Traditionally oriented **elementary** **differential** **equations** 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

Show more
806 Read more

Anti-periodic **boundary** **value** **problems** 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 **boundary** **value** problem for nonlinear fractional diﬀerential equation:

15 Read more

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.

Show more
13 Read more

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.

13 Read more

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 [–].

12 Read more

where < α ≤ . The existence results were obtained by the nonlinear alternative theorem. Inspired by above work, the author will be concerned with the **boundary** **value** 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 **boundary** **value** problem. In addition, the conditions imposed in this paper are easily veriﬁed.

Show more
19 Read more

Recently, there have been many papers focused on **boundary** **value** **problems** of frac- tional ordinary diﬀerential **equations** [–] and an initial **value** problem of fractional functional diﬀerential **equations** [–]. But the results dealing with the **boundary** **value** **problems** 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

Show more
15 Read more

Of concern is the existence of solutions for a class of **boundary** **value** **problems** 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.

17 Read more

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 (.).

Show more
18 Read more

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 **boundary** **value** **problems** 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 **boundary** **value** **problems** appear in physics in a variety of situations 33, 34. Recently, the existence results were extended to antiperiodic **boundary** **value** **problems** for first-order impulsive diﬀerential **equations** 35, 36. Very recently, Wang and Shen 37 investigated the antiperiodic **boundary** **value** problem for a class of second-order diﬀerential **equations** by using Schauder’s fixed point theorem and the lower and upper solutions method.

Show more
14 Read more

We have proved the existence of solutions for two classes of fractional diﬀerential equa- tions with periodic **boundary** **value** 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.

27 Read more

Fractional order **differential** **equations** are generalizations of integer order **differential** **equations**. Using fractional order **differential** **equations** can help us to reduce the errors arising from the neglected parameters in modeling real life phenomena. Fractional **differential** **equations** have many applications see Chapter 10 in [63], books [41, 63, 66].

68 Read more

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 **boundary** **value** 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.

Show more
11 Read more

Compared with the existing literature, this paper has the following two new features. First, diﬀerent from [], inﬁnite-point **boundary** **value** 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.

Show more
11 Read more

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 **boundary** **value** **problems**. Our conclusions and conditions are di ﬀ erent from the ones used in [1–10] for single **equations**.

Show more
12 Read more

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-**differential** **equations**. 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 **boundary** **value** **problems** of Fredholm functional integro-**differential** **equations**.

Show more
Mathematical modelling of real-life **problems** usually results in functional **equations**, like ordinary or partial **differential** **equations**, integral and integro- **differential** **equations**, stochastic **equations**. Many mathematical formulation of physical phenomena contain integro-**differential** **equations**, these **equations** arises in many fields like fluid dynamics, biological models and chemical kinetics integro-**differential** **equations** are usually difficult to solve analytically so it is required to obtain an efficient approximate solution. Consider the following **boundary** **value** **problems** of functional integro-**differential** **equations** with the nonlocal **boundary** conditions.

Show more
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.

20 Read more

[3] 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. [4] 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 **differential** **equations** on terms of special functions. Recently, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional **differential** **equations** **boundary** **value** **problems**, see [8]-[14].