Exact Controllability for Nonlinear Implicit Caputo Fractional Models with Hybrid Structure

This paper focuses on a class of Hadamard‐type fractional differential equation with nonlocal boundary conditions on an infinite interval. New existence, uniqueness, and multiplicity results of positive solutions are obtained by using Schauder's fixed point theorem, Banach's contraction mapping principle, the monotone iterative method, and the Avery‐Peterson fixed point theorem. Examples are included to illustrate our main results.

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

This work aims to investigate a delay hematopoiesis model where the delay depends on both the time and the current density of mature blood cells. Based on the Banach contraction principle, the Schauder's fixed point theorem and some properties of a Green's function, we establish several interesting existence and uniqueness results of positive periodic solutions for the proposed model. The derived results are new and generalize some previous studies.

This paper is concerned with the existence and uniqueness of solutions for boundary value problems with p-Laplacian delay differential equations on the half-line. The existence of solutions is derived from the Schauder fixed point theorem, whereas the uniqueness of solution is established by the Banach contraction principle. As an application, an example is given to demonstrate the main results.MSC:34K10, 34B18, 34B40.

By employing the Srivastava-Owa fractional operators, we consider a class of fractional differential equation in the unit disk. The existence of the univalent solution is founded by using the Schauder fixed point theorem while the uniqueness is obtained by using the Banach fixed point theorem. Moreover, the integral mean of these solutions is studied by applying the concept of the subordination.

In this thesis we investigate the existence of solutions to boundary value problems (BVPs) for nonlinear systems of first-order ordinary differential equations (ODEs) and difference equations. The thesis is divided into two parts. Part one is devoted to the study of difference equations where two types of problems are studied. First, we derive the existence of solutions to first-order systems of difference equations that arise when one applies the trapezoidal rule to approximate solutions of second-order scalar ODEs. Strict discrete lower solutions are used with maximum principle arguments in the discrete problem to obtain a priori bounds on solutions. The a priori bounds on difference quotients for solutions of the discrete problem are obtained from the Bernstein-Nagumo condition. We use homototopy to find solutions of the discrete amroximations. Second, motivated by Ma [60] we consider existence and uniqueness of solutions to difference equations which arise as discrete approximations to three-point BVPs for systems of first-order ODES after employing the Euler method. Several existence and uniqueness results for three-point BVPs are established, using the contraction mapping theorem and the Brouwer fixed point theorem in Rn. In part two, two types of problems for systems of first-order ODES are analyzed. First, we study the existence of a unique solution to a system of first-order differentia1 equations under multi-point BVPs with nonlinear boundary conditions. We prove the existence and uniqueness of solutions for a system of first-order three-point BVP under which the properties of linear three-point BVP are preserved under small nonlinear perturbations of both the differentia1 equation and the boundary conditions. The method of proof of the existence of solutions using the Banach contraction mapping principle is similar to that used by Rodriguez [75], who studied the discrete analogue of this BVP. Second, motivated by Cronin 120,211 we investigate the existence of solutions to three-point BVPs in perturbed systems of first-order ODEs at resonance where the associated homogeneous linear problem has nontrivial solutions. Based on the second part of the investigation of three-point BVPs in perturbed systems of first-order ODEs at resonance, four different results were obtained: (i) The existence of solutions via a version of the Brouwer Fixed Point Theorem which is due to Miranda. (ii) The existence of solutions to the problem through the application of the Implicit Function Theorem. These two results extend the work of Feng and Webb [34], and Gupta [39] for the problems at resonance in Euclidean 2-space. (iii) Existence results for three-point BVPs at resonance for general BVPs through the application of Brouwer degree theory; we show the degree is non zero through the application of Borsuk's Theorem for one problem and through a result of Cronin involving pairs of polynomials in two variables whose terms of highest order have no common factors. (iv) The existence of solutions to the entrainment of frequency problem in Euclidean 2-space. We will investigate conditions under which entrainment of frequency for three-point as well as two-point BVPs by adapting the work of Cronin in periodic case. We will describe such conditions and then show the degree is non zero through a result of Cronin involving pairs of polynomials in two variables whose terms of lowest order have no common factors.

In this paper, the fuzzy multiterm fractional differential equation involving Caputo-type fuzzy fractional derivative of order 0<α<1 is considered. The uniqueness of solution is established by using the contraction mapping principle and the existence of solution is obtained by Schauder fixed point theorem.

This paper is concerned with a class of fractional partial functional differential equations with nonlocal conditions in Banach spaces. By using the theories of resolvent operator, the Schauder fixed point theorem and the Banach contraction mapping principle, we obtain the sufficient conditions for the local and global existence of the mild solution.

In this paper, we study a integral boundary value problem of fractional differential equation with the nonlinearity depending on fractional derivatives of lower order on an infinite interval. We establish a proper compactness criterion in a special function space. By using the Schauder fixed point theorem and Banach contraction mapping principle, we show the existence and uniqueness results of solutions. Two examples are also provided to illustrate the main results. The results obtained generalize and include some known results.

Sufficient conditions for the controllability of nonlinear stochastic fractional boundary control systems are established. The equivalent integral equations are derived for both linear and nonlinear systems, and the control function is given in terms of the pseudoinverse operator. The Banach contraction mapping theorem is used to obtain the result. A controllability result for nonlinear stochastic fractional integrodifferential systems is also attained. Examples are included to illustrate the theory.

In this paper, by using both Banach fixed point theorem and Schauder's fixed point theorem we obtain the existence and uniqueness results for a kind of Katugampola fractional differential equation.

<abstract><p>In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e.,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \label{fredholm} w(\zeta_1) = y(\zeta_1)+w(\zeta_1)\int_{\bf J}\kappa(\zeta_1, \varphi)(T_1w)(\varphi)d\varphi+z(\zeta_1)\int_{\bf J}h(\varphi, (T_2w)(\varphi))d\varphi, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>for $ \zeta_1\in{\bf J} = [0, 1], $ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.</p></abstract>

In this paper, we obtain an equivalent nonlinear integral equation to the stochastic neutral fractional system with bounded operator. Using the integral equation, the sufficient conditions for ensuring the complete controllability of the stochastic fractional neutral systems with Wiener and Lévy noise are obtained. Banach's fixed point theorem is used to obtain the results. Examples are provided to illustrate the theory.

In this paper, a general class of nonautonomous Lotka-Volterra dispersal system with discrete and continuous infinite delays is investigated. This class of Lotka-Volterra systems model the diffusion of a single species into n patches by discrete dispersal. By using Schauder's fixed point theorem, we prove the existence of positive periodic solutions of system. The global asymptotical stability of positive periodic solution is discussed and the sufficient conditions for exponential stability are also given. we give an example to illustrate the validity of the results in the end. The conditions we obtained are more general and it can be extended to several special systems.

Positive periodic solutions of delayed periodic Lotka–Volterra systems