Research Article Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations

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1 Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID , 13 pages doi: /2010/ Research Article Singular Cauchy Initial Value Problem for Certain Classes of Integro-Differential Equations Zdeněk Šmarda Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic Correspondence should be addressed to Zdeněk Šmarda, Received 30 December 2009; Accepted 10 March 2010 Academic Editor: Ağacik Zafer Copyright q 2010 Zdeněk Šmarda. 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. The existence and uniqueness of solutions and asymptotic estimate of solution formulas are studied for the following initial value problem: g t y t ay t 1 f t, y t, K t, s, y t,y s ds, y 0, t 0,,wherea>0isaconstant and > 0. An approach which combines topological method of T. Ważewski and Schauder s fixed point theorem is used. 1. Introduction and Preliminaries The singular Cauchy problem for first-order differential and integro-differential equations resolved or unresolved with respect to the derivatives of unknowns 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 Cauchy problems were widely considered by using various methods see, e.g., 1 13, 16 18, the method used here is based on a different approach. In particular, we use a combination of the topological method of T. Ważewski see, e.g., 19, 20 and Schauder s fixed point theorem 21. Our technique leads to the existence and uniqueness of solutions with asymptotic estimates in the right neighbourhood of a singular point. Consider the following problem: [ g t y t ay t 1 f t, y t, y 0, K t, s, y t,y s ds ], 1.1

2 2 Advances in Difference Equations where f C J R R, R, K C J J R R, R, J 0,, > 0. Denote f t o g t as t if there is valid lim t f t /g t 0, f t g t as t if there is valid lim t f t /g t 1. The functions g,f,k will be assumed to satisfy the following. i a>0isaconstant, g t C 1 J, g t > 0, g 0, g t ψ t g λ t as t,λ>0, ψ t g τ t o 1 as t for each τ>0, ψ C J, R. ii f t, u, v u v, K t, s, y t,y s ds r t y, 0 < r t C J, r t φ t, C o 1 as t, where φ t, C C exp a/ ds is the general solution of the equation g t y t ay t. In the text we will apply the topological method of Ważewski 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 R n, Ω 0 an open set of Ω, Ω 0 the boundary of Ω 0 with respect to Ω, and Ω 0 the closure of Ω 0 with respect to Ω. Consider the system of ordinary differential equations y f t, y. 1.2 Definition 1.1 see 19. The point,y Ω Ω 0 is called an egress or an ingress point of Ω 0 with respect to system 1.2 if for every fixed solution of system 1.2, y y 0, there exists an ɛ>0such that t, y t Ω 0 for ɛ t< <t ɛ. An egress point ingress point,y of Ω 0 is called a strict egress point strict ingress point of Ω 0 if t, y t / Ω 0 on interval <t ɛ 1 ɛ 1 t< for an ɛ 1. Definition 1.2 see 19. An open subset Ω 0 of the set Ω is called a u, v -subset of Ω with respect to system 1.2 if the following conditions are satisfied. 1 There exist functions u i t, y C 1 Ω, R, i 1,...,m, and v j t, y C Ω, R, j 1,...,n, m n>0 such that Ω { t, y Ω : u i t, y < 0, vj t, y < 0 i, j } u α t, y < 0 holds for the derivatives of the functions u α t, y, α 1,...,m, along trajectories of system 1.2 on the set U α { t, y Ω : u α t, y 0, ui t, y 0, vj t, y 0, j, i / α } v β t, y > 0 holds for the derivatives of the functions v β t, y, β 1,...,n, along trajectories of system 1.2 on the set V β { t, y Ω : v β t, y 0, ui t, y 0, vj t, y 0, i, j / β }. 1.5 The set of all points of egress strict egress is denoted by Ω 0 e Ω 0 se.

3 Advances in Difference Equations 3 Lemma 1.3 see 19. Let the set Ω 0 be a u, v -subset of the set Ω with respect to system 1.2. Then m n Ω 0 se Ω 0 e U α \ V β. α 1 β Definition 1.4 see 19. LetX be a topological space and B X. Let A B. Afunctionr C B, A such that r a a for all a A is a retraction from B to A in X. The set A B is a retract of B in X if there exists a retraction from B to A in X. Theorem 1.5 Ważewski s theorem 19. Let Ω 0 be some u, v -subset of Ω with respect to system 1.2.LetS be a nonempty compact subset of Ω 0 Ω 0 e such that the set S Ω 0 e is not a retract of S but is a retract Ω 0 e. Then there is at least one point,y S Ω 0 such that the graph of a solution y t of the Cauchy problem y y 0 for 1.2 lies in Ω 0 on its right-hand maximal interval of existence. Theorem 1.6 Schauder s theorem 21. Let E be a Banach space and S its nonempty convex and closed subset. If P is a continuous mapping of S into itself and PS is relatively compact then the mapping P has at least one fixed point. 2. Main Results Theorem 2.1. Let assumptions i and ii hold, then for each C / 0, there exists one solution y t, C of initial problem 1.1 such that y i t, C φ i i, t, C δ φ 2 t, C i 0, 1, 2.1 for t 0,t Δ, where 0 <t Δ,δ>1 is a constant, and t Δ depends on δ, C. Proof. 1 Denote E the Banach space of continuous functions h t on the interval 0, with the norm h t max t 0, h t. 2.2 The subset S of Banach space E will be the set of all functions h t from E satisfying the inequality h t φ t, C δφ 2 t, C. 2.3 The set S is nonempty, convex and closed. 2 Now we will construct the mapping P. Leth t S be an arbitrary function. Substituting h t,h s instead of y t,y s into 1.1, weobtainthedifferential equation [ g t y t ay t 1 f t, y t, ] K t, s, h t,h s ds. 2.4

4 4 Advances in Difference Equations Set y t φ t, C C exp ds Y t, y t φ t, C 1 g t C exp ds Y 1 t, where 0 < α < 1 is a constant and new functions Y t and Y 1 t satisfy the differential equation g t Y t α 1 ay 0 t Y 1 t. 2.7 From 2.3, it follows that h t φ t, C H t, H t δφ 2 t, C. 2.8 Substituting 2.5, 2.6 and 2.8 into 2.4 we get αa Y 1 t ay t ac exp ds ay t ds Y t, K t, s, φ t, C H t,φ s, C H s ds. 2.9 Substituting 2.9 into 2.7 we get g t Y t αay αa 0 t ac exp ds ay t ds Y t, K t, s, φ t, C H t,φ s, C H s ds In view of 2.5, 2.6 it is obvious that a solution of 2.10 determines a solution of 2.4.

5 Advances in Difference Equations 5 NowwewilluseWażewski s topological method. Consider an open set Ω R R. Investigate the behaviour of integral curves of 2.10 with respect to the boundary of the set Ω 0 Ω, Ω { t, Y : 0 <t<,u t, Y < 0}, 2.11 where 2 u t, Y Y 2 0 δc 1 α a exp ds Calculating the derivative u t, Y along the trajectories of 2.10 on the set Ω { t, Y :0<t<,u t, Y 0}, 2.13 we obtain u t, Y 2a [αy 20 g t t αa Y t C exp ds Y 2t ds Y t, K t, s, φ t, C H t,φ s, C H s ds ] δ 2 1 α C α a exp ds Since lim t 0 ψ t gτ t 0 g t ψ t g λ t for any τ>0, for t,λ>0, 2.15 then there exists a positive constant M such that g t <M, t 0,. 2.16

6 6 Advances in Difference Equations Consequently, ds < 1 M g s dt 1 M ln g t g if t From here lim t φ t, C 0 and by L Hospital s rule φ τ t, C g σ t o 1 for t,σ is an arbitrary real number. These both identities imply that the powers of φ t, C affect the convergence to zero of the terms in 2.14, in decisive way. Using the assumptions of Theorem 2.1 and the definition of Y t, φ t, C, we get that the first term αy 2 0 t in 2.14 has the form αy 2 t αδ2 C α a exp ds, 2.18 and the second term Y t C exp αa t ds Y 2t ds Y t, K t, s, φ t, C H t,φ s, C H s ds 2.19 is bounded by terms with exponents which are greater than 2 1 α a ds From here, we obtain sgn u t, Y sgn δ 2 C α a 1 α exp ds for sufficiently small t, depending on C, δ, 0 <t. The relation 2.21 implies that each point of the set Ω 0 is a strict ingress point with respect to Change the orientation of the axis t into opposite. Now each point of the set Ω 0 is a strict egress point with respect to the new system of coordinates. By Ważewski s topological method, we state that there exists at least one integral curve of 2.10 lying in Ω 0 for t 0,t. It is obvious that this assertion remains true for an arbitrary function h t S.

7 Advances in Difference Equations 7 Now we will prove the uniqueness of a solution of 2.10.LetY t be also the solution of Putting Z Y 0 Y 0 and substituting into 2.10, weobtain g t Z αaz αa ac exp ds a t Z t [ ds Z t Y t, K t, s, φ t, C H t,φ s, C H s ds f t, φ t, C C exp ds Y t, K t, s, φ t, C H t,φ s, C H s ] ds Let Ω 1 δ { t, Z : 0 <t<t,u 1 t, Z < 0}, 2.23 where 2 u 1 t, Z Z 2 0 δc 1 α μ a exp ds, 0 <μ<α Using the same method as above, we have sgn u 1 t, Z for t 0,t. It is obvious that Ω 0 Ω 1 δ for t 0,t. Let Z t be any nonzero solution of 2.14 such that t 1, Z t 1 Ω 1 for 0 <t 1 <t. Let δ 0,δ be such a constant that t 1, Z t 1 Ω 1 δ. If the curve Z t lays in Ω 1 δ for 0 <t<t 1, then t 1, Z t 1 would have to be a strict egress point of Ω 1 δ with respect to the original system of coordinates. This contradicts the relation Therefore, there exists only the trivial solution Z t 0 of 2.22, soy Y t is the unique solution of From 2.5, weobtain yt, C φ t, C δφ 2 t, C, 2.26

8 8 Advances in Difference Equations where y t, C is the solution of 2.4 for t 0,t. Similarly, from 2.6, 2.9 we have y t, C φ t, C 1 t g t C exp ds Y 1 t 1 t g t C exp ds 1 α a aδc exp ds aδc exp 2δa g t C2 exp 2a ds. δ φ 2 t, C 1 α a ds 2.27 It is obvious after a continuous extension of y t for t 0andy 0 that P : h 0 y 0 maps S into itself and PS S. 3 We will prove that PS is relatively compact and P is a continuous mapping. It is easy to see, by 2.26 and 2.27, thatps is the set of uniformly bounded and equicontinuous functions for t 0,t. By Ascoli s theorem, PS is relatively compact. Let {h r t } be an arbitrary sequence functions in S such that h r t h t ɛ r, lim r ɛ r 0, h t S The solution Y k t of the equation g t Y t αay αa 0 t ac exp ds ay t ds Y t, K t, s, φ t, C H k t,φ s, C H k s ds 2.29 corresponds to the function h k t and Y k t Ω 0 for t 0,t. Similarly, the solution Y t of 2.10 corresponds to the function h t. We will show that Y k t Y t 0 uniformly on 0,t Δ, where 0 <t Δ t, t Δ is a sufficiently small constant which will be specified later. Consider the region Ω 0k { t, Y : 0 <t<t,u 0k t, Y < 0}, 2.30

9 Advances in Difference Equations 9 where u 0k t, Y 2 t 2 1 α ν a Y t Y t ɛ k C exp ds, 0 <ν<α, k There exists sufficiently small constant t Δ t such that Ω 0 Ω 0k for any k, t 0,t Δ. Investigate the behaviour of integral curves of 2.29 with respect to the boundary Ω 0k,t 0,t Δ. Using the same method as above, we obtain for trajectory derivatives sgn u 0k t, Y for t 0,t Δ and any k. ByWażewski s topological method, there exists at least one solution Y k t lying in Ω 0k, 0 <t<t Δ. Hence, it follows that 1 α ν a Y k t Y t ɛ k C exp ds Mɛ k, 2.33 and M>0 is a constant depending on C, t Δ.From 2.5, weobtain yk t y t Yk C exp ds t Y t mɛ k, 2.34 where m>0 is a constant depending on t Δ,C, M.This estimate implies that P is continuous. We have thus proved that the mapping P satisfies the assumptions of Schauder s fixed point theorem and hence there exists a function h t S with h t P h t. The proof of existence of a solution of 1.1 is complete. Now we will prove the uniqueness of a solution of 1.1. Substituting 2.5, 2.6 into 1.1, weget αa Y 1 t ay t ac exp ds a t Y t ds Y t, K t, s, φ t, C C exp du Y t, s φ s, C C exp du Y s ds. 2.35

10 10 Advances in Difference Equations Equation 2.7 may be written in the following form: g t Y t αay αa 0 t ac exp ds ay t ds Y t, K t, s, φ t, C C exp du Y t, s φ s, C C exp du Y s ds Now we know that there exists the solution y t, C of 1.1 satisfying 2.1 such that y t, C φ t, C C exp ds U t, 2.37 where U t is the solution of Denote W t Y t U t and substituting it into 2.36, weobtain g t W t αaw αa 0 t a C exp ds W t [ ds W t U t, K t, s, φ t, C C exp du W t U t, s φ s, C C exp du W s U s ds f t, φ t, C C exp ds U t, K t, s, φ t, C C exp du U t, s ] φ s, C C exp du U s ds. 2.38

11 Advances in Difference Equations 11 Let Ω 0 { } t, W : 0 <t<t Δ,u 0t, W < 0, 2.39 where { 2 1 α μ a u 0t, W W 2 0 δc exp ds}, 0 <μ<α If 2.38 had only the trivial solution lying in Ω 00, then Y t U t would be the only solution of 2.38 and from here, by 2.36, y t, C would be the only solution of 1.1 satisfying 2.1 for t 0,t Δ. We will suppose that there exists a nontrivial solution W t of 2.38 lying in Ω 00. Substitute W s instead of W t into 2.38, weobtainthedifferential equation g t W t αaw αa 0 t a C exp ds W t [ ds W t U t, K t, s, φ t, C C exp du W t U t, u φ s, C C exp du W s U s ds f t, φ t, C C exp ds U t, K t, s, φ t, C C exp du U t, s ] φ s, C C exp du U s ds Calculating the derivative u 0t, W along the trajectories of 2.41 on the set Ω 00,we get sgn u 0t, W 1fort 0,t Δ. By the same method as in the case of the existence of a solution of 1.1,weobtainthat in Ω 00 there is only the trivial solution of The proof is complete. Example 2.2. Consider the following initial value problem: t 2 y t t 2e s 2 y t t 3y t 1 y t 1 t2 s 3 1 y 2 t y 2 s ds, y

12 12 Advances in Difference Equations In our case a general solution of the equation t 2 y t 3y t 2.43 has the form φ t, C Cexp 3t 1 0 3t 1 and g t t 2, a 3, ψ t 2, λ 1/2, ψ t g τ t 2t 2τ o 1 as t. Further f t, u, v t t 2e s 2 y t y t 1 t2 0 s 3 1 y 2 t y 2 s ds 2.44 t y t 2e s 2 y t 0 s 3 1 y 2 t y 2 s ds, r t exp t 2, exp t 2 C exp 3t 1 0 3t 1 o 1 as t and 0 2e s 2 y t s 3 1 y 2 t y 2 s ds exp t 2 y t According to Theorem 2.1, there exists for every constant C / 0 the unique solution y t, C of 2.42 such that y i t, C 3 C exp 3 i [ 3 δ C exp 3 2 ] i, i 0, 1, 2.46 t t for t 0, t. Acknowledgment The author was supported by the Council of Czech Government Grants MSM and MSM 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, no. 3, pp , V. A. Čečik, Investigation of systems of ordinary differential equations with a singularity, Trudy Moskovskogo Matematičeskogo Obščestva, vol. 8, pp , 1959 Russian. 3 I. Diblík, Asymptotic behavior of solutions of a differential equation partially solved with respect to the derivative, Siberian Mathematical Journal, vol. 23, no. 5, pp , 1982 Russian. 4 I. Diblík, On the existence of solutions of a real system of ordinary differential equations entering into a singular point, Ukrainskii Matematicheskii Zhurnal, vol. 38, no. 6, pp , 1986 Russian. 5 J. Baštinec and J. Diblík, On existence of solutions of a singular Cauchy-Nicoletti problem for a system of integro-differential equations, Demonstratio Mathematica, vol. 30, no. 4, pp , I. Diblík, On the existence of 0-curves of a singular system of differential equations, Mathematische Nachrichten, vol. 122, pp , 1985.

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