Second order Volterra-Fredholm functional integrodifferential equations

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1 Malaya Journal of Matematik )22) 7 Second order Volterra-Fredholm functional integrodifferential equations M. B. Dhakne a and Kishor D. Kucche b, a Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-43 4, Maharashtra, India. b Department of Mathematics, Shivaji University, Kolhapur-464, Maharashtra, India. Abstract his paper deals with the study of global existence of solutions to initial value problem for second order nonlinear mixed Volterra-Fredholm functional integrodifferential equations in Banach spaces. he technique used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori boun of solution. Keywor: Global solution; Volterra-Fredholm functional integrodifferential equation; Leray-Schauder alternative; Fixed point; priori boun. 2 MSC: 5G, 47H, 35R. c 22 MJM. All rights reserved. Introduction Let X be a Banach space with the norm. Let C = C r,, X), < r <, be the Banach space of all continuous functions ψ : r, X endowed with supremum norm ψ C = sup{ ψθ) : r θ }. Let B = C r,, X), >, be the Banach space of all continuous functions x : r, X with the supremum norm x B = sup{ xt) : r t }. For any x B and t, we denote x t the element of C given by x t θ) = xt θ) for θ r,. In this paper we prove the global existence for second order abstract nonlinear mixed Volterra-Fredholm functional integrodifferential equation of the form ϱt)x t)) = f t, x t, at, s)gs, x s ), bt, s)hs, x s ), t,,.) xt) = φt), r t, x ) = δ,.2) where f :, C X X X, a, b :,, R, g, h :, C X are continuous functions, ϱt) is real valued positive sufficiently smooth function on,, φ C and δ X are given. Equation of the form.)-.2) and their special forms serve as an abstract formulation of many partial differential equations or partial integrodifferential equations which arising in heat flow in materials with memory, viscoelasticity and many other physical phenomena see 6, 8, 2 and the references given therein. he problem of existence, uniqueness and other properties of solutions of the special forms of.)-.2) have been studied by many authors by using different techniques, see 4, 7, 9, 4 7 and some of the references given therein. In an interesting paper 3, Ntouyas have investigated the global existence for Volterra functional integro-differential equations in R n by using classical application of Leray-Schauder alternative. he Corresponding author. addresses: mbdhakne@yahoo.com M. B. Dhakne), kdkucche@gmail.com Kishor D. Kucche)

2 2 M. B. Dhakne et al. / Existence results... present paper generalizes the result of 3. he aim of this paper is to study the existence of global solutions of.)-.2). he main tool used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative, rely on a priori boun of solutions. he interesting and useful aspect of the method employed here is that it yiel simultaneously the global existence of solutions and the maximal interval of existence. 2 Preliminaries and Main Results Firstly, we shall set forth some preliminaries and hypotheses that will be used in our subsequent discussion. Definition 2.. A function x : r, X is called solution of initial value problem.)-.2) if x C r,, X) C 2,, X) and satisfies.)-.2) on r,. Our results are based on the following lemma, which is a version of the topological transversality theorem given by Granas 5, p. 6. Lemma 2.. Let S be a convex subset of a normed linear space E and assume S. Let F : S S be a completely continuous operator, and let hen either εf ) is unbounded or F has a fixed point. εf ) = {x S : x = λf x for some < λ < }. We list the following hypotheses for our convenience. H ) here exists a continuous function p :, R =, ) such that for every t,, ψ C and x, y X. ft, ψ, x, y) pt) ψ C x y ), H 2 ) here exists a continuous function m :, R such that gt, ψ) mt)g ψ C ), for every t, and ψ C, where G : R, ) is continuous nondecreasing function. H 3 ) here exists a continuous function n :, R such that ht, ψ) nt)h ψ C ), for every t, and ψ C, where H : R, ) is continuous nondecreasing function. H 4 ) here exists a constants K and L such that at, s) K, for t s, and bt, s) L, for s, t,. H 5 ) For each t, the function ft,.,.,.) : C X X X is continuous and for each ψ, x, y) C X X the function f., ψ, x, y) :, X is strongly measurable. H 6 ) For each t, the functions gt,.), ht,.) : C X are continuous and for each ψ C the functions g., ψ), h., ψ) :, X are strongly measurable. With these preparations we state and prove our main results. heorem 2.. Suppose that the hypothesis H )-H 6 ) hol. hen the initial-value problem.)-.2) has a solution x on r, if satisfies Ms) < s Gs), 2.) where { Mt) = max R α } ps), Kmt), Lnt), t,,

3 M. B. Dhakne et al. / Existence results... 3 α = β φ C δ ϱ) R = min {ϱt) : t, } and β is constant such that J :, R. ϱs), Ms)HJs)) β for any continuous function Proof. First we establish the priori boun on the solutions of the initial value problem ) ϱt)x t)) = λf t, x t, at, s)gs, x s ), bt, s)hs, x s ), t,, 2.2) with the initial condition.2) for λ, ). Let xt) be a solution of the problem 2.2)-.2) then it satisfies the equivalent integral equation xt) = φ) δϱ) λ ϱs) ϱs) f τ, x τ, aτ, η)gη, x η )dη, bτ, η)hη, x η )dη dτ, t, 2.3) xt) = φt), r t, x ) = δ. 2.4) Using 2.3), hypotheses H ) H 4 ) and the fact that λ, ), for t, we have xt) φ C δ ϱ) λ ϱs) φ C δ ϱ) ϱs) s ϱs) f τ, x τ, pτ) ϱs) x τ C aτ, η)gη, x η )dη, bτ, η)hη, x η )dη) dτ Kmη)G x η C )dη Lnη)H x η C )dη dτ. 2.5) Consider the function Z given by Zt) = sup{ xs) : r s t}, t,. Let t r, t be such that Zt) = xt ). If t, t then from 2.5), we have Zt) φ C δ ϱ) ϱs) φ C δ ϱ) ϱs) pτ) pτ) ϱs) x τ C ϱs) Zτ) Kmη)G x η C )dη Lnη)H x η C )dη dτ Mη)GZη))dη Mη)HZη))dη dτ. 2.6) If t r, then Zt) = φ C and the previous inequality 2.6) obviously hol. Denoting by ut) the right-hand side of the inequality 2.6), we have and u t) = ϱt) R R u) = φ C δ ϱ) ps) ps) Zs) us) ps) us), Zt) ut), t, ϱs) Mτ)GZτ))dτ Mτ)HZτ))dτ Mτ)Guτ))dτ Mτ)Guτ))dτ β. Mτ)Huτ))dτ

4 4 M. B. Dhakne et al. / Existence results... Let wt) = ut) Mτ)Guτ))dτ β. hen we have ut) wt), t,. Since ut) is increasing, wt) is also increasing on, and w) = β φ C δ ϱ) w t) = u t) Mt)Gut)) R wt) R ϱs) = α. Now, ps)ws) Mt)Gut)) ps) Mt)Gut)) Mt)wt) Gwt)). herefore w t) Mt), t,. wt) Gwt)) Integrating from to t and using change of variables t s = wt) and the condition 2.), we obtain wt) α s Gs) Ms) Ms) < α, t,. 2.7) s Gs) From the inequality 2.7) there exists a constant γ, independent of λ, ) such that wt) γ for t,. Hence Zt) ut) wt) γ, t,. Since for every t,, x t C Zt), we have x B = sup{ xt) : t r, } γ. Now, we rewrite initial value problem.)-.2) as follows: For φ C, define φ B, B = C r,, X) by φt) = { φt) if r t φ) δϱ) ϱs) if t. If y B and xt) = yt) φt), t r, then it is easy to see that yt) satisfies yt) = y = ; r t and s yt) = f τ, y τ ϱs) φ τ, if and only if xt) satisfies xt) = φ) δϱ) ϱs) ϱs) f τ, x τ, aτ, η)gη, y η φ η )dη, bτ, η)hη, y η φ η )dη dτ, t,, aτ, η)gη, x η )dη, bτ, η)hη, x η )dη dτ, t,, 2.8) xt) = φt), r t, x ) = δ. 2.9) We define the operator F : B B, B = {y B : y = } by if r t F y)t) = ϱs) f τ, y τ φ τ, aτ, η)gη, y η φ η )dη, bτ, η)hη, y η φ η )dη ) dτ, if t,. 2.) From the definition of operator F equations 2.8)-2.9) can be written as y = F y, and the equations 2.3)-2.4) can be written as y = λf y. Now, we prove that F is completely continuous. First, we prove that F : B B is continuous. Let {u m } be a sequence of elements of B converging to u in B. hen by using hypothesis H 5 ) and H 6 ) we have f t, u mt φ t, at, s)gs, u ms φ s ), bt, s)hs, u ms φ s )

5 f t, u t φ t, M. B. Dhakne et al. / Existence results... 5 at, s)gs, u s φ s ), bt, s)hs, u s φ s ) for each t,. hen by dominated convergence theorem, we have F u m )t) F u)t) s ϱs) f τ, u mτ φ τ, f τ, u τ φ τ, as n, t,. aτ, η)gη, u mη φ η )dη, bτ, η)hη, u mη φ η )dη bτ, η)hη, u η φ η )dη) dτ aτ, η)gη, u η φ η )dη, Since, F u m F u B = sup t r, F u m )t) F u)t), it follows that F u m F u B as n which implies F u m F u in B as u m u in B. herefore, F is continuous. We prove that F maps a bounded set of B into a precompact set of B. Let B k = {y B : y B k} for k. We show that F B k is uniformly bounded. Let M = sup{mt) : t, } and φ C = c. hen from the definition of F in 2.) and using hypotheses H ) H 4 ) and the fact that y B k, y B k implies y t C k, t, we obtain F y)t) ϱs) ϱs) pτ) pτ) y τ φ τ C k c k c M Gk c) M Hk c) k c M Gk c) M Hk c) Kmη)G y η φ η C )dη Lnη)H y η φ η C )dη dτ Mη)Gk c)dη R Ms). Mη)Hk c)dη pτ)dτ his implies that the set {F y)t) : y B k, r t } is uniformly bounded in X and hence F B k is uniformly bounded. Next we show that F maps B k into an equicontinuous family of functions with values in X. Let y B k and t, t 2 r,. hen from the equation 2.) and using the hypotheses H ) H 4 ) we have three cases: Case : Suppose t t 2 dτ F y)t 2 ) F y)t ) 2 s t ϱs) f τ, y τ φ τ, 2 s pτ) y τ t ϱs) φ τ C 2 s pτ) k c t ϱs) aτ, η)gη, y η φ η )dη, k c M Gk c) M Hk c) ) bτ, η)hη, y η φ η dη dτ Kmη)G y η φ η C )dη Lnη)H y η φ η C )dη Mη)Gk c)dη 2 t R 2 k c M Gk c) M Hk c) Ms). t Mη)Hk c)dη pτ)dτ dτ dτ

6 6 M. B. Dhakne et al. / Existence results... Case 2 : Suppose r t t 2. Proceeding as in Case, we get F y)t 2 ) F y)t ) 2 s ϱs) f τ, y τ φ τ, k c M Gk c) M Hk c) aτ, η)gη, y η φ η )dη, 2 Ms). ) bτ, η)hη, y η φ η dη dτ Case 3 : Suppose r t t 2. hen F y)t 2 ) F y)t ) =. From Cases -3, we see that F y)t 2 ) F y)t ) as t 2 t ) and we conclude that F B k is an equicontinuous family of functions with values in X. We have already shown that F B k is an equicontinuous and uniformly bounded collection. o prove the set F B k is precompact in B, it is sufficient, by Arzela-Ascoli s argument, to show that the set {F y)t) : y B k } is precompact in X for each t r,. Since F y)t) = for t r, and y B k, it is sufficient to show this for < t. Let < t be fixed and ɛ a real number satisfying < ɛ < t. For y B k, we define F ɛ y)t) = ɛ ϱs) f τ, y τ φ τ, aτ, η)gη, y η φ η )dη, bτ, η)hη, y η φ η )dη dτ. Since the set F B k is bounded in B, the set Y ɛ t) = {F ɛ y)t) : y B k } is precompact in X for every ɛ, < ɛ < t. Moreover for every y B k, we have s F y)t) F ɛ y)t) = f τ, y τ t ɛ ϱs) φ τ, aτ, η)gη, y η φ η )dη, bτ, η)hη, y η φ η )dη dτ. By making use of hypotheses H ) H 4 ) and the fact that y B k, y B k implies y t C k, t,, we have F y)t) F ɛ y)t)) s pτ) y τ t ɛ ϱs) φ τ C Kmη)G y η φ η C )dη Lnη)H y η φ η C )dη dτ s pτ) k c Mη)Gk c)dη Mη)Hk c)dη dτ ϱs) t ɛ k c M Gk c) M Hk c) k c M Gk c) M Hk c) t ɛ t ɛ ϱs) Ms). pτ)dτ his shows that there exists precompact sets arbitrarily close to the set {F y)t) : y B k } hence the set {F y)t) : y B k } is precompact in X. hus we have shown that F is completely continuous operator. Moreover, the set εf ) = {y B : y = λf y, < λ < }, is bounded in B, since for every y in εf ), the function xt) = yt) φt) is a solution of initial value problem 2.2)-.2) for which we have proved that x B γ and hence y B γ c. Now, by virtue of Lemma 2., the operator F has a fixed point ỹ in B. hen x = ỹ φ is a solution of the initial value problem.)-.2). his completes the proof of the heorem 2.. In concluding this paper, we remark that one can easily extend the ideas of this paper to study the global existence of solutions to second order nonlinear mixed Volterra-Fredholm functional integrodifferential equation of the form ϱt)x t)) = f t, x t, x t), at, s)gs, x s, x s)), bt, s)hs, x s, x s)), t,,

7 M. B. Dhakne et al. / Existence results... 7 xt) = φt), r t, x ) = δ. with conditions given in H ) H 6 ) and suitable condition similar to that given in 2.). he precise formulation of this result is very close to that of the result given in our heorem 2. with suitable modification and hence we omit details. References. A. Burton, Volterra integral and differential equations, Academic Press, New York, M. B. Dhakne, S. D. Kendre, On abstrct nonlinear mixed Volterra- Fredhom integrodifferential equations, Communications on Applied Nonlinear Analysis, 34) 26), -. 3 M. B. Dhakne, G. B. Lambh, On an abstrct nonlinear second order integrodifferential equation, Journal of Function Space and Application, 52) 27), M. B. Dhakne, H. L. idke, On global existance of solution of abstrct nonlinear mixed integrodifferential equation with nonlocal condition, Communications on Applied Nonlinear Analysis, 6)29), J. Dugundji, A. Granas, Fixed Point heory, Vol. I, Monographie Matematycane, PNW Warsawa, W. Fitzgibbon, Semilinear integrodifferential equations in Banach space, Nonlinear Analysis, 498), J. Hale, heory of Functinal Differential Equations, Springer-Verlag, New York, M. Hussain, On a nonlinear integrodifferential equations in Banach space, Indian J. Pure Appl. Math., 9988), V. Lakshmikantham, S. Leela, Nonlinear Differential Dquations in Abstract Spaces, Pergamon Press, New York, 98. J. Lee, D. O Regon, opological transversality, application to initial value problems, Ann. Pol. Math., 48988), J. Lee, D. O Regon, Existance results for differential delay equations, Inter. Journal. Differential Equations, 2993), R. MacCamy, An integro-differential equation with applications in heat flow, Quart. Appl. Math., 35977/78), S. K. Ntouyas,Global existance for volterra functional integro-differential equations, Panam. Math. J., 8998), S. K. Ntouyas, Initial and boundary value problems for functional differential equations via the topological transversality method, Bull. Greek Math. Soc., 4 998), S. K. Ntouyas, Y. Sficas, P. samatos, An existance principle for boundary value problems for second order functinal integro-differential equations, Nonlinear Analysis, 2993), S. K. Ntouyas, P. samatos, Inital and boundary value problems for functinal Integro-differential equations, J. Appl. Math. Stoch. Analysis, 7994), B. G. Pachpatte, Applications of the Leray-Shauder Alternative to some Volterra integral and integrodifferential equations, Indian J. Pure. Appl. Math., 262)995), Received: July 3, 22; Accepted: September, 22 UNIVERSIY PRESS