The existence of positive solutions for a new coupled system of multiterm singular fractional integrodifferential boundary value problems. (English) Zbl 1294.45005

Baleanu, Dumitru; Zahra Nazemi, Sayyedeh; Rezapour, Shahram

Abstr. Appl. Anal. 2013, Article ID 368659, 15 p. (2013).

Summary: We discuss the existence of positive solutions for the coupled system of multiterm singular fractional integrodifferential boundary value problems \[ D_{0^+}^\alpha u(t) + f_1(t,u(t),v(t),(\phi_1u)(t),(\psi_1v)(t),D_{0^+}^p u(t),D_{0^+}^{u_1} v(t),D_{0^+}^{u_2} v(t),\dots,D_{0^+}^{u_m} v(t)) = 0, \]
\[ D_{0^+}^\beta v(t) + f_2(t,u(t),v(t),(\phi_2u)(t),(\psi_2v)(t),D_{0^+}^q u(t),D_{0^+}^{v_1} u(t),D_{0^+}^{v_2} u(t),\dots,D_{0^+}^{v_m} u(t)) = 0, \] \(u^{(i)} = 0\) and \(v^{(i)} = 0\) for all \(0 \leq i \leq n-2\), \([D_{0^+}^{\delta_1}u(t)]_{t=1} = 0\) for \(2<\delta_1<n-1\) and \(\alpha-\delta_1 \geq 1\), \([D_{0^+}^{\delta_2}v(t)]_{t=1} = 0\) for \(2<\delta_2<n-1\) and \(\beta-\delta_2 \geq 1\), where \(n\geq 4\), \(n-1<\alpha,\beta<n\), \(0<p,q<1\), \(1<\mu_i,\nu_i<2\) \((i=1,2,\dots,m)\), \(\gamma_j,\lambda_j:[0,1] \times [0,1] \to (0,\infty)\) are continuous functions \((j=1,2)\) and \((\phi_j u)(t) = \int_0^t \gamma_j(t,s)u(s)ds\), \((\psi_j v)(t) = \int_0^t \lambda_j(t,s)v(s)ds\). Here \(D\) is the standard Riemann-Liouville fractional derivative, \(f_j\) \((j=1,2)\) is a Caratheodory function, and \(f_j(t,x,y,z,w,v,u_1,u_2,\dots,u_m)\) is singular at the value 0 of its variables.

MSC:

45M20	Positive solutions of integral equations
45G05	Singular nonlinear integral equations
45G15	Systems of nonlinear integral equations
26A33	Fractional derivatives and integrals

Keywords:

positive solutions; singular fractional integro-differential boundary value problem; Riemann-Liouville fractional derivative; Carathéodory function

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