In this paper, using Riemann-Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane-Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results.
KEYWORDSCaputo derivative, existence of solution, Lane-Emden equation, multi-point problem, singular equation MSC CLASSIFICATION 30C45; 39B72; 39B82 Recently, Caputo-Fabrizio fractional derivative (CFD for short) has been introduced. 16,17 This derivative is defined in the space H 1 (0, b), b > 0 as the convolution of an ordinary derivative and an exponential function. The beauty of CFD is that it has a nonsingular kernel! It has added a new important dimension to the study of fractional differential equations and systems. Many results with the CFD have been developed in the last 5 years. For more details about this interesting approach, and for some important applications, we cite the papers 18-20 where we find some very interesting ideas on applications of Caputo-Fabrizio theory in differential equations, integro-differential equations, and differential inclusions. We cite also the important papers 21-24 for some new elegant extensions and generalizations for CFD (e.g., the two types of high order derivations CFD and the extended fractional CFD, for the case 0 ≤ < 1, C R[0,1] , and their applications on fractional differential equations and fractional differential inclusions). The papers 6,25 are also of an important interest; the first Math Meth Appl
In this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.
We focus on a new type of nonlinear integro-differential equations with
nonlocal integral conditions. The considered problem has one
nonlinearity with time variable singularity. It involves also some
convergent series combined to Riemann-Liouville integrals. We prove a
uniqueness of solutions for the proposed problem, then, we provide some
examples to illustrate this result. Also, we discuss the Ulam-Hyers
stability for the problem. Some numerical simulations, using Rung Kutta
method, are discussed too. At the end, a conclusion follows.
A coupled system of singular fractional differential equations involving Riemann–Liouville integral and Caputo derivative is considered in this paper. The question of existence and uniqueness of solutions is studied using Banach contraction principle. Furthermore, the question of existence of at least one solution is discussed. At the end, an illustrative example is given in details.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.