Existence of solutions for fractional q-integro-difference inclusions with fractional q-integral boundary conditions
© Ahmad et al.; licensee Springer 2014
Received: 15 July 2014
Accepted: 30 September 2014
Published: 13 October 2014
In this paper, we investigate the existence of solutions for a nonlocal boundary value problem of fractional q-integro-difference inclusions of two fractional orders with Riemann-Liouville fractional q-integral boundary conditions. A new existence result is obtained by making use of a nonlinear alternative for contractive maps and is well illustrated with the aid of an example.
Nonlocal nonlinear boundary value problems of fractional order have been extensively investigated in recent years. Several results of interest ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional differential equations are available in the literature on the topic. The introduction of fractional derivative in the mathematical modeling of many real world phenomena has played a key role in improving the integer-order mathematical models. One of the important factors accounting for the popularity of the subject is that differential operators of fractional-order help to understand the hereditary phenomena in many materials and processes in a better way than the corresponding integer-order differential operators. For examples and details in physics, chemistry, biology, biophysics, blood flow phenomena, control theory, wave propagation, signal and image processing, viscoelasticity, identification, fitting of experimental data, economics, etc., we refer the reader to the texts [1–4]. Some recent work on fractional differential equations can be found in a series of papers [5–17] and the references cited therein.
Fractional q-difference equations, known as fractional analogue of q-difference equations, have recently been discussed by several researchers. For some recent work on the topic, see [18–32]. In a recent paper , the authors obtained some existence results for the Langevin type q-difference (integral) equation with two fractional orders and four-point nonlocal integral boundary conditions.
where is the fractional q-derivative of the Caputo type, is a multivalued map, is the family of all subsets of ℝ and (). However, the study of multivalued problems in the setting of fractional q-difference equations is still at an initial stage and needs to be explored further.
Here we emphasize that the multivalued (inclusion) problem at hand is new in the sense that it involves fractional q-integro-difference inclusions of two fractional orders with four-point nonlocal Riemann-Liouville fractional q-integral boundary conditions (in contrast to the problem considered in ). Also our method of proof is different from the one employed in . The paper is organized as follows. An existence result for problem (1.1), based on a nonlinear alternative for contractive maps, is established in Section 3. The background material for the problem at hand can be found in the related literature. However, for quick reference, we outline it in Section 2.
2 Preliminaries on fractional q-calculus
Definition 2.1 ()
For more details on q-integral and fractional q-integral, see Section 1.3 and Section 4.2 respectively in .
Before giving the definition of fractional q-derivative, we recall the concept of q-derivative.
Definition 2.3 ()
where is the smallest integer greater than or equal to β.
To define the solution for problem (1.1), we need the following lemma.
Lemma 2.4 ()
Let denote the Banach space of all continuous functions from endowed with the norm defined by . Also by we denote the Banach space of measurable functions which are Lebesgue integrable and normed by .
In order to prove our main existence result, we make use of the following form of the nonlinear alternative for contractive maps [, Corollary 3.8].
is a contraction, and
is upper semi-continuous (u.s.c.) and compact.
ℋ has a fixed point in , or
there is a point and with .
is measurable for each ,
is upper semi-continuous for almost all , and
- (iii)for each real number , there exists a function such that
for all with .
Lemma 2.7 (Lasota and Opial )
is a closed graph operator in .
3 Existence result
Before presenting the main results, we define the solutions of the boundary value problem (1.1).
Theorem 3.2 Assume that
(H1) is an -Carathéodory multivalued map;
for all and , where is given by (3.1) and H denotes the Hausdorff metric;
(H3) is an -Carathéodory multivalued map;
for all ;
where , , , are given by (3.1), (3.2) and (3.3) respectively, and .
Then problem (1.1) has a solution on .
We shall show that the operators and satisfy all the conditions of Theorem 2.5 on . For the sake of clarity, we split the proof into a sequence of steps and claims.
Step 1. We show that is a multivalued contraction on .
Step 2. We shall show that the operator is u.s.c. and compact. It is well known [, Proposition 1.2] that a completely continuous operator having a closed graph is u.s.c. Therefore we will prove that is completely continuous and has a closed graph. This step involves several claims.
Claim I: maps bounded sets into bounded sets in .
and hence is bounded.
Claim II: maps bounded sets into equicontinuous sets.
Obviously the right-hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Claim III: Next we prove that has a closed graph.
for some .
Hence has a closed graph (and therefore has closed values). In consequence, is compact valued.
where , , , , , , , .
We found , , , , , , , , .
it is found that , where . Thus, all the assumptions of Theorem 3.2 are satisfied. Hence, the conclusion of Theorem 3.2 applies to problem (3.6).
This paper was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon; 1993.MATHGoogle Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
- Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095-1100. 10.1016/j.camwa.2009.05.010MathSciNetView ArticleMATHGoogle Scholar
- Baleanu D, Mustafa OG, Agarwal RP:On -solutions for a class of sequential fractional differential equations. Appl. Math. Comput. 2011, 218: 2074-2081. 10.1016/j.amc.2011.07.024MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 36Google Scholar
- Ahmad B, Ntouyas SK, Alsaedi A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011., 2011: Article ID 107384Google Scholar
- Sudsutad W, Tariboon J: Existence results of fractional integro-differential equations with m -point multi-term fractional order integral boundary conditions. Bound. Value Probl. 2012., 2012: Article ID 94Google Scholar
- O’Regan D, Stanek S: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 2013, 71: 641-652. 10.1007/s11071-012-0443-xMathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013., 2013: Article ID 320415Google Scholar
- Ahmad B, Nieto JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013., 2013: Article ID 149659Google Scholar
- Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56.MathSciNetView ArticleMATHGoogle Scholar
- Liu X, Jia M, Ge W: Multiple solutions of a p -Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013., 2013: Article ID 126Google Scholar
- Ahmad B, Ntouyas SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013., 2013: Article ID 20Google Scholar
- Zhai C, Xu L: Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul. 2014, 19: 2820-2827. 10.1016/j.cnsns.2014.01.003MathSciNetView ArticleGoogle Scholar
- Punzo F, Terrone G: On the Cauchy problem for a general fractional porous medium equation with variable density. Nonlinear Anal. 2014, 98: 27-47.MathSciNetView ArticleMATHGoogle Scholar
- Ferreira R: Nontrivial solutions for fractional q -difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 70Google Scholar
- Goodrich CS: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 2011, 61: 191-202. 10.1016/j.camwa.2010.10.041MathSciNetView ArticleMATHGoogle Scholar
- Ma J, Yang J: Existence of solutions for multi-point boundary value problem of fractional q -difference equation. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 92Google Scholar
- Abdeljawad T, Baleanu D: Caputo q -fractional initial value problems and a q -analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 4682-4688. 10.1016/j.cnsns.2011.01.026MathSciNetView ArticleMATHGoogle Scholar
- Jarad F, Abdeljawad T, Gundogdu E, Baleanu D: On the Mittag-Leffler stability of q -fractional nonlinear dynamical systems. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 2011, 12: 309-314.MathSciNetGoogle Scholar
- Graef JR, Kong L: Positive solutions for a class of higher order boundary value problems with fractional q -derivatives. Appl. Math. Comput. 2012, 218: 9682-9689. 10.1016/j.amc.2012.03.006MathSciNetView ArticleMATHGoogle Scholar
- Li X, Han Z, Sun S: Existence of positive solutions of nonlinear fractional q -difference equation with parameter. Adv. Differ. Equ. 2013., 2013: Article ID 260Google Scholar
- Alsaedi A, Ahmad B, Al-Hutami H: A study of nonlinear fractional q -difference equations with nonlocal integral boundary conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 410505Google Scholar
- Jarad F, Abdeljawad T, Baleanu D: Stability of q -fractional non-autonomous systems. Nonlinear Anal., Real World Appl. 2013, 14: 780-784. 10.1016/j.nonrwa.2012.08.001MathSciNetView ArticleMATHGoogle Scholar
- Abdeljawad T, Baleanu D, Jarad F, Agarwal RP: Fractional sums and differences with binomial coefficients. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 104173Google Scholar
- Ferreira R: Positive solutions for a class of boundary value problems with fractional q -differences. Comput. Math. Appl. 2011, 61: 367-373. 10.1016/j.camwa.2010.11.012MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto JJ, Alsaedi A, Al-Hutami H: Existence of solutions for nonlinear fractional q -difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 2014, 351: 2890-2909. 10.1016/j.jfranklin.2014.01.020MathSciNetView ArticleGoogle Scholar
- Agarwal RP, Ahmad B, Alsaedi A, Al-Hutami H: Existence theory for q -antiperiodic boundary value problems of sequential q -fractional integrodifferential equations. Abstr. Appl. Anal. 2014., 2014: Article ID 207547Google Scholar
- Zhang L, Baleanu D, Wang G:Nonlocal boundary value problem for nonlinear impulsive -integrodifference equation. Abstr. Appl. Anal. 2014., 2014: Article ID 478185Google Scholar
- Ahmad B, Ntouyas SK, Alsaedi A, Al-Hutami H: Nonlinear q -fractional differential equations with nonlocal and sub-strip type boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2014., 2014: Article ID 26Google Scholar
- Ahmad, B, Nieto, JJ, Alsaedi, A, Al-Hutami, H: Boundary value problems of nonlinear fractional q-difference (integral) equations with two fractional orders and four-point nonlocal integral boundary conditions. Filomat (in press)Google Scholar
- Abbasbandy S, Nieto JJ, Alavi M: Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos Solitons Fractals 2005, 26: 1337-1341. 10.1016/j.chaos.2005.03.018MathSciNetView ArticleMATHGoogle Scholar
- Frigon M: Systems of first order differential inclusions with maximal monotone terms. Nonlinear Anal. 2007, 66: 2064-2077. 10.1016/j.na.2006.03.002MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, Rodriguez-Lopez R: Euler polygonal method for metric dynamical systems. Inf. Sci. 2007, 177: 4256-4270. 10.1016/j.ins.2007.05.002MathSciNetView ArticleMATHGoogle Scholar
- Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay. Nonlinear Anal. 2009, 70: 2091-2105. 10.1016/j.na.2008.02.111MathSciNetView ArticleMATHGoogle Scholar
- Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 2009, 49: 605-609. 10.1016/j.mcm.2008.03.014MathSciNetView ArticleMATHGoogle Scholar
- Cernea A: On the existence of solutions for nonconvex fractional hyperbolic differential inclusions. Commun. Math. Anal. 2010, 9(1):109-120.MathSciNetMATHGoogle Scholar
- Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 2010, 109: 973-1033. 10.1007/s10440-008-9356-6MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Ntouyas SK: Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 71Google Scholar
- Ntouyas SK: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opusc. Math. 2013, 33: 117-138. 10.7494/OpMath.2013.33.1.117MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Ntouyas SK: An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions. Abstr. Appl. Anal. 2014., 2014: Article ID 705809Google Scholar
- Ahmad B, Ntouyas SK: Existence of solutions for nonlinear fractional q -difference inclusions with nonlocal Robin (separated) conditions. Mediterr. J. Math. 2013, 10: 1333-1351. 10.1007/s00009-013-0258-0MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060View ArticleMathSciNetMATHGoogle Scholar
- Rajkovic PM, Marinkovic SD, Stankovic MS: On q -analogues of Caputo derivative and Mittag-Leffler function. Fract. Calc. Appl. Anal. 2007, 10: 359-373.MathSciNetMATHGoogle Scholar
- Annaby MH, Mansour ZS Lecture Notes in Mathematics 2056. In q-Fractional Calculus and Equations. Springer, Berlin; 2012.View ArticleGoogle Scholar
- Petryshyn WV, Fitzpatric PM: A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact maps. Trans. Am. Math. Soc. 1974, 194: 1-25.View ArticleGoogle Scholar
- Lasota A, Opial Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 781-786.MathSciNetMATHGoogle Scholar
- Kuratowski K, Ryll-Nardzewski C: A general theorem on selectors. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1965, 13: 397-403.MathSciNetMATHGoogle Scholar
- Gorniewicz L: Topological Fixed Point Theory of Multivalued Mappings. Springer, Dordrecht; 2006.MATHGoogle Scholar
- Deimling K: Multivalued Differential Equations. de Gruyter, Berlin; 1992.View ArticleMATHGoogle Scholar
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