,xm,Dqγ1x1,,Dqγmxm)+hi(t,x1,,xm,Dqγ1x1,,Dqγmxm)=0 with boundary conditions $x_{i}(0) = x_{i}' (1) = 0$xi(0)=xi(1)=0 and $x_{i}^{(k)}(t) = 0$xi(k)(t)=0 whenever $t=0$t=0, here $2\leq k \leq n-1$2kn1, where $n= [\alpha_{i}]+ 1$n=[αi]+1, $\alpha_{i} \geq2$αi2, $\gamma_{i} \in(0,1)$γi(0,1), $D_{q}^{\alpha}$Dqα is the Caputo fractional q-derivative of order α, here $q \in(0,1)$q(0,1), function $g_{i}$gi is of Carathéodory type, $h_{i}$hi satisfy the Lipschitz condition and $g_{i} (t , x_{1}, \ldots, x_{2m})$gi(t,x1,,x2m) is singular at $t=0$t=0, for $1 \leq i \leq m$1im. By means of Krasnoselskii’s fixed point theorem, the Arzelà-Ascoli theorem, Lebesgue dominated theorem and some norms, the existence of positive solutions is obtained. Also, we give an example to illustrate the primary effects.

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Algebra and Number Theory,Analysis

Reference38 articles.

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