Solvability of Fractional Multi-Point Boundary Value Problems with Nonlinear Growth at Resonance
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rential Equations
Solvability of Fractional Multi-Point Boundary Value Problems with Nonlinear Growth at Resonance Zidane Baitiche1* , Kaddour Guerbati1** , Mouffak Benchohra2, 3*** , and Yong Zhou4, 5**** 1
2
Ghardaia University, Ghardaia, Algeria University of Sidi Bel-Abbes, Sidi Bel-Abbes, Algeria 3 King Saud University, Riyadh, Saudi Arabia 4 Xiangtan University, Xiangtan, China 5 King Abdulaziz University, Jeddah, Saudi Arabia
Received November 21, 2018; revised March 22, 2019; accepted April 25, 2019
Abstract—This work is concerned with the solvability of multi-point boundary value problems for fractional differential equations with nonlinear growth at the resonance. Existence results are obtained with the use of the coincidence degree theory. As an application, we discuss an example to illustrate the obtained results. MSC2010 numbers : 34A08, 34B15 DOI: 10.3103/S1068362320020041 Keywords: fractional differential equations; fractional Caputo derivative; multi-point boundary value problem; resonance; coincidence degree theory.
1. INTRODUCTION This paper is devoted to the solvability of the following fractional multi-point boundary value problems (BVPs) at the resonance ⎧ ⎪ ⎨ φ(t)C D0α+ u(t) = f t, u(t), u (t), u (t), C D0α+ u(t) , t ∈ I = [0, 1], m l (1.1) C D α u(0) = 0, ⎪ u (0) = ai u (ξi ), u (1) = bj u (ηj ), ⎩u(0) = 0, 0+ i=1
j=1
where C D0α+ is the Caputo fractional derivative, 2 < α ≤ 3, 0 < ξ1 < · · · < ξm < 1, 0 < η1 < · · · < ηl < 1, ai , bj ∈ R, (i = 1, . . . , m, j = 1, . . . , l), φ(t) ∈ C 1 ([0, 1]), and μ = mint∈I φ(t) > 0. The nonlinearity is such that the following conditions are satisfied: ´ function, that is, (H0 ) f : [0, 1] × R4 −→ Ris a Caratheodory (i) for each x ∈ R4 , the function t −→ f t, x is Lebesgue measurable; (ii) for almost every t ∈ [0, 1], the function t −→ f t, x is continuous on R4 ; (iii) for each r > 0, there exists ϕr (t) ∈ L1 [0, 1], R such that for a.e. t ∈ [0, 1] and every |x| ≤ r, we have |f t, x | ≤ ϕr (t). *
E-mail: [email protected] E-mail: guerbati\[email protected] *** E-mail: [email protected] **** E-mail: [email protected] **
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SOLVABILITY OF FRACTIONAL
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The resonant conditions of (1.1) are as follows: (H1 )
m
i=1 ai
= 1,
l
j=1 bj
l
= 1,
j=1 bj ηj
= 1.
This means that the linear operator Lu = φC D0α+ u corresponding to the problem (1.1) has a nontrivial solution or, in a functional framework, L is not invertible, that is, dim kerL ≥ 1. In order to be sure that the linear operator Q (to be specified later on) is well defined, we assume, in addition, that (H2 ) There exist p, q ∈ Z+ , q ≥ p + 1 such that Δ(p, q) = d11 d22 − d12 d21 , where d11 =
m
ξi ai
i=1
sp (ξi − s)α−3 ds, pφ(s)
d21 =
d12 =
sp (1 − s)α−2 ds − bj pφ(s) l
j=1
0
1 d22 =
sq (1 − s)α−2 ds − bj qφ(s) l
j=1
0
ξi ai
i=1
0
1
m
ηj
sq (ξi − s)α−3 ds, qφ(s)
0
sp (ηj − s)α−2 ds, pφ(s)
0
ηj
sq (ηj − s)α−2 ds. qφ(s)
0
Notice that Δ(p, q) = 0 (see [19, 23]). Fractional calculus is an extension of t
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