This thesis is devoted to the rapidly developing area of research for the qualitative theory of fractional differential equations. In particular, our main interest lies in the existence of mild solutions of some classes of fractional differential equations with non-instantaneous impulses.
Some notations and definitions
Let C be the class of functions from J to the normed space X. C is uniformly bounded if there exists M > 0 independently of f ∈ C such that. The function f :J →X is said to be H¨older continuous with the exponent ν, 0< ν <1 on J, if there exists a constant L such that
Basic fractional calculus
Gamma function
Beta function
Mittag-Leffler function
A frequently occurring definition of an integral of the fractional order is via an integral transformation called the Riemann-Liouville integral. By replacing the integer n with the real number β and the discrete factorial (n−1)!with the continuous gamma functionΓ(n), the Riemann-Liouville fractional integral is obtained. 119] The Riemann-Liouville fractional integral of order β > 0 for a function f is defined as. i) The Caputo derivative of constant k is equal to zero, whereas the Riemann-Liouville derivative of constant k is not equal to zero.
Impulsive differential equations
Depending on the moments of change with jumps, impulsive differential equations are broadly classified into two categories. In the second impulsive class, jump moments occur when certain space-time relations h(t, y(t)) = 0 are satisfied.
Delay differential equations
Calculus of Banach space valued function
A function f : J → X is said to be strongly measurable in J if there exists a sequence {fn(t)}∞n=1 of countably valued functions (strongly) that converges almost everywhere in J to f(t) . A function f : J → X is Bochner integrable if and only if f(t) is strongly measurable and kf(t)k is Lebesgue measurable.
Methods
Semigroup of bounded linear operators
We say that f is strongly continuously differentiable at t1 if the derivative f0 of f is strongly continuous at t1. The set of all complex numbers λ such that λI −A is invertible is called the solvent setρ(A) of a linear operator A.
Measure of noncompactness
One of the most important measures of compactness is the Hausdorff measure of compactness (χ) and is defined as, for any bounded set B of X,. A continuous map T : B ⊂ X → X is said to be a t-contraction χ if there exists a positive constant ν <1 such that χ(TU)≤νχ(U) for every bounded closed subset U ⊆B.
Fixed point theorems
If F1 is a contraction and F2 is completely continuous, then the equation w=F1w+F2w has a solution in the fixed point theorem of E. Schauder) [120] Let E ⊂ X be a nonempty, closed, subgroup of bounded, convex of a Banach spaceX, and T :E →E a compact operator. -Kirk's Fixed Point Theorem) [35] Let X be a Banach space and F1, F2 be two satisfying operators. i) F1 is a contraction and (ii) F2 is completely continuous.
Mild solutions for abstract fractional Cauchy problem
For the second method, we take Laplace transform on both sides of equation (1.8), which results in.
Review of literature
Kilbas and Marzan [70] investigated the existence of solution of the nonlinear FDE of the form. Balachandran and Park [18] investigated the existence of a solution of nonlocal Cauchy problem for abstract FEEs of the form
Motivation
Although solutions of many applied problems lead to integral equations that are not easily in Abel's form, they can nevertheless be transformed into Abel's integral equation to obtain the solution immediately. For the material density (ρ) concentrated at a single point t = ξ, f(t) = ρ2δ(t−ξ), where δ(t−ξ) is the Dirac delta function, the equation represents the classical diffusion equation. 1.74) Equation (1.74) has the form of a time fractional diffusion equation. Viscoelasticity: The fractional derivatives and integrals are widely used in the mathematical modeling of viscoelastic materials.
The classical advection distribution equation contains first-order time derivative and second-order spatial derivative.
Introduction
Nirmalkumar and Murugesu [89] discussed an approximate controllability problem in Volterra-Fredholm FEE with infinite delay in Banach spaces. After the introduction of non-instantaneous impulses by Hern'andez and O'Regan [60], who treated a dynamic system that was affected by an impulsive action starting from a random fixed point and remaining dynamic for a certain interval, many mathematics - ematicians have investigated various classes of differential equations in both the classical and partial configurations. The same problem as above, without Fredholm's arguments, was investigated by Anguraj and Kanjanadevi [8] and they studied the existence results using the fixed point theorem for map condensation and the resolving operator.
Motivated by the above literature, we consider here the following impulsive functional differential equation with non-instantaneous impulses:.
Preliminaries
Concept of mild solution
Existence of mild solutions
Main Results
In the next result, we establish the existence of mild solution via Krasnoselskii's fixed point theorem. Krasnoselskii's fixed point theorem therefore ensures that T has a fixed point which gives rise to a mild solution.
Examples
Now to validate the result obtained in Theorem 2.5.2, let us take and define β = 13.
Conclusion
In this chapter we establish sufficient conditions for the existence and uniqueness of the integral solution to a non-densely defined, non-instantaneous impulsive evolution equation on a Banach space with a Caputo fractional derivative. Da Prato and Sinestrari [41] started the study of evolution equations with a non-densely defined linear operator. Some results about the existence of an integral solution of a non-densely defined evolution equation without momentum have been proven under suitable hypotheses for any X-value continuous function f and anyx0 ∈D(A).
Zhang and Liu [124] corrected an error in the formulation of the integral solution for the fuzzy defined partial differential.
Preliminaries
We use the principle of contraction mapping and Krasnoselskii's fixed point theorem to prove the existence of the integral solution of the problem. 56] By the integral solution y(t) of the non-homogeneous fractional order evolution system (with continuous source f).
Integral solution to a nonlinear Cauchy problem
Assuming that the conditions (H1)-(H6) are satisfied with the exception of condition (H3), the functions Gi(.,0) are bounded and. It therefore follows from Step IV that F yn →F y is uniform on (ti, ti+1] as n → ∞ and therefore F is continuous. Then Krasnoselskii's fixed point theorem ensures that T has a fixed point which gives rise to a light point solution.
Application
Conclusion
6] studied the existence of the solution of a class of fractional neutral functional differential equation with finite delay of the form. 25] established sufficient conditions for the existence and uniqueness of the solution of semilinear functional differential equations with finite delay. In this paper, we establish sufficient conditions for the existence of the smooth solution for a class of finite-delay impulsive fractional functional differential equation.
In section 4.3 we give sufficient conditions for the existence of a mild solution of the system.
Preliminaries
Existence of PC-mild solution
From the hypothesis (H3) and the work in [40], it is easy to verify that the operator is well-defined. The first term on the right-hand side tends to zero as τ2 →τ1, since S(t) is compact for t > 0 and hence continuous in the uniform operator topology. From the compactness of the operator Q(βδ), we say that the group Vδ(l) ={(Fi2)δy(l) : y∈Bη} is relatively compact in X.
Example
Conclusion
In this chapter, we extend the result obtained in the previous chapter to a class of impulsive fractional functional evolution equations with infinite delay. 39] discussed the controllability of a first-order impulsive functional differential system with infinite delay in Banach spaces. 65] established the existence of mild solution of a class of Riemann-Liouville fractional evolution equation with non-local conditions and infinite delay in a Banach space, where the linear part.
Some interesting results on the existence of solution of impulsive fractional evolution equations with infinite delay are obtained in.
Preliminaries
Mahmudov and Zorlu [80] investigated the approximate controllability of fractional evolution equations with a compact analytical semigroup. To find the solution of a differential equation at any time with infinite delay requires knowledge not only of the current state, but also of the past state. The choice of phase space is thus one of the most important features in solving such equations.
64], it is a common practice to take the phase space as a semi-normed space satisfying some axioms.
Definition of PC-mild solution and assumptions
Existence of PC-mild solution
In the previous theorem, we established the existence and uniqueness of the PC-mild solution by applying Lipschitz conditions to both the source and impulse functions. In our next result, we relax the Lipschitz condition on the source function and use the Burton-Kirk fixed point theorem to establish the existence of a PC-mild solution. To use the Burton-Kirk fixed point theorem, we split our operator ˜F : ´PT → P´T introduced in the previous theorem into two parts given by .
According to the Burton-Kirk fixed point theorem, the operator ˜F has a fixed point sincey(t) =z(t) + Φ(t), t∈(−∞, T).Theny is a fixed point of the operator T that's a mild solution to the problem.
Example
Conclusion
In this section, we establish a set of sufficient conditions for the existence of a mild solution of the FDE class with non-instantaneous impulses. The results are obtained using Banach's Fixed Point Theorem and Krasnoselski's Fixed Point Theorem.
Introduction
Mild solution for non-instantaneous impulsive FDE with non-local conditions is mostly an unaddressed topic in the literature. Motivated by the work of Wang and Li [113], we discuss the existence and uniqueness of the following problem:
Preliminaries
Main Results
Using Krasnoselski's fixed point theorem, T =T1+T2 has a fixed point which is a solution to the problem.
Application
Since all the assumptions in Theorem 6.3.1 are satisfied, our results can thus be applied to the problem given by Eqs.
Conclusion
This chapter is about the existence of a mild solution of a class of abstract FEE with a nearly sectoral operator. 116] established the existence of a classical and mild solution of the linear and semilinear abstract fractional Cauchy problem using a nearly sectoral operator. Fang Li [77] investigated the existence of a mild solution to postpone FDEs with a nearly sectoral operator.
In this chapter, we prove the existence and uniqueness of mild solution of a class of semilinear impulsive Cauchy problem with an almost sectoral operator of the following form:.
Preliminaries
The existence theorems for mild solutions for the Riemann Liouville fractional Cauchy problem with an almost sectoral have been studied by Zhang and Zhou [121].
Existence and uniqueness of mild solutions
Existence of solutions to fractional impulsive neutral functional infinite delay integro-differential equations with non-local states. Existence and uniqueness of solutions of fractional quasilinear mixed integro-differential equations with non-local state in banach spaces. Soft solutions to abstract fractional differential equations with almost sectoral operators and infinite delay.
The existence of mild solutions to Sobolev-type fractional integrodifferential equations with nonlocal conditions.
