AN EXTENSION OF PICARD-LINDELOFF THEOREM TO FRACTIONAL DIFFERENTIAL EQUATIONS

The method of contraction mapping defined on a complete metric space is used to prove the existence and uniqueness of solutions of the system of differential equations: together with the initial condition y(xo)=yo, where y(α is the Riemann-Liouville derivative of non integer order a of a real valued vector function y(x), under usual continuity and Lipschitz conditions on the function f. The classical Picard-Lindeloff theorem for ordinary differential equations of integer order is a special case of the main result when α = 1. Some examples are given. Finally, consequences for the linear case are obtained.

[1]  H. L. Arora,et al.  A Global Existence and Uniqueness Theorem for Ordinary Differential Equations of Generalized Order , 1978, Canadian Mathematical Bulletin.

[2]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[3]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[4]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[5]  Matsusaburô Fujiwara,et al.  On the Integration and Differentiation of an Arbitrary Order , 1933 .

[6]  L. Campos.  ON THE SOLUTION OF SOME SIMPLE FRACTIONAL DIFFERENTIAL EQUATIONS , 1990 .

[7]  J. H. Barrett,et al.  Differential Equations of Non-Integer Order , 1954, Canadian Journal of Mathematics.

[8]  H. Srivastava,et al.  Some fractional differintegral equations , 1985 .

[9]  R. Gorenflo,et al.  Abel Integral Equations: Analysis and Applications , 1991 .

[10]  M.A. Al-Bassam,et al.  Some existence theorems on differential equations of generalized order. , 1965 .

[11]  Shusen Ding,et al.  A Class of Analytic Functions Defined by Fractional Derivation , 1994 .

[12]  Luigi Rodino,et al.  Existence and Uniqueness for a Nonlinear Fractional Differential Equation , 1996 .

[13]  B. Ross,et al.  The development of fractional calculus 1695–1900 , 1977 .

[14]  Elena Castiñeira,et al.  Ecuaciones diferenciales ordinarias , 1995 .

[15]  R. Bagley,et al.  On the Fractional Calculus Model of Viscoelastic Behavior , 1986 .

[16]  Ahmad Zain,et al.  Local existence theorems for ordinary differential equations of fractional order , 1982 .

[17]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[18]  W. E. Sewell,et al.  Existence theorems for solutions of differential equations of non-integral order , 1938 .

[19]  G. H. Hardy,et al.  Riemann's Form of Taylor's Series , 1945 .