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Existence of positive solutions for the nonlinear fractional differential equation D s u(x) = f (x, u(x)), 0 < s < 1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804–812), where D s denotes Riemann–Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential(More)
and Applied Analysis 3 Definition 2.2. For f, g ∈ E the order interval 〈f, g〉 is defined as 17 〈 f, g 〉 { h ∈ E : f ≤ h ≤ g. 2.1 Definition 2.3. A subset E ⊂ Π is called order bounded if E is contained in some order interval. Definition 2.4. A coneK is called normal if there exists a positive constant μ such that f, g ∈ V and θ ≺ f ≺ g implies that ‖f‖ ≤(More)
Basic theory on a class of initial value problem of some fractional differential equation involving Riemann-Liouville differential operators is discussed by employing the classical approach from the work of Lakshmikantham and A. S. Vatsala 2008 . The theory of inequalities, local existence, extremal solutions, comparison result and global existence of(More)
and Applied Analysis 3 Definition 2.2 see 15 . A cone K is called normal, if there exists a positive constant r such that f, g ∈ K and θ ≺ f ≺ g implies ‖f‖ ≤ r‖g‖, where θ denotes the zero element of K. Definition 2.3 see 16, 17 . Let f : a, b → R, and f ∈ L1 a, b . The left-sided RiemannLiouville fractional integral of f of order α is defined as I af x 1(More)
and Applied Analysis 3 [0, b]. If there exist positive constants a and α ∈ (0, 1) such that V(t) ≤ w(t) + a ∫ t 0 (t − s) V(s)ds, then there exists a constant K = K(α) such that V(t) ≤ w(t)+Ka∫t 0 w(s)(t− s) −α ds, for all t ∈ [0, b]. In this paper we use the alternative Leray-Schauder’s theorem and Banach’s contraction principle for getting the main(More)
In this article, we discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations D α − ρtD β x(t) = f (t, x(t), D γ x(t)) , t ∈ (0, 1) with boundary conditions x(0) = x 0 , x(1) = x 1 or satisfying the initial conditions x(0) = 0, x (0) = 1 where D denotes Caputo fractional derivative, ρ is constant, 1 < α <(More)
and Applied Analysis 3 Let E be a real Banach space with a cone K. K introduces a partial order ≤ in E as x ≤ y if y − x ∈ K. Definition 2.1. For x, y ∈ E, the order interval 〈x, y〉 is defined as 〈 x, y 〉 { z ∈ E : x ≤ z ≤ y. 2.1 Theorem 2.2 Leray-Schauder Theorem 17 . Let E be a Banach space with C ⊆ E closed and convex. Assume U is relatively open subset(More)
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