Some new versions of fractional boundary value problems with slit-strips conditions
© Ahmad and Agarwal; licensee Springer. 2014
Received: 27 May 2014
Accepted: 1 July 2014
Published: 24 September 2014
We discuss the existence and uniqueness of solutions for a fractional differential equation of order with slit-strips type boundary conditions. The slit-strips type boundary condition states that the sum of the influences due to finite strips of arbitrary lengths is related to the value of the unknown function at an arbitrary position (nonlocal point) in the slit (a part of the boundary off the two strips). The desired results are obtained by applying standard tools of the fixed point theory and are well illustrated with the aid of examples. We also extend our discussion to the cases of arbitrary number of nonlocal points in the slit, the nonlocal multi-substrips conditions and Riemann-Liouville type slit-strips boundary conditions.
MSC: 34A12, 34A40.
where denotes the Caputo fractional derivative of order , is a given continuous function, and and are real positive constants.
In the problem (1.1)-(1.2), the integral boundary condition describes that the contribution due to finite strips of arbitrary lengths occupying the positions and on the interval is related to the value of the unknown function at a nonlocal point () located at an arbitrary position in the aperture (slit) - the region of the boundary off the strips. Examples of such boundary conditions include scattering by slits –, silicon strips detectors for scanned multi-slit X-ray imaging , acoustic impedance of baffled strips radiators , diffraction from an elastic knife-edge adjacent to a strip , sound fields of infinitely long strips , dielectric-loaded multiple slits in a conducting plane , lattice engineering , heat conduction by finite regions (Chapter 5, ), etc.
where , with , , .
Boundary value problems for nonlinear differential equations arise in a variety of areas such as applied mathematics, physics, and variational problems of control theory. In recent years, the study of boundary value problems of fractional order has attracted the attention of many scientists and researchers and the subject has been developed in several disciplines. Significant development of the topic over the past few years clearly indicates its popularity. As a matter of fact, the literature on the topic is now well enriched with a variety of results covering theoretical as well as application aspects of the subject. In consequence, fractional calculus has evolved as an interesting topic of research and its tools have played a key role in improving the mathematical modeling of many physical and engineering phenomena. The nonlocal nature of fractional-order operators is one of the salient features accounting for the practical utility of the subject. With the aid of fractional calculus, it has now become possible to trace the history of many important materials and processes. For examples and application details in physics, chemistry, biology, biophysics, blood flow phenomena, control theory, wave propagation, signal and image processing, viscoelasticity, percolation, identification, fitting of experimental data, economics, etc., we refer the reader to the books –. For some recent work on the topic, see – and the references therein.
We emphasize that the problems considered in this paper are new and important from application point of view. The existence theory developed for the fractional differential equation (1.1) subject to the boundary conditions (1.2), (1.3), (1.4) and (1.5) will lead to a useful and significant contribution to the existing material on nonlocal fractional-order boundary value problems.
2 Existence results for the problem (1.1)-(1.2)
First of all, we recall some basic definitions of fractional calculus.
provided the integral exists.
where denotes the integer part of the real number .
Next we present an auxiliary lemma which plays a key role in proving the main results for the problem (1.1)-(1.2).
where are arbitrary constants.
Substituting the values of in (2.4), we get (2.2). This completes the proof. □
Let denote the Banach space of all continuous functions from to ℝ endowed with the norm: .
where is given by (2.3). Observe that the problem (1.1)-(1.2) has solutions if and only if the operator ℋ has fixed points.
Letbe a continuous function satisfying the Lipschitz condition:
():, , , .
Then the problem (1.1)-(1.2) has a unique solution if, whereis given by (2.5).
which implies that , where we have used (2.5).
Since by the given assumption, therefore the operator ℋ is a contraction. Thus, by Banach’s contraction mapping principle, there exists a unique solution for the problem (1.1)-(1.2). This completes the proof. □
The next result is based on Krasnoselskii’s fixed point theorem .
Letbe a continuous function satisfying () and
():, , and.
For , it is easy to show that , which implies that .
which tends to zero independent of as . This implies that is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus all the assumptions of Krasnoselskii’s fixed point theorem are satisfied. So the problem (1.1)-(1.2) has at least one solution on . This completes the proof. □
Our next result is based on the following fixed point theorem .
Letbe a Banach space. Assume thatis a completely continuous operator and the setis bounded. Thenhas a fixed point in .
Assume that exists a positive constantsuch thatfor all, . Then there exists at least one solution for the problem (1.1)-(1.2).
Therefore, ℋ is equicontinuous on . Thus, by the Arzelá-Ascoli theorem, the operator ℋ is completely continuous.
and . Hence, , , . So is bounded. Thus, Theorem 2.3 applies and in consequence the problem (1.1)-(1.2) has at least one solution. This completes the proof. □
Our final result is based on the Leray-Schauder nonlinear alternative.
(Nonlinear alternative for single valued maps )
has a fixed point in , or
there is a (the boundary of in ) and with .
Letbe a continuous function. Assume that
():there exist a functionand a nondecreasing functionsuch that, ;
Then the problem (1.1)-(1.2) has at least one solution on.
Clearly, the right-hand side tends to zero independently of as . Thus, by the Arzelá-Ascoli theorem, the operator ℋ is completely continuous.
In view of (), there exists such that . Let us choose .
Observe that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by Lemma 2.2, we deduce that the operator ℋ has a fixed point which is a solution of the problem (1.1)-(1.2). This completes the proof. □
Clearly . Thus all the conditions of Theorem 2.1 are satisfied and, consequently, there exists a unique solution for the problem (2.7).
we find that . Thus, by Theorem 2.5, there exists at least one solution for the problem (2.7) with given by (2.8).
3 Nonlocal multi-point case on the aperture
4 Nonlocal multi-substrips case
where is given by (4.2). With the above setting, the existence results, analogous to the ones given in Section 2, can be obtained for the problem (1.1) and (1.4) in a similar way.
5 Riemann-Liouville slit-strips boundary conditions
The existence results for the problem (1.1) and (1.5) can be obtained by employing the procedure used in Section 2.
This paper was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.
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