Traditionally oriented elementarydifferentialequations texts are occasionally criticized as being col- lections of unrelated methods for solving miscellaneous problems. To some extent this is true; after all, no single method applies to all situations. Nevertheless, I believe that one idea can go a long way toward unifying some of the techniques for solving diverse problems: variation of parameters. I use variation of parameters at the earliest opportunity in Section 2.1, to solve the nonhomogeneous linear equation, given a nontrivial solution of the complementary equation. You may find this annoying, since most of us learned that one should use integrating factors for this task, while perhaps mentioning the variation of parameters option in an exercise. However, there’s little difference between the two approaches, since an integrating factor is nothing more than the reciprocal of a nontrivial solution of the complementary equation. The advantage of using variation of parameters here is that it introduces the concept in its simplest form and
Anti-periodic boundaryvalueproblems occur in the mathematic modeling of a variety of physical processes and have recently received considerable attention. For examples and details of anti-periodic fractional boundary conditions, see [–]. In , Agarwal and Ahmad studied the solvability of the following anti-periodic boundaryvalue problem for nonlinear fractional diﬀerential equation:
It has been noticed that most of the work on the topic is based on Riemann-Liouville and Caputo type fractional diﬀerential equations. Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in , which diﬀers from the preceding ones in the sense that the kernel of the integral (in the deﬁnition of Hadamard derivative) contains a logarithmic function of arbitrary exponent. Details and properties of the Hadamard fractional derivative and integral can be found in [, –]. However, this calculus with Hadamard derivatives is still studied less than that of Riemann-Liouville.
8 T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces, Nova Science, New York, NY, USA, 2007. 9 T. Diagana, G. M. Mophou, and G. M. N’Gu´er´ekata, “On the existence of mild solutions to some semilinear fractional integro-diﬀerential equations,” Electronic Journal of Qualitative Theory of Diﬀerential Equations, vol. 2010, no. 58, pp. 1–17, 2010.
Impulsive diﬀerential equations describe processes which experience a sudden change of their state at certain moments. The theory of impulse diﬀerential equations has been a signiﬁcant development in recent years and played a very important role in modern applied mathematical models of real processes arising in phenomena studied in physics, population dynamics, chemical technology and biotechnology; see [–].
where < α ≤ . The existence results were obtained by the nonlinear alternative theorem. Inspired by above work, the author will be concerned with the boundaryvalue prob- lem (BVP for short in the sequel) (.)-(.). To the best of our knowledge, no contribution exists concerning the existence of solutions for BVP (.)-(.). In the present paper, by ap- plying a new ﬁxed point theorem on cone and Krasnoselskii’s ﬁxed point theorem, some existence results of positive solution for BVP (.)-(.) are obtained. It is worth to point out that the results in this paper are also new even for α = relative to the correspond- ing literature with regard to the fourth-order boundaryvalue problem. In addition, the conditions imposed in this paper are easily veriﬁed.
Recently, there have been many papers focused on boundaryvalueproblems of frac- tional ordinary diﬀerential equations [–] and an initial value problem of fractional functional diﬀerential equations [–]. But the results dealing with the boundaryvalueproblems of fractional functional diﬀerential equations with delay are relatively scarce [–]. It is well known that in practical problems, the behavior of systems not only depends on the status just at the present, but also on the status in the past. Thus, in many
Of concern is the existence of solutions for a class of boundaryvalueproblems for impulsive fractional diﬀerential equations involving the Caputo fractional derivative in a Banach space. Our approach is based upon the techniques of noncompactness measures and the ﬁxed point theory. Two examples are presented to illustrate the results.
The monotone iterative technique coupled with the method of lower and upper solu- tions provides an eﬀective way to obtain two sequences which approximate the extremal solutions between lower and upper solutions of nonlinear diﬀerential and impulsive dif- ferential (fractional or non-fractional ordered) equations; see, for instance, [–]. To the best of the authors’ knowledge, this is the ﬁrst paper establishing the impulsive frac- tional diﬀerential equations via the conformable fractional calculus developed by . By means of a new maximal principle and new deﬁnitions of lower and upper solutions, the monotone iterative technique will be used in our investigation of the problem (.).
Impulsive diﬀerential equations, which arise in biology, physics, population dynamics, economics, and so forth, are a basic tool to study evolution processes that are subjected to abrupt in their states see 1–4. Many literatures have been published about existence of solutions for first-order and second-order impulsive ordinary diﬀerential equations with boundary conditions 5–19, which are important for complementing the theory of impulsive equations. In recent years, the solvability of the antiperiodic boundaryvalueproblems of first-order and second-order diﬀerential equations were studied by many authors, for example, we refer to 20–32 and the references therein. It should be noted that antiperiodic boundaryvalueproblems appear in physics in a variety of situations 33, 34. Recently, the existence results were extended to antiperiodic boundaryvalueproblems for first-order impulsive diﬀerential equations 35, 36. Very recently, Wang and Shen 37 investigated the antiperiodic boundaryvalue problem for a class of second-order diﬀerential equations by using Schauder’s fixed point theorem and the lower and upper solutions method.
We have proved the existence of solutions for two classes of fractional diﬀerential equa- tions with periodic boundaryvalue conditions, where certain nonlinear growth conditions of the nonlinearity need to be satisﬁed. The problem is issued by applying the Leggett– Williams norm-type theorem for coincidences. We also provide examples to make our results clear.
Fractional order differentialequations are generalizations of integer order differentialequations. Using fractional order differentialequations can help us to reduce the errors arising from the neglected parameters in modeling real life phenomena. Fractional differentialequations have many applications see Chapter 10 in , books [41, 63, 66].
In this paper, we study boundary condition () in the case when the function g(x, y) is nondecreasing with respect to its second argument y. So, the anti-periodic boundaryvalue problem is a partial case of boundary condition (). Note that similar problems are investi- gated for ordinary diﬀerential equations , delay diﬀerential equations  and impulsive diﬀerential equations , and some approximate methods are suggested. The presence of the maximum of the unknown function requires additionally some new comparison results, existence results as well as a new algorithm for constructing successive approxi- mations to the exact unknown solution.
Compared with the existing literature, this paper has the following two new features. First, diﬀerent from , inﬁnite-point boundaryvalue conditions are considered in this paper. At the same time, the nonlinearity f in this paper permits singularities with respect to both the time and the space variables which is seldom considered at present. Second, the purpose of this paper is to investigate the existence of multiple positive solutions for BVP (). As to multiple positive solutions, it is worth pointing out that conditions imposed on f are diﬀerent from that in . To achieve this goal, ﬁrst we convert the expression of the unique solution into an integral form and then get the Green function BVP (). After further discussion of the properties of the Green function, a suitable cone is constructed to obtain the main result in this paper by means of the Guo-Krasnoselskii ﬁxed point theorem.
Fourth-order nonlinear diﬀerential equations have many applications such as balancing condition of an elastic beam which may be described by nonlinear fourth-order ordinary diﬀerential equations. Concerning the studies for singular and nonsingular case, one can refer to [1–10]. However, there are not many results on the system for nonlinear fourth- order di ﬀ erential equations. In this paper, by using topological degree theory and cone theory, we study the existence of the positive solutions and the multiple positive solutions for singular and nonsingular system of nonlinear fourth-order boundaryvalueproblems. Our conclusions and conditions are di ﬀ erent from the ones used in [1–10] for single equations.
Mathematical modelling of real-life problems usually results in functional equations, of various types appear in many applications that arise in the fields of mathematical analysis, nonlinear functional analysis, mathematical physics, and engineering. An interesting feature of functional integral equations is their role in the study of many problems of functional integro-differentialequations. Several different techniques were proposed to study the existence of solutions of the functional integral equations in appropriate function spaces. Although all of these techniques have the same goal, they differ in the function spaces and the fixed point theorems to be applied. Consider the following boundaryvalueproblems of Fredholm functional integro-differentialequations.
Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differentialequations, integral and integro- differentialequations, stochastic equations. Many mathematical formulation of physical phenomena contain integro-differentialequations, these equations arises in many fields like fluid dynamics, biological models and chemical kinetics integro-differentialequations are usually difficult to solve analytically so it is required to obtain an efficient approximate solution. Consider the following boundaryvalueproblems of functional integro-differentialequations with the nonlocal boundary conditions.
In this paper, we have discussed the achievement of sufficient conditions for the existence of solutions of fractional BVP (1.1) which is more general than problems in literatures review [1, 3, 20], by applying Krasnoselskii’s fixed point theorem, nonlinear alternative of Leray-Schauder type, H¨ older inequality and fractional calculus. Examples provided to illustrate main results.
 A. Alibeigloo, H. Jafarian, ”Three-Dimensional Static and Free Vibra- tion Analysis of Carbon Nano Tube Reinforced Composite Cylindrical Shell Using Differential Quadrature Method,” International Journal of Applied Mechanics, DOI: 10.1142/S1758825116500332, 2016.  M. M. KIPNIS, I. S. LEVITSKAYA, ”STABILITY OF DELAY D-
It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differentialequations on terms of special functions. Recently, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional differentialequationsboundaryvalueproblems, see -.