Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition

Abstract

We study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra- g s x s ds, where t ∈ 0, 1 , 0 < α < 1, and f is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.

On the existence of solutions of fractional integro-differential equations

“…In [11], Clément et al proved the existence of Hölder continuous solutions for a partial fractional differential equation and in [18], Kilbas et al studied the existence of solutions of several classes of ordinary fractional differential equations. Also, Samko et al [35], Anguraj et al [4], Baleanu and Mustafa [9], Diethelm and Ford [15], Kilbas and Marzan [17], Kosmatov [21], Tian and Bai [39], Wei et al [40], Aghajani et al [3], Pilipović and Stojanović [30], Yuste and Acedo [41], Idczak and Kamocki [16], and Kostić [22], between so many more, have investigated the existence of solutions for various types of fractional differential and integral equations. Furthermore, several analytical and numerical methods have been proposed for approximate solutions of fractional differential equations, e.g.…”

Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus

“…In the recent years, there have appeared many papers about differential and integrodifferential equations and inclusions which are valuable tools in the modeling of many phenomena in various fields of science [18][19][20][21][22][23][24][25]. In 2012, Ahmad et al [26] discussed the existence and uniqueness of solutions for the fractional q-difference equations c D α q u(t) = T(t, u(t)), α 1 u(0)β 1 D q u(0) = γ 1 u(η 1 ) and α 2 u(1)β 2 D q u(1) = γ 2 u(η 2 ), for t ∈ I, where α ∈ (1, 2], α i , β i , γ i , η i are real numbers, for i = 1, 2 and T ∈ C(J × R, R).…”

Existence of solutions for equations and inclusions of multiterm fractional q-integro-differential with nonseparated and initial boundary conditions

“…The field of fractional calculus has countless applications, and the subject of fractional differential equations ranges from the theoretical views of existence and uniqueness of solutions to the analytical and mathematical methods for finding solutions (for instance, see [1][2][3][4]). There has been an intensive development in fractional differential equations and inclusion (for example, see [5][6][7][8][9][10][11]). During the last two decades, the fractional differential equations and inclusions, both differential and q-differential, were developed intensively by many authors for a variety of subjects (for instance, consider [12][13][14][15][16][17][18][19][20]).…”

“…In this article, motivated by [7,21,27], among these achievements and following results, we are working to stretch out the analytical and computational methods of checking of positive solutions for fractional q-integro-differential equation c D α q u(t) = f t, u(t), u (t), u (t), u (t), ϕ 1 u(t), ϕ 2 u(t),…”