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THE UNIQUE EXISTENCE OF SOLUTION IN THE q-INTEGRABLE SPACE FOR THE NONLINEAR q-FRACTIONAL DIFFERENTIAL EQUATIONS

Author

Listed:
  • TIE ZHANG

    (Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, P. R. China)

  • YUZHONG WANG

    (Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, P. R. China)

Abstract

In this paper, we study the solution theory of the nonlinear q-fractional differential equation of Caputo type cD qαy(t) = f(t,y(t)) with given initial values Dqky(0 +) = dk,k = 0, 1,…,n − 1 where α > 0 is the order, n = [α] and 0 < q < 1 is the scale index. For 0 ≤ β < α − n + 1, by assuming that function tβf(t,y) is bounded and satisfies the Lipschitz condition on variable y, we prove that this problem admits a unique solution in the q-integrable function space 𠒟q(n)(0,b) and this solution is absolutely stable in the L∞-norm. This unique existence condition allows that f(t,y) is singular at t = 0 and discontinuous for t ∈ (0,b]. Finally, a successive approximation method is presented to find out the analytic approximation solution of this problem.

Suggested Citation

  • Tie Zhang & Yuzhong Wang, 2021. "THE UNIQUE EXISTENCE OF SOLUTION IN THE q-INTEGRABLE SPACE FOR THE NONLINEAR q-FRACTIONAL DIFFERENTIAL EQUATIONS," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(03), pages 1-13, May.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:03:n:s0218348x2150050x
    DOI: 10.1142/S0218348X2150050X
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