A Study of Nonlinear Fractional Differential Equations of Arbitrary Order with Riemann-Liouville Type Multistrip Boundary Conditions

Abstract

We develop the existence theory for nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type boundary conditions involving nonintersecting finite many strips of arbitrary length. Our results are based on some standard tools of fixed point theory. For the illustration of the results, some examples are also discussed.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Bashir Ahmad 1 and Sotiris K. Ntouyas 2 and Ahmed Alsaedi 1

Recommended by José Tenreiro Machado

1, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2, Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 28 November 2012; Accepted 19 January 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The subject of fractional calculus has recently developed into a hot topic for the researchers in view of its numerous applications in the field of physics, mechanics, chemistry, engineering, and so forth. One can find the systematic progress of the topic in the books ([ 1- 6]). A significant characteristic of a fractional-order differential operator distinguishing it from the integer-order differential operator is that it is nonlocal in nature, that is, the future state of a dynamical system or process involving fractional derivative depends on its current state as well its past states. In fact, this feature of fractional-order operators has contributed towards the popularity of fractional-order models, which are recognized as more realistic and practical than the classical integer-order models. In other words, we can say that the memory and hereditary properties of various materials and processes can be described by differential equations of arbitrary order. There has been a rapid development in the theoretical aspects such as periodicity, asymptotic behavior, and numerical methods for fractional equations. For some recent work on the topic, see ([ 7- 23]) and the references therein. In particular, Ahmad et al. [ 22] studied nonlinear fractional differential equations and inclusions of arbitrary order with multistrip boundary conditions.

In this paper, we continue the study initiated in [ 22] and consider a boundary value problem of fractional differential equations of arbitrary order q ∈ (n -1 ,n ] , n ...5;2 with finite many multistrip Riemann-Liouville type integral boundary conditions: [figure omitted; refer to PDF] where ^{c D q} denotes the Caputo fractional derivative of order q , f is a given continuous function, ^{I _{β i}} is the Riemann-Liouville fractional integral of order _{β i} >0 , i =1,2 , ... ,m , 0 < _{ζ 1} < _{η 1} < _{ζ 2} < _{η 2} < ... < _{ζ m} < _{η m} <T , and _{γ i} ∈ ... are suitable chosen constants.

Regarding the motivation of the problem, we know that the strip conditions appear in the mathematical modeling of certain real world problems, for instance, see [ 24, 25]. In [ 22], the authors considered the nonlocal strip conditions of the form: [figure omitted; refer to PDF] In the problem ( 1), we have introduced Riemann-Liouville type multistrip integral boundary conditions which can be interpreted as the controller at the right-end of the interval under consideration is influenced by a discrete distribution of finite many nonintersecting sensors (strips) of arbitrary length expressed in terms of Riemann-Liouville type integral boundary conditions. For some engineering applications of strip conditions, see ([ 26- 32]).

The main objective of the present study is to develop some existence results for the problem ( 1) by using standard techniques of fixed point theory. The paper is organized as follows. In Section 2we discuss a linear variant of the problem ( 1), which plays a key role in developing the main results presented in Section 3. For the illustration of the theory, we have also included some examples.

2. Preliminary Result

Let us begin this section with some basic definitions of fractional calculus [ 2- 4].

Definition 1.

If g (t ) ∈A ^{C n} [a ,b ] , then the Caputo derivative of fractional order q is defined as [figure omitted; refer to PDF] where [q ] denotes the integer part of the real number q . For details, see Theorem 2.1 ([ 4, page 92]). Here A ^{C n} [a ,b ] denote the space of real valued functions g (t ) which have continuous derivatives up to order n -1 on [a ,b ] such that ^{g n -1} (t ) ∈AC [a ,b ] .

Definition 2.

The Riemann-Liouville fractional integral of order q is defined as [figure omitted; refer to PDF] provided the integral exists.

The following result associated with a linear variant of problem ( 1) plays a pivotal role in establishing the main results.

Lemma 3.

For h ∈C [0 ,T ] , the fractional boundary value problem [figure omitted; refer to PDF] has a unique solution x (t ) ∈A ^{C n} [0 ,T ] given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

The general solution of fractional differential equations in ( 5) can be written as [figure omitted; refer to PDF] Using the given boundary conditions, it is found that _{c 0} =0 , _{c 1} =0 , ... , _{c n -2} =0 . Applying the Riemann-Liouville integral operator ^{I _{β i}} on ( 8), we get [figure omitted; refer to PDF] Using the condition x (T ) = ^{∑ i =1 m} ... _{γ i} [ ^{I _{β i}} x ( _{η i} ) - ^{I _{β i}} x ( _{ζ i} ) ] , together with the fact that [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] which yields [figure omitted; refer to PDF] where λ is given by ( 7). Substituting the values of _{c 0} , _{c 1} , ... , _{c n -2} , _{c n -1} in ( 8), we obtain ( 6). This completes the proof.

3. Main Results

Let ...9E; : =C ( [0 ,T ] , ... ) denotes the Banach space of all continuous functions defined on [0 ,T ] × ... endowed with a topology of uniform convergence with the norm ||x || = _{sup t ∈ [0 ,T ]} |x (t ) | .

By Lemma 3, we define an operator ...AB; : ...9E; [arrow right] ...9E; as [figure omitted; refer to PDF]

Observe that the problem ( 1) has a solution if and only if the associated fixed point problem ...AB;x =x has a fixed point.

In the first result we prove an existence and uniqueness result by means of Banach's contraction mapping principle. For the sake of convenience, we set [figure omitted; refer to PDF]

Theorem 4.

Suppose that f : [0 ,T ] × ... [arrow right] ... is a continuous function and satisfies the following assumption:

( _{A 3} ) : [figure omitted; refer to PDF]

Then the boundary value problem ( 1) has a unique solution provided [figure omitted; refer to PDF] where Λ is given by ( 14).

Proof.

With r ...5;M Λ / (1 -L Λ ) , we define _{B r} = {x ∈ ...9E; : ||x || ...4;r } , where M = _{sup t ∈ [0 ,T ]} |f (t ,0 ) | < ∞ and Λ is given by ( 14). Then we show that ...AB; _{B r} ⊂ _{B r} . For x ∈ _{B r} , by means of the inequality |f (s ,x (s ) ) | ...4; |f (s ,x (s ) ) -f (s ,0 ) | + |f (s ,0 ) | ...4;L ||x || +M ...4;Lr +M , it can easily be shown that [figure omitted; refer to PDF] Now, for x ,y ∈ ...9E; and for each t ∈ [0 ,T ] , we obtain [figure omitted; refer to PDF] Note that Λ depends only on the parameters involved in the problem. As L Λ <1 , therefore ...AB; is a contraction. Hence, by Banach's contraction mapping principle, the problem ( 1) has a unique solution on [0 ,T ] .

Example 5.

Let us consider the following 4 -strip nonlocal boundary value problem: [figure omitted; refer to PDF] where q =9 /2 , n =5 , _{ζ 1} =1 /4 , _{η 1} =1 /2 , _{ζ 2} =2 /3 , _{η 2} =1 , _{ζ 3} =5 /4 , _{η 3} =4 /3 , _{ζ 4} =3 /2 , _{η 4} =7 /4 , _{γ 1} =5 , _{γ 2} =10 , _{γ 3} =15 , _{γ 4} =25 , _{β 1} =5 /4 , _{β 2} =7 /4 , _{β 3} =9 /4 , _{β 4} =11 /4 .

With the given values of the parameters involved, we find that [figure omitted; refer to PDF] Let us choose [figure omitted; refer to PDF] Clearly L =1 /2 as |f (t ,x ) -f (t ,y ) | ...4; (1 /2 ) |x -y | and L <1 / Λ , where Λ ≈1.406972 . Therefore all the conditions of Theorem 4hold and consequently there exists a unique solution for the problem ( 19) with f (t ,x (t ) ) given by ( 21).

In case of the following unbounded nonlinear function: [figure omitted; refer to PDF] we have L =9 /14 and L <1 / Λ ( Λ ≈1.406972 ). As before, the problem ( 19) with f (t ,x (t ) ) given by ( 22) has a unique solution.

In the second result we use the Leray-Schauder alternative.

Theorem 6 ((Leray-Schauder alternative) [ 33, page 4]).

Let X be a Banach space. Assume that T :X [arrow right]X is completely continuous operator and the set [figure omitted; refer to PDF] is bounded. Then T has a fixed point in X .

Theorem 7.

Assume that there exists a positive constant _{L 1} such that |f (t ,x ) | ...4; _{L 1} for t ∈ [0 ,T ] , x ∈ ... . Then the problem ( 1) has at least one solution.

Proof.

First of all, we show that the operator ...AB; is completely continuous. Note that the operator ...AB; is continuous in view of the continuity of f . Let [Bernoulli] ⊂ ...9E; be a bounded set. By the assumption that |f (t ,x ) | ...4; _{L 1} , for x ∈ [Bernoulli] , we have [figure omitted; refer to PDF] which implies that || ( ...AB;x ) || ...4; _{L 2} . Further, we find that [figure omitted; refer to PDF] Hence, for _{t 1} , _{t 2} ∈ [0 ,T ] , we have [figure omitted; refer to PDF] This implies that ...AB; is equicontinuous on [0 ,T ] . Thus, by the Arzelá-Ascoli theorem, the operator ...AB; : ...9E; [arrow right] ...9E; is completely continuous.

Next, we consider the set [figure omitted; refer to PDF] and show that the set V is bounded. Let x ∈V , then x = μ ...AB;x ,0 < μ <1 . For any t ∈ [0 ,T ] , we have [figure omitted; refer to PDF]

Thus, ||x || ...4; _{M 1} for any t ∈ [0 ,T ] . So, the set V is bounded. Thus, by the conclusion of Theorem 6, the operator ...AB; has at least one fixed point, which implies that ( 1) has at least one solution.

Example 8.

Consider the boundary value problem of Example 5with [figure omitted; refer to PDF] Observe that |f (t ,x ) | ...4; _{L 1} with _{L 1} = ^{e 2 2} (1 +ln25 ) . Thus the hypothesis of Theorem 7is satisfied. Hence by the conclusion of Theorem 7, the problem ( 19) with f (t ,x (t ) ) given by ( 29) has at least one solution.

In the next we prove one more existence result for problem ( 1), based on the following known result.

Theorem 9 (see [ 34]).

Let X be a Banach space. Assume that Ω is an open bounded subset of X with θ ∈ Ω and let T : Ω ¯ [arrow right]X be a completely continuous operator such that [figure omitted; refer to PDF] Then T has a fixed point in Ω ¯ .

Theorem 10.

Let there exist a small positive number τ such that |f (t ,x ) | ...4; ν |x | for 0 < |x | < τ , with 0 < ν ...4;1 / Λ , where Λ is given by ( 14). Then the problem ( 1) has at least one solution.

Proof.

Let us define _{[Bernoulli] τ} = {x ∈ ...9E; |" ||x || < τ } and take x ∈ ...9E; such that ||x || = τ , that is, x ∈ ∂ _{[Bernoulli] τ} . As before, it can be shown that ...AB; is completely continuous and [figure omitted; refer to PDF] which in view of the given condition ( ν Λ ...4;1 ) , gives || ...AB;x || ...4; ||x || , x ∈ ∂ _{[Bernoulli] τ} . Therefore, by Theorem 9, the operator ...AB; has at least one fixed point, which in turn implies that the problem ( 1) has at least one solution.

Example 11.

Consider the boundary value problem of Example 5and let us consider [figure omitted; refer to PDF] For sufficiently small x (ignoring ^{x 2} and higher powers of x ), we have [figure omitted; refer to PDF] Choosing b ...4;1 / Λ , all the assumptions of Theorem 10hold. Therefore, the conclusion of Theorem 10implies that the problem ( 19) with f (t ,x (t ) ) given by ( 32) has at least one solution.

Our final existence result is based on Leray-Schauder nonlinear alternative.

Lemma 12 ((Nonlinear alternative for single valued maps) [ 33, page 135]).

Let E be a Banach space, C a closed, convex subset of E , U an open subset of C and 0 ∈U . Suppose that F : U ¯ [arrow right]C is a continuous, compact (i.e., F ( U ¯ ) is a relatively compact subset of C ) map. Then either

(i) F has a fixed point in U ¯ , or

(ii) there is a u ∈ ∂U (the boundary of U in C ) and λ ∈ (0,1 ) with u = λF (u ) .

Theorem 13.

Assume that

( _{A 1} ) : there exist a function σ ∈C ( [0,1 ] , ^{... +} ) , and a nondecreasing function ψ : ^{... +} [arrow right] ^{... +} such that |f (t ,x ) | ...4; σ (t ) ψ ( ||x || ) , for all (t ,x ) ∈ [0 ,T ] × ... ;

( _{A 2} ) : there exists a constant M >0 such that [figure omitted; refer to PDF]

Then the boundary value problem ( 1) has at least one solution on [0 ,T ] .

Proof.

Consider the operator ...AB; : ...9E; [arrow right] ...9E; defined by ( 13). We show that ...AB; maps bounded sets into bounded sets in C ( [0 ,T ] , ... ) . For a positive number r , let _{B r} = {x ∈C ( [0 ,T ] , ... ) : ||x || ...4;r } be a bounded set in C ( [ 0 ,T ] , ... ) . Then [figure omitted; refer to PDF] Next we show that F maps bounded sets into equicontinuous sets of C ( [0,1 ] , ... ) . Let ^{t [variant prime]} , ^{t [variant prime][variant prime]} ∈ [0,1 ] with ^{t [variant prime]} < ^{t [variant prime][variant prime]} and x ∈ _{B r} , where _{B r} is a bounded set of C ( [0,1 ] , ... ) . Then we obtain [figure omitted; refer to PDF] Obviously the right hand side of the above inequality tends to zero independently of x ∈ _{B r} as ^{t [variant prime][variant prime]} - ^{t [variant prime]} [arrow right]0 . As ...AB; :C ( [0 ,T ] , ... ) [arrow right]C ( [0 ,T ] , ... ) satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that ...AB; is completely continuous.

Let x be a solution. Then, for t ∈ [0 ,T ] , and following the similar computations as before, we find that [figure omitted; refer to PDF] In consequence, we have [figure omitted; refer to PDF] Thus, by ( _{A 2} ) , there exists M such that ||x || ...0;M . Let us set [figure omitted; refer to PDF] Note that the operator ...AB; : V ¯ [arrow right]C ( [0 ,T ] , ... ) is continuous and completely continuous. From the choice of V , there is no x ∈ ∂V such that x = μ ...AB; (x ) for some μ ∈ (0,1 ) . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 12), we deduce that ...AB; has a fixed point x ∈ V ¯ which is a solution of the problem ( 1). This completes the proof.

Example 14.

Consider the boundary value problem of Example 5with [figure omitted; refer to PDF] Then σ (t ) =1 / t +4 and ψ ( ||x || ) =2 . Using || σ || =1 /2 , Λ ≈1.406972 , we find by the condition ( _{A 2} ) that M > Λ . Thus all the assumptions of Theorem 13are satisfied. Hence, it follows by Theorem 13that the problem ( 19) with f (t ,x (t ) ) defined by ( 40) has at least one solution.

If we choose an unbounded nonlinearity as follows: [figure omitted; refer to PDF] Then f (t ,x (t ) ) ...4; σ (t ) ψ ( ||x || ) with σ (t ) =1 / t +4 and ψ ( ||x || ) =2 + ||x || /2 . Using the earlier arguments, with || σ || =1 /2 , Λ ≈1.406972 , we find that M > _{M 1} , _{M 1} [approximate]2.170392 . Hence the problem ( 19) with f (t ,x (t ) ) given by ( 41) has at least one solution.

Acknowledgment

The authors are grateful to the anonymous referees for their valuable comments.

References

[1] K. S. Miller, B. Ross An Introduction to the Fractional Calculus and Differential Equations , of A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.

[2] S. G. Samko, A. A. Kilbas, O. I. Marichev Fractional Integrals and Derivatives , Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.

[3] I. Podlubny Fractional Differential Equations , vol. 198, of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo Theory and Applications of Fractional Differential Equations , vol. 204, of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.

[5] V. Lakshmikantham, S. Leela, J. V. Devi Theory of Fractional Dynamic Systems , Cambridge Academic, Cambridge, UK, 2009.

[6] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo Fractional Calculus Models and Numerical Methods , vol. 3, of Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012.

[7] M. Benchohra, S. Hamani, S. K. Ntouyas, "Boundary value problems for differential equations with fractional order and nonlocal conditions," Nonlinear Analysis: Theory, Methods & Applications , vol. 71, no. 7-8, pp. 2391-2396, 2009.

[8] R. P. Agarwal, M. Benchohra, S. Hamani, "A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions," Acta Applicandae Mathematicae , vol. 109, no. 3, pp. 973-1033, 2010.

[9] K. Balachandran, J. J. Trujillo, "The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces," Nonlinear Analysis: Theory, Methods & Applications , vol. 72, no. 12, pp. 4587-4593, 2010.

[10] D. Baleanu, O. G. Mustafa, "On the global existence of solutions to a class of fractional differential equations," Computers & Mathematics with Applications , vol. 59, no. 5, pp. 1835-1841, 2010.

[11] W. Zhong, W. Lin, "Nonlocal and multiple-point boundary value problem for fractional differential equations," Computers & Mathematics with Applications , vol. 59, no. 3, pp. 1345-1351, 2010.

[12] V. Gafiychuk, B. Datsko, "Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems," Computers & Mathematics with Applications , vol. 59, no. 3, pp. 1101-1107, 2010.

[13] B. Ahmad, J. J. Nieto, "Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,", article 36 Boundary Value Problems , vol. 2011, 2011.

[14] B. Ahmad, S. K. Ntouyas, "A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order," Electronic Journal of Qualitative Theory of Differential Equations , vol. 2011, no. 22, 2011.

[15] C. Cuevas, H. Soto, A. Sepúlveda, "Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations," Applied Mathematics and Computation , vol. 218, no. 5, pp. 1735-1745, 2011.

[16] N. J. Ford, M. L. Morgado, "Fractional boundary value problems: analysis and numerical methods," Fractional Calculus and Applied Analysis , vol. 14, no. 4, pp. 554-567, 2011.

[17] B. Ahmad, S. K. Ntouyas, "A note on fractional differential equations with fractional separated boundary conditions," Abstract and Applied Analysis , vol. 2012, 2012.

[18] A. Aghajani, Y. Jalilian, J. J. Trujillo, "On the existence of solutions of fractional integro-differential equations," Fractional Calculus and Applied Analysis , vol. 15, no. 1, pp. 44-69, 2012.

[19] W. Sudsutad, J. Tariboon, "Existence results of fractional integro-differential equations with m-point multi-term fractional order integral boundary conditions,", article 94 Boundary Value Problems , vol. 2012, 2012.

[20] B. Ahmad, S. K. Ntouyas, "Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions,", article 55 Boundary Value Problems , vol. 2012, 2012.

[21] B. Ahmad, J. J. Nieto, "Riemann-liouville fractional differential equations with fractional boundary conditions," Fixed Point Theory , vol. 13, pp. 329-336, 2012.

[22] B. Ahmad, J. J. Nieto, A. Alsaedi, M. El-Shahed, "A study of nonlinear Langevin equation involving two fractional orders in different intervals," Nonlinear Analysis: Real World Applications , vol. 13, no. 2, pp. 599-606, 2012.

[23] A. Cabada, G. Wang, "Positive solutions of nonlinear fractional differential equations with integral boundary value conditions," Journal of Mathematical Analysis and Applications , vol. 389, no. 1, pp. 403-411, 2012.

[24] B. Ahmad, T. Hayat, S. Asghar, "Diffraction of a plane wave by an elastic knife-edge adjacent to a rigid strip," The Canadian Applied Mathematics Quarterly , vol. 9, no. 4, pp. 303-316, 2001.

[25] S. Asghar, B. Ahmad, M. Ayub, "Diffraction from an absorbing half plane due to a finite cylindrical source," Acustica-Acta Acustica , vol. 82, pp. 365-367, 1996.

[26] G. S. Wang, A. F. Blom, "A strip model for fatigue crack growth predictions under general load conditions," Engineering Fracture Mechanics , vol. 40, no. 3, pp. 507-533, 1991.

[27] H. Jiang, F. Liu, I. Turner, K. Burrage, "Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain," Journal of Mathematical Analysis and Applications , vol. 389, no. 2, pp. 1117-1127, 2012.

[28] E. Yusufoglu, I. Turhan, "A mixed boundary value problem in orthotropic strip containing a crack," Journal of the Franklin Institute , vol. 349, no. 9, pp. 2750-2769, 2012.

[29] Dh. Chang, A. Kotousov, "A strip yield model for two collinear cracks," Engineering Fracture Mechanics , vol. 90, pp. 121-128, 2012.

[30] B. Lebental, F. Bourquin, "Visco-acoustic modelling of a vibrating plate interacting with water confined in a domain of micrometric size," Journal of Sound and Vibration , vol. 331, no. 8, pp. 1870-1886, 2012.

[31] T. V. Renterghem, D. Botteldooren, K. Verheyen, "Road traffic noise shielding by vegetation belts of limited depth," Journal of Sound and Vibration , vol. 331, no. 10, pp. 2404-2425, 2012.

[32] J. Poblet-Puig, A. Rodrguez-Ferran, "Modal-based prediction of sound transmission through slits and openings between rooms," Journal of Sound and Vibration , vol. 332, pp. 1265-1287, 2012.

[33] A. Granas, J. Dugundji Fixed Point Theory , of Springer Monographs in Mathematics, Springer, New York, NY, USA, 2005.

[34] D. R. Smart Fixed Point Theorems , Cambridge University Press, Cambridge, UK, 1980.

[]