Existence and Uniqueness of Solution of Nonlinear Fractional Differential Equations Involving k-Riemann-Liouville Derivative

Document Type : Regular research papers

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Dept. of Mathematics,Shrikrishna Mahavidyalaya, Gunjoti, Dist.Dharashiv (M.S) INDIA

Abstract

The present study deals with the existence and uniqueness of solution of nonlinear fractional differential equations involving $k$-Riemann-Liouville fractional derivative with boundary conditions. Green's function and Banach contraction principle approach is used to prove solution of nonlinear fractional differential equations involving $k$-Riemann-Liouville fractional derivative with boundary conditions . Fractional differential equation with boundary conditions is reduced to the problem of Volterra integral equations. The equivalence of solution of fractional differential equations involving $k$-Riemann-Liouville fractional derivative with boundary conditions and Volterra integral equations is also proved. The properties of $k$-gamma functions , $k$-beta functions and $k-$ Riemann Liouville fractional deirvatives are considered. The Green's function is obtained to prove the existence and uniqueness of solution of the nonlinear boundary value problem involving $k-$ Riemann Liouville fractional deirvatives. Some properties of the Green's theorem for the existence and uniqueness of solution of nonlinear fractional differential equations involving $k$-Riemann-Liouville derivative with boundary conditions are considered.

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