Abstract
We investigate the existence of solutions for new boundary value problems of Caputo-type sequential fractional differential equations and inclusions supplemented with nonlocal integro-multipoint boundary conditions. We apply the modern techniques of functional analysis to obtain the main results. We emphasize that the results presented in this paper are new and specialize to some known theorems with an appropriate choice of the parameters involved in the problems at hand.
Highlights
In this paper, we investigate the existence criteria for the solutions of Caputo-type sequential fractional differential equations and inclusions: cDq + μcDq–1 x(t) = f t, x(t), cDκ x(t), t ∈ [0, 1], cDq + μcDq–1 x(t) ∈ F t, x(t), cDκ x(t), t ∈ [0, 1], (1.1) (1.2)supplemented with nonlocal integro-multipoint boundary conditions:⎧ ⎪⎪⎨ρ1x(0) + ρ2x(1) = m–2 i=1 αix(σi ) +p–2 j=1 rj ηj ξj x(s) ds,⎪⎪⎩0ρ3
4 The case of inclusions we investigate the existence of solutions for the multivalued boundary value problem (1.2) and (1.3)
4.2 The case of Lipschitz maps we prove the existence of solutions for the boundary value problem (1.2) and (1.3) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler [35]
Summary
Supplemented with the boundary conditions (1.3) is given by m–2 σi x(t) = φi(t) eμ(s–σi) Iq–1h(s) ds i=1 p–2 ηj Main results for the problem (1.1) and (1.3) Let X = {x|x ∈ C([0, 1], R) and cDκ x ∈ C([0, 1], R)} be a space equipped with the norm x X = supt∈[0,1] |x(t)| + supt∈[0,1] |cDκ x(t)| = x + cDκ x , where cDκ denotes the standard Caputo fractional derivative of order 0 < κ ≤ 1. The case of inclusions we investigate the existence of solutions for the multivalued (inclusion) boundary value problem (1.2) and (1.3).
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