Singular Initial Value Problem for a System of Integro-Differential Equations

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 918281, 18 pages doi:10.1155/2012/918281 Research Article Singular Initial Value Problem for a System of Integro-Differential Equations Zdenˇ ek ˇ Smarda1and Yasir Khan2 1Department of Mathematics, Brno University of Technology, 61600 Brno, Czech Republic 2Department of Mathematics, Zhejiang University, Hangzhou 310027, China Correspondence should be addressed to Zdenˇ ek ˇ Smarda, smar[email protected].cz Received 29 October 2012; Accepted 15 November 2012 Academic Editor: Juntao Sun Copyright q2012 Z. ˇ Smarda and Y. Khan. 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. Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem: gity itaiyit1fit, yt,t 0Kit, s, yt, ysds,y i00,t∈0,t 0,whereyy1,...,y n,a i>0,i1,...,nare constants and t0>0. An approach which combines topological method of T. Wa˙ zewski and Schauder’s fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point. 1. Introduction and Preliminaries Singular initial value problem for ordinary differential and integro-differential equations is fairly well studied see, e.g., 1–16, but the asymptotic properties of the solutions of such equations are only partially understood. Although the singular initial value problems were widely considered using various methods see, e.g., 1–13,16, our approach to this problem is essentially different from others known in the literature. In particular, we use a combination of the topological method of T. Wa˙ zewski 8and Schauder’s fixed point theorem 11.Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right-hand neighbourhood of a singular point. Asymptotic expansions of solutions are constructed for certain classes of systems of integrodifferential equations as well. Consider the following problem: gity itaiyit1fit, yt,t 0 Kit, s, yt,ysds,1.1 2 Abstract and Applied Analysis yi00,t∈0,t 0,1.2 where yy1,...,y n,ai>0 are constants, fi∈C0J×Rn×R,R,Ki∈C0J×J×Rn×Rn,R, J0,t 0,t0>0, i1,...,n. Denote iftOgt as t→0if there is a right-hand neighbourhood U0and a constant K>0 such that ft/gt ≤Kfor t∈U0. iiftogt as t→0if there is valid limt→0ft/gt0. iiift∼gtas t→0if there is valid limt→0ft/gt1. Definition 1.1. The sequence of functions φnt is called an asymptotic sequence as t→0 if φn1toφntas t→01.3 for all n. Definition 1.2. The series cnφnt,c n∈R, is called an asymptotic expansion of the function ftup to Nth term as t→0if aφnt is an asymptotic sequence, b ft− N  n1 cnφntoφNt,as t→0.1.4 The functions gi,f i,and Kiwill be assumed to satisfy the following: igit∈C1J,git>0, gi00, g it∼ψitgλi itas t→0,λi>0, ψitgτ it o1as t→0for each τ>0, i1,...,n, ii|fit, u, v|≤|u||v|,|t 0Kit, s, yt,ysds|≤rit|y|,0<r it∈CJ,rit ϕit, Cio1as t→0where ϕit, CiCiexpt t0ai/gisdsis the general solution of the equation gity itaiyit. In the text, we will apply topological method of Wa˙ zewski and Schauder’s theorem. Therefore we give a short summary of them. Let ft, ybe a continuous function defined on an open t, yset Ω⊂R×Rn,Ω0an open set of Ω,∂Ω0the boundary of Ω0,andΩ0the closure of Ω0. Consider the following system of ordinary differential equations: yft, y.1.5 Definition 1.3 see 17. The point t0,y0∈Ω∩∂Ω0is called an egress or an ingress point of Ω0with respect to system 1.5if for every fixed solution of the problem yt0y0, there Abstract and Applied Analysis 3 exists an >0 such that t, yt ∈Ω0for t0−≤t<t 0t0<t≤t0. An egress point ingress pointt0,y0of Ω0is called a strict egress point strict ingress pointof Ω0if t, yt /∈Ω0on interval t0<t≤t01t0−1≤t<t 0for an 1. Definition 1.4 see 18. An open subset Ω0of the set Ωis called an u, vsubset of Ωwith respect to system 1.5if the following conditions are satisfied. 1There exist functions uit, y∈C1Ω,R,i 1,...,m and vjt, y∈CΩ,Rj 1,...,n,mn>0 such that Ω0t, y∈Ω:uit, y<0,v jt, y<0∀i, j.1.6 2˙uαt, y<0 holds for the derivatives of the functions uαt, y,α1,...,m along trajectories of system 1.5on the set Uαt, y∈Ω:uαt, y0,u it, y≤0,v jt, y≤0,∀jand i/ α.1.7 3˙vβt, y>0 holds for the derivatives of the functions vβt, y,β1,...,n along trajectories of system 1.5on the set Vβt, y∈Ω:uβt, y0,u it, y≤0,v jt, y≤0,∀iandj/ β.1.8 The set of all points of egress strict egressis denoted by Ω0 eΩ0 se. Lemma 1.5 see 18.Let the set Ω0be a u, vsubset of the set Ωwith respect to system 1.5. Then Ω0 se Ω 0 e m  α1 Uα\ n  β1 Vβ.1.9 Definition 1.6 see 18.LetXbe a topological space and B⊂X. Let A⊂B.Afunctionr∈CB,Asuch that raafor all a∈Ais a retraction from Bto Ain X. The set A⊂Bis a retract of Bin Xif there exists a retraction from Bto Ain X. Theorem 1.7 Wa ˙ zewski’s theorem 18.Let Ω0be some u, vsubset of Ωwith respect to system 1.5.LetSbe a nonempty compact subset of Ω0∪Ω0 esuch that the set S∩Ω0 eis not a retract of Sbut is a retract Ω0 e. Then there is at least one point t0,y0∈S∩Ω0such that the graph of a solution yt of the Cauchy problem yt0y0for 1.5lies on its right-hand maximal interval of existence. Theorem 1.8 Schauder’s theorem 19.Let Ebe a Banach space and Sits nonempty convex and closed subset. If Pis a continuous mapping of Sinto itself and PS is relatively compact then the mapping Phas at least one fixed point. 4 Abstract and Applied Analysis 2. Main Results Theorem 2.1. Let assumptions (i) and (ii) hold, then for each Ci/ 0there is one solution yt, C y1t, C1,y 2t, C2,...,y nt, Cn,CC1,...,C nof initial problem 1.1and 1.2such that yj it, Ci−ϕj it, Ci≤δϕ2 it, Cij,j0,1,2.1 for t∈0,t Δ,where0<t Δ≤t0,δ>1is a constant, and tΔdepends on δ, Ci,i1,...,n. Proof. 1Denote Ethe Banach space of vector-valued continuous functions hton the interval 0,t 0with the norm htmax t∈0,t0|hit|,i1,...,n. 2.2 The subset Sof Banach space Ewill be the set of all functions htfrom Esatisfying the inequality hit−ϕit, Ci≤δϕ2 it, Ci.2.3 The set Sis nonempty, convex, and closed. 2Now we will construct the mapping P.Leth0t∈Sbe an arbitrary function. Substituting h0t,h0sinstead of yt,ysinto 1.1, we obtain the following differential equation: gity itaiyit1fit, yt,t 0 Kit, s, h0t,h0sds,i1,...,n. 2.4 Put yitϕit, Ciϕ1−μ it, CiY0it,2.5 y itϕ it, C1 gitϕ1−μ it, CiY1it,2.6 where 0 <μ<1 is a constant and new functions Y0it,Y1itsatisfy the differential equations as gitY 0itμ−1aiY0itY1it,i1,...,n. 2.7 From 2.3, it follows h0itϕit, CiH0it,|H0it|≤δϕ2 it, Ci.2.8 Abstract and Applied Analysis 5 Substituting 2.5,2.6,and2.8into 2.4,weget Y1itaiY0itaiϕμ it, CiaiY0it ×fit, ϕ1t, C1ϕ1−μ 1t, C1Y01t,...,ϕ nt, Cnϕ1−μ nt, CnY0nt, t 0 Kit, s, ϕ1t, C1H01t,...,ϕ nt, CnH0nt,ϕ 1s, C1 H01s,...ϕ ns, CnH0nsds. 2.9 Substituting 2.9into 2.7,weget gitY 0itμaiY0itaiϕμ it, CiaiY0it ×fit, ϕ1t, C1ϕ1−μ 1t, C1Y01t,...,ϕ nt, Cnϕ1−μ nt, CnY0nt, t 0 Kit, s, ϕ1t, C1H01t,...,ϕ nt, CnH0nt,ϕ 1s, C1 H01s,...ϕ ns, CnH0nsds. 2.10 In view of 2.5and 2.6, it is obvious that a solution of 2.10determines a solution of 2.4. NowweuseWa ˙ zewski’s topological method. Consider an open set Ω⊂R×Rn. Denote Y0Y01,...,Y 0n. Define an open subset Ω0⊂Ωas follows: Ω0{t, Y0:uit, Y0<0,vt, Y0<0,i1,...,n }, Uα{t, Y0:uαt, Y00,u it, Y0≤0,vt, Y0≤0,i1,...,n,i/ α}, VβVt, Y0:vt, Y00,u jt, Y0≤0,i1,...,n , 2.11 where uit, Y0Y2 0i−δϕ1μ it, Ci2,v t, Y0t−t0,i1,...,n. 2.12 6 Abstract and Applied Analysis Calculating the derivatives ˙uαt, Y0,˙vt, Y0along the trajectories of 2.10on the set Uα,V, α1,...,nwe obtain ˙uαt, Y02aα gαtμY2 0αtY0αtϕμ αt, CαY2 0αt ×fαt, ϕ1t, C1ϕ1−μ 1t, C1Y01t,...,ϕ nt, Cnϕ1−μ nt, CnY0nt, t 0 Kit, s, ϕ1t, C1H01t,...,ϕ nt, Cn H0nt,ϕ 1s, C1H01s,...ϕ ns, CnH0nsds. −δ21μϕ21μ αt, Cα. 2.13 Since lim t→0ψitgτ it0 for any τ>0,i1,...,n g it∼ψitgλi itas t→0,λ i>0,i1,...,n, 2.14 then there exists a positive constant Misuch that g it<M i,t∈0,t 0,i1,...,n. 2.15 Consequently, t t0 ds gis<1 Mit t0 g isdt gis1 Mi ln git git0−→ −∞ as t−→ 0,i1,...,n. 2.16 From here limt→0ϕit, Ci0 and by L’Hospital’s rule ϕτ it, Cigσ ito1,fort→0, i1,...,n,σ is an arbitrary real number. These both identities imply that the powers of ϕit, Ciaffect the convergence to zero of the terms in 2.13, in a decisive way. Using the assumptions of Theorem 2.1 and the definition of Y0t,ϕit, Ci,i1,...,n, we get that the first term μY2 0αt, Cαin 2.13has the following form: μY2 0αtμδ2ϕ21μ αt, Cα,2.17 Abstract and Applied Analysis 7 and the second term Y0αtϕμ αt, CαY2 0αt ×fαt, ϕ1t, C1ϕ1−μ 1t, C1Y01t,...,ϕ nt, Cn ϕ1−μ nt, CnY0nt,t 0 Kαt, s, ϕ1t, C1H01t,...,ϕ nt, Cn H0nt,ϕ 1s, C1H01s,...ϕ ns, Cn H0nsds. 2.18 is bounded by terms with exponents which are greater than ϕ21μ αt, Cα,α1,...,n.From here, we obtain sgn ˙uαt, Y0−δ21μϕ21μ αt, Cα−12.19 for sufficiently small t∗, depending on Cαα1,...,n,δ,0<t ∗≤t0. It is obvious that sgn ˙vt, Y01. Change the orientation of the axis t into opposite. Then, with respect to the new system of coordinates, the set Ω0is the u, vsubset with respect to system 2.10.ByWa ˙ zewski’s topological method, we state that there exists at least one integral curve of 2.10lying in Ω0 for t∈0,t ∗. It is obvious that this assertion remains true for an arbitrary function h0t∈S. Now we prove the uniqueness of a solution of 2.10.LetY0tY01t,...,Y0nt be also the solution of 2.10. Putting Z0iY0i−Y0i,i1,...,n 2.20 and substituting into 2.10,weobtain gitZ 0itμaiY0itaiϕμ it, CiaiZ0it ×fit, ϕ1t, C1ϕ1−μ 1t, C1Z0itY01t,...,ϕ nt, Cn ϕ1−μ nt, CnZ0ntY0nt, t 0 Kit, s, ϕ1t, C1H01t,...,ϕ nt, Cn H0nt,ϕ 1s, C1H01s,...ϕ ns, CnH0nsds. 2.21 8 Abstract and Applied Analysis Define Ω1δ{t, Z0:0<t<t ∗,u 1it, Z0<0,v 1t, Z0<0,0<t<t ∗,i1,...,n } U1α{t, Z0:u1αt, Z00,u 1it, Z0≤0,v 1t, Z0≤0,i1,...,n,i/ α}, V1βVt, Z0:v1t, Z00,u jt, Z0≤0,i1,...,n , 2.22 where u1it, Z0Z2 0i−δϕ1μ−γ i2,0<γ<μ, v 1t, Z0t−t∗.2.23 Using the same method as above, we have sgn ˙u1it, Z0−1,sgn ˙v1t, Z01,i1,...,n 2.24 for sufficiently small t♦,0<t ♦≤t∗. It is obvious that Ω0⊂Ω1δfor t∈0,t ♦.LetZ0t Z01t,...,Z0nt be any nonzero solution of 2.10such that t1,Z0t1 ∈Ω1for 0 <t 1<t ♦. Let δ∈0,δbe such a constant that t1,Z0t1 ∈∂Ω1δ. If the curve Z0tlay in Ω1δfor 0<t<t 1, then t1,Z0t1 would have to be a strict egress point of ∂Ω1δwith respect to the original system of coordinates. This contradicts the relation 2.24. Therefore there exists only the trivial solution Z0t≡0of2.21,soY0Y0tis the unique solution of 2.10. From 2.5we obtain yit, Ci−ϕit, Ci≤δϕ2 it, Ci,i1,...,n, 2.25 where y1t, C1,...,y nt, Cn is the solution of 2.4for t∈0,t ♦. Similarly, from 2.6and 2.9, we have y it, Ci−ϕ it, Ci 1 gitϕ1−μ it, CiY1it ≤ 1 gitϕ1−μ it, Ci2aiδϕ1−μ it, Ci δϕ2 it, Ci. 2.26 It is obvious after a continuous extension of yt, Cfor t0, y00that P:h0→ymaps Sinto itself and PS ⊂S. 3We will prove that PSis relatively compact and Pis a continuous mapping. It is easy to see, by 2.25and 2.26,thatPS is the set of uniformly bounded and equicontinuous functions for t∈0,t ♦. By Ascoli’s theorem, PSis relatively compact. Let {hkt}be an arbitrary sequence vector-valued functions in Ssuch that hkt−h0tk,lim k→∞k0,h0t∈S. 2.27 Abstract and Applied Analysis 9 The solution YktYk1,...,Yknof the following equation: gitY 0itμaiY0itaiϕμ it, CiaiY0it ×fit, ϕ1t, C1ϕ1−μ 1t, C1Y01t,...,ϕ nt, Cnϕ1−μ nt, CnY0nt, t 0 Kit, s, ϕ1t, C1H01t,...,ϕ nt, Cn H0nt,ϕ 1s, C1Hk1s,...ϕ ns, CnHknsds. 2.28 corresponds to the function hktand Ykt∈Ω0for t∈0,t ♦. Similarly, the solution Y0t of 2.10corresponds to the function h0t. We will show that |Ykt−Y0t|→0 uniformly on 0,t Δ, where 0 <t Δ≤t♦,tΔis a sufficiently small constant which will be specified later. Consider the following region: Ω0kt, Y0:0<t<t ♦,u 0kit, Y0<0,v 0t, Y0<0,i1,...,n ,2.29 where u0kit, Y0Y0it−Y0it2−kϕ1μ−ν it, Ci2,0<ν<α, i1,...,n, k≥1, v0t, Y0t−t♦. 2.30 There exists sufficiently small constant tΔ≤t♦such that Ω0⊂Ω0kfor any k,t∈0,t Δ. Investigate the behaviour of integral curves of 2.28with respect to the boundary ∂Ω0k,t∈ 0,t Δ. Using the same method as above, we obtain the following trajectory derivatives: sgn ˙u0kt, Y0−1,sgn ˙v0t, Y012.31 for t∈0,t Δand any k.ByWa˙ zewski’s topological method, there exists at least one solution Yktlying in Ω0k,0<t<t Δ. Hence, it follows that Ykit−Y0it≤kϕ1μ−ν i≤Nik,2.32 Ni>0, i1,...,nare constants depending on Ci,tΔ.From2.5,weobtain ykit−y0itϕ1−μ it, CiYkit−Y0it≤nik,2.33 where ni>0, i1,...,nare constants depending on tΔ,Ci,Ni. This estimate implies that P is continuous. 16 Abstract and Applied Analysis Hence we can choose a constant λ21>1/2 and similarly >1/2. By Theorem 3.1., we have u11 2t2Ot2ν1,ν 1∈1,3 2.3.21 Second equation 3.19is different from 3.18only in the constant a−1. Thus u2t2Ot2ν2,ν 2∈1,3 2.3.22 Substituting solutions 3.21and 3.22into 3.15instead of integral terms, we obtain for unknown coefficients f12,f 22 the following differential equations: t2f 12 −f12 1 2tOt2ν1−3,3.23 t2f 22 −f22 t5/2Ot2ν21/2.3.24 For 3.23, we can put a−1,b 0t1 2,g λtt2−1/2,λ−1 2,b 1t1, ν1−1,g tt22gt1/2⇒λ11 2,b  0t0·gλ2t. 3.25 Then we can choose a constant λ21>1/2. By Theorem 3.1., we get f12t1 2tOt2ν12 ,f  12tOt2ν12−2,ν 12 ∈−1 2,0.3.26 Similarly for 3.24, we can put a−1, b0t1, gλtt25/4,λ5/4,b 1t1, ν2−1, gtt22gt1/2⇒λ11 2,b  0t0·gλ2t.3.27 Then we can choose a constant λ21>1/2. By Theorem 3.1., we have f22tt5/2Ot2ν22 ,f  22tOt2ν22−2,ν 22 ∈5 4,7 4.3.28 Abstract and Applied Analysis 17 Substituting coefficients f12,f22 into 3.16and using the same method as in the calculation of coefficients f12,f22, we have f13t1 12t2Ot2ν13 ,f  13tOt2ν13−1,ν 13 ∈−1,−1 2, f23t1 4t3/2Ot2ν23 ,f  23tOt2ν23−1,ν 23 ∈3 4,5 4. 3.29 Thus the solution of system 3.11has for h3 the following asymptotic expansions: y1≈φt, C1 2tOt2ν12  φ2t, C1 12t2Ot2ν13  φ3t, C, y2≈φt, Ct5/2Ot2ν22 φ2t, C1 4t3/2Ot2ν23  φ3t, C. 3.30 Acknowledgments The first author is supported by Grant FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology and Grant P201/11/0768 of the Czech Grant Agency Prague. References 1R. P. Agarwal, D. O’Regan, and O. E. Zernov, “A singular initial value problem for some functional differential equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2004, no. 3, pp. 261– 270, 2004. 2V. A. ˇ Ceˇ cik, “Investigation of systems of ordinary differential equations with a singularity,” Trudy Moskovskogo Matematiˇ ceskogo Obˇ sˇ cestva, vol. 8, pp. 155–198, 1959 Russian. 3I. Dibl´ ık, “Asymptotic behavior of solutions of a differential equation partially solved with respect to the derivative,” Siberian Mathematical Journal, vol. 23, no. 5, pp. 654–662, 1982 Russian. 4J. Dibl´ ık, “Existence of solutions of a real system of ordinary differential equations entering into a singular point,” Ukrainian Mathematical Journal, vol. 38, no. 6, pp. 588–592, 1986 Russian. 5J. Baˇ stinec and J. Dibl´ ık, “On existence of solutions of a singular Cauchy-Nicoletti problem for a system of integro-differential equations,” Demonstratio Mathematica, vol. 30, no. 4, pp. 747–760, 1997. 6J. Dibl´ ık, “On the existence of n k1ak1tak2xxkb1tb2xft, x, x,x00-curves of a singular system of differential equations,” Mathematische Nachrichten, vol. 122, pp. 247–258, 1985 Russian. 7J. Dibl´ ık and C. Nowak, “A nonuniqueness criterion for a singular system of two ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 64, no. 4, pp. 637–656, 2006. 8J. Dibl´ ıkandM.R ˚ uˇ ziˇ ckov´ a, “Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 3, pp. 923–944, 2004. 9J. Dibl´ ık and M. R. R˚ uˇ ziˇ ckov´ a, “Inequalities for solutions of singular initial problems for Caratheodory systems via Wa˙ zewski’s principle,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 12, pp. 657–656, 2008. 10Z. ˇ Smarda, “On the uniqueness of solutions of the singular problem for certain class of integrodifferential equations,” Demonstratio Mathematica, vol. 25, no. 4, pp. 835–841, 1992. 11Z. ˇ Smarda, “On a singular initial value problem for a system of integro-differential equations depending on a parameter,” Fasciculi Mathematici, no. 25, pp. 123–126, 1995. 12Z. ˇ Smarda, “On an initial value problem for singular integro-differential equations,” Demonstratio Mathematica, vol. 35, no. 4, pp. 803–811, 2002. 18 Abstract and Applied Analysis 13Z. ˇ Smarda, “Implicit singular integrodifferential equations of Fredholm type,” Tatra Mountains Mathematical Publications, vol. 38, pp. 255–263, 2007. 14A. E. Zernov and Yu. V. Kuzina, “Qualitative investigation of the singular Cauchy problem n k1ak1tak2xxkb1tb2xft, x, x,x00,” Ukrainian Mathematical Journal, vol. 55, no. 10, pp. 1419–1424, 2003 Russian. 15A. E. Zernov and Yu. V. Kuzina, “Geometric analysis of a singular Cauchy problem,” Nonlinear Oscillations, vol. 7, no. 1, pp. 67–80, 2004 Russian. 16A. E. Zernov and O. R. Cha˘ ıchuk, “Asymptotic behavior of solutions of a singular Cauchy problem for a functional-differential equation,” Journal of Mathematical Sciences, vol. 160, no. 1, pp. 123–135, 2009. 17R. Srzednicki, “Wa˙ zewski method and Conley index,” in Handbook of Differential Equations: Ordinary Differential Equations, A. Canada, P. Drabek, and A. Fonda, Eds., vol. 1, pp. 591–684, Elsevier, Amsterdam, The Netherlands, 2004. 18P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964. 19E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, vol. 108 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1999. Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Function Spaces Abstract and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Algebra Discrete Dynamics in Nature and Society Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Decision Sciences Advances in Discrete Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Stochastic Analysis International Journal of