Springer Iranian Journal of Science 2731-8095 38 4 2014 12 22 Existence solutions for nonlocal fractional differential equation with nonlinear boundary conditions 455 461 2562 10.22099/ijsts.2014.2562 EN N. Nyamoradi Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran H. Alaei Dizaji Department of Mathematics, Payame Noor University, Iran Journal Article 2013 03 06 Β  <span style="font-size: xx-small;">In this paper, by employing the Guo-Krasnoselskii fixed point theorem in a cone, we study the existence of positive solutions to the following nonlocal fractional boundary value problems </span> <span style="font-size: xx-small;">{ </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝐷</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">0</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">+ </span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝛼 </span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝑒(𝑑) = 𝑓(𝑑, 𝑒(𝑑)), 𝑑 ∈ (0,1), 𝑒𝑑 + 𝑒′(0) = 12 [𝐻1(πœ‘π‘’) + ∫ 𝐻2 𝐸(𝑠 𝑒𝑠)𝑑𝑠 , 𝑒1) + 𝑒′(1) = 0, </span></span><span style="font-size: xx-small;">c </span> <span style="font-size: xx-small;">where </span><span style="font-size: xx-small;">c</span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝐷0</span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">+ </span></span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝛼</span></span><span style="font-size: xx-small;">is the standard Caputo derivative of order </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝛼 1 < 𝛼< 2, πΈβŠ† (0,1) </span></span><span style="font-size: xx-small;">is some measurable set. We provide conditions on </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">𝑓 𝐻1, 𝐻2 </span></span><span style="font-size: xx-small;">and </span><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA"><span style="font-family: Cambria Math,Cambria Math; font-size: xx-small;" lang="JA">πœ‘</span></span><span style="font-size: xx-small;">such that the problem exhibits at least one positive solution. </span> Cone fixed point theorem standard Caputo Derivative https://ijsts.shirazu.ac.ir/article_2562_89c639061537ef57883fe24a62f10c88.pdf