In this paper, we investigate the approximate controllability of Hilfer fractionalneutral stochastic differential equations. Firstly, the existence and uniqueness of\ mild solutions for these equations are obtained by means of the Banach\ contraction mapping principle. Then, combining the techniques of stochastic\ analysis theory, fractional calculations and operator semigroup theory, a new set\ of sufficient conditions for approximate controllability of these equations is\ formulated. At last, an example is presented to illustrate the obtained results.

\

In this work, we have developed Picard{\textquoteright}s iterative method to prove the existence and uniqueness of the solution of the nonlinear Caputo fractional reaction diffusion equation in one dimensional space. The order of the fractional time derivative q is such that $0.5\leq q\leq 1$. The existence result has been proved by a priori assuming the solution is bounded. Thus, we refer to this method as existence of solution in the large. The method can be \ extended to the Caputo fractional reaction diffusion system also.

}, keywords = {34A08, 34R11}, issn = {1056-2176}, doi = {10.12732/dsa.v27i4.10}, url = {https://acadsol.eu/dsa/articles/27/4/10.pdf}, author = {PRADEEP G. CHHETRI and AGHALAYA S. VATSALA} } @article {403, title = {ON THE OSCILLATION OF THREE DIMENSIONAL alpha-FRACTIONAL DIFFERENTIAL SYSTEMS}, journal = {Dynamic Systems and Applications}, volume = {27}, year = {2018}, month = {11/2018}, pages = {22}, chapter = {873}, abstract = {In this article, we consider the three dimensional $\alpha$-fractional nonlinear differential system of the form $$ D^{\alpha}\left(x(t)\right)=a(t)f\left(y(t)\right), $$ $$ D^{\alpha}\left(y(t)\right)=-b(t)g\left(z(t)\right), $$ $$ D^{\alpha}\left(z(t)\right)=c(t)h\left(x(t)\right),\quad t \geq t_0, $$ where $0 \< \alpha \leq 1$, $D^{\alpha}$ denotes the Katugampola fractional derivative of order $\alpha$. We establish some new sufficient conditions for the oscillation of the solutions of the differential system, using the generalized Riccati transformation and inequality technique. Examples illustrating the results are also given.

}, keywords = {34A08, 34A34, 34K11}, issn = {1056-2176}, doi = {10.12732/dsa.v27i4.12}, url = {https://acadsol.eu/dsa/articles/27/4/12.pdf}, author = {G.E. CHATZARAKIS and M. DEEPA and N. NAGAJOTHI and V. SADHASIVAM} } @article {317, title = {A VARIATIONAL APPROACH OF THE STURM-LIOUVILLE PROBLEM IN FRACTIONAL DIFFERENCE CALCULUS}, journal = {Dynamic Systems and Applications}, volume = {27}, year = {2018}, month = {01/2018}, pages = {12}, chapter = {137}, abstract = {In this article, we formulate and analyze a nabla fractional difference Sturm Liouville problem (SLP) with the nabla left Caputo fractional difference\ and the nabla right Riemann-Liouville fractional difference. The discrete fractional\ variational calculus is used to study the eigenvalues and eigenfunctions of the formulated SLP by presenting a new nabla fractional difference isoperimetric variational\ problem.

}, keywords = {34A08, 35R11}, issn = {1056-2176}, doi = {10.12732/dsa.v27i1.7}, url = {https://www.acadsol.eu/dsa/articles/27/1/7.pdf}, author = {RAZIYE MERT and LYNN ERBE and THABET ABDELJAWAD} } @article {333, title = {AN EXISTENCE AND UNIQUENESS RESULT FOR LINEAR SEQUENTIAL FRACTIONAL BOUNDARY VALUE PROBLEMS (BVPS) VIA LYAPUNOV TYPE INEQUALITY}, journal = {Dynamic Systems and Applications}, volume = {26}, year = {2017}, month = {2017}, pages = {10}, chapter = {147}, abstract = {A sufficient condition for the existence and uniqueness of solution of nonhomogenous fractional boundary value problem involving sequential fractional derivative of Riemann Liouville type is established by using a new Lyapunov type inequality and disconjugacy criterion. Green{\textquoteright}s function and some of its properties are also presented. Our approach is quite new and to the best of our knowledge, the uniqueness of solution of nonhomogenous fractional boundary value problems is proved by employing Lyapunov type inequality for the first time and this Lyapunov type inequality improves and generalizes the previous ones.

}, keywords = {34A08, 34B05, 34C10}, issn = {1056-2176}, doi = {10.12732/dsa.v26i1.8}, url = {https://acadsol.eu/dsa/articles/26/1/8.pdf}, author = {ZEYNEP KAYAR} } @article {110, title = {ON A FRACTIONAL DIRICHLET PROBLEM OF HIGHER ORDER}, journal = {Dynamic Systems and Applications}, volume = {24}, year = {2015}, month = {2015}, pages = {12}, chapter = {479}, abstract = {We consider a fractional version of some 2nth order Dirichlet problem. In the paper a sufficient condition for the existence of solution to the aforementioned problem is proved. The proved is based on some variational methods and application of a fractional counterpart of the du Bois-Reymond lemma for the order α ∈( n - 1/2 , n) (see [1])

}, keywords = {26A33, 34A08, 34B15}, issn = {1056-2176}, url = {https://acadsol.eu/dsa/articles/24/38-DSA-479-490.pdf}, author = {MAREK MAJEWSKI and STANISLAW WALCZAK} } @article {106, title = {GENERALIZED MONOTONE METHOD FOR ORDINARY AND CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS}, journal = {Dynamic Systems and Applications}, volume = {24}, year = {2015}, month = {2015}, pages = {9}, chapter = {429}, keywords = {34A08, 34A12}, issn = {1056-2176}, url = {https://acadsol.eu/dsa/articles/24/34-DSA-429-438.pdf}, author = {A S VATSALA and M. SOWMYA and D S STUTSON} } @article {65, title = {GENERALIZED MONOTONE METHOD FOR CAPUTO FRACTIONAL DIFFERENTIAL SYSTEMS VIA COUPLED LOWER AND UPPER SOLUTIONS}, journal = {Dynamic Systems and Applications}, volume = {20}, year = {2011}, month = {2011}, pages = {9}, chapter = {495}, abstract = {Monotone method combined with the method of upper and lower solutions yields monotone sequences which converge uniformly and monotonically to minimal and maximal solutions of the nonlinear systems, when the forcing function is quasi monotone nondecreasing. In this paper we develop genearalized monotone method for N system of Caputo fractional differential equations when the forcing function is the sum of an increasing and decreasing functions. In generalized monotone method we use coupled upper and lower solutions and the method yields two monotone sequences which converge uniformly and monotonically to coupled minimal and maximal solutions. This method is applicable to the Lotka-Volterra equation with Caputo fractional derivative of order q when 0 \< q <= 1. This provides an opportunity to provide better results or improve on the existing results with integer derivatives. Finally, under uniqueness condition we obtain the unique solution of the Caputo fractional differential system.

}, keywords = {26A33, 34A08, 34A445}, issn = {1056-2176}, url = {https://acadsol.eu/dsa/articles/20/32-DSA-31-19.pdf}, author = {DONNA STUTSON and A. S. VATSALA} } @article {34, title = {MONOTONE METHOD FOR NONLINEAR CAPUTO FRACTIONAL BOUNDARY VALUE PROBLEMS}, journal = {Dynamic Systems and Applications}, volume = {20}, year = {2011}, month = {2011}, pages = {16}, chapter = {73}, abstract = {In this paper, by using upper and lower solutions, we develop monotone method for the nonlinear Caputo fractional boundary value problem of order α where 1 \< α \< 2. We construct two sequences which converge uniformly and monotonically to the extremal solutions of the nonlinear Caputo fractional boundary value problem.

}, keywords = {26A33, 34A08, 34B15}, issn = {1056-2176}, url = {https://acadsol.eu/dsa/articles/20/06-DSA-30-15.pdf}, author = {J. D. RAM{\textasciiacute}IREZ and A. S. VATSALA} } @article {141, title = {A FITE TYPE RESULT FOR SEQUENTIAL FRACTIONAL DIFFERENTIAL EQUATIONS}, journal = {Dynamic Systems and Applications}, volume = {19}, year = {2010}, month = {2010}, pages = {12}, chapter = {383}, abstract = {Given the solution f of the sequential fractional differential equation

_{a}D^{α}_{t} (\ _{a}D^{α}_{t} f ) + P( t ) f = 0, t ∈ [b, c],\ \ where\ \ -$\infty$ \< a \< b \< c \< +$\infty$, α ∈ (1/ 2 , 1)\ \ and\ \ P : [a, +$\infty$) {\textrightarrow} [ 0, P_{$\infty$}], P_{$\infty$} \< +$\infty$,\ \ is continuous. Assume that there exist\ \ t_{1}, t_{2} ∈ [b, c]\ such that\ f( t_{1\ }) = (_{a}D^{α}_{t }f )( t_{2\ }) = 0. Then, we establish here a positive lower bound for c - a which depends solely on α, P_{$\infty$}. Such a result might be useful in discussing disconjugate fractional differential equations and fractional interpolation, similarly to the case of (integer order) ordinary differential equations.