SOLUTIONS TO QUASI-LINEAR DIFFERENTIAL EQUATIONS WITH ITERATED DEVIATING ARGUMENTS

Electronic Journal of Differential Equations, Vol. 214 (214), No. 249, pp. 1 13. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTIONS TO QUASI-LINEAR DIFFERENTIAL EQUATIONS WITH ITERATED DEVIATING ARGUMENTS RAJIB HALOI Abstract. We establish sufficient conditions for the existence and uniqueness of solutions to quasi-linear differential equations with iterated deviating arguments, complex Banach space. The results are obtained by using the semigroup theory for parabolic equations and fixed point theorems. The main results are illustrated by an example. 1. Introduction Let (X, ) be a complex Banach space. For each t, t T < and x X, let A(t, x) : D(A(t, x)) X X be a linear operator on X. We study the following problem in X: du dt + A(t, u(t))u(t) = f(t, u(t), u(w 1(t, u(t)))), t > ; u() = u, (1.1) where u : R + X, u X, w 1 (t, u(t)) = h 1 (t, u(h 2 (t,..., u(h m (t, u(t)))... ))), f : R + X X X and h j : R + X R +, j = 1, 2, 3,..., m are continuous functions. The non-linear functions f and h j satisfy appropriate conditions in terms of their arguments (see Section 2). The class of quasi-linear differential equations is one of the most important classes that arise in the study of gas dynamics, continuum mechanics, traffic flow models, nonlinear acoustics, and groundwater flows, to mention only a few of their application in real world problems (see [2, 15, 16]). Thus the theory of quasi-linear differential equations and their generalizations become as one of the most rapid developing areas in applied mathematics. In this article we consider one such generalization. We establish the existence and uniqueness theory for a class quasi-linear differential equation to a class quasi-linear differential equation with iterated deviating arguments. The main results of this article are new and complement to the existing ones that generalize some results of [7, 8, 28]. Different sufficient conditions for the existence and uniqueness of a solution to quasi-linear differential equations can be found in [1, 3, 8, 16, 17, 18, 22, 26]. Further, we refer to [4, 7, 14, 13] for a 2 Mathematics Subject Classification. 34G2, 34K3, 35K9, 47N2, 39B12. Key words and phrases. Deviating argument; analytic semigroup; fixed point theorem; iterated argument. c 214 Texas State University - San Marcos. Submitted September 15, 214. Published December 1, 214. 1

2 R. HALOI EJDE-214/249 nice introductions to the theory of differential equations with deviating arguments and references cited therein for more details. The plentiful applications of differential equations with deviating arguments has motivated the rapid development of the theory of differential equations with deviating arguments and their generalization in the recent years (see [7, 8, 9, 1, 11, 19, 27, 28, 29, 3]). Stević [28] has proved some sufficient conditions for the existence of a bounded solution on the whole real line for the system u (t) = Au(t) + G(t, u(t), u(v 1 (t)), u (g(t))), where v 1 (t) = f 1 (t, x(f 2 (t,..., x(f k (t, x(t)),... ))), A = diag(a 1, A 2 ) is an n n real constant matrix with A 1 and A 2 matrices of dimensions p p and q q, respectively, p + q = n, and G : R R n R n R n R n, f j : R R n R, j = 1, 2, 3,..., k and g : R R. The results are obtained when G(, x, y, z) satisfies Lipschitz condition in x, y, z; f j (, x) satisfies Lipschitz condition in x and A satisfies some stability condition [28]. Recently, Kumar et al [19] established sufficient conditions for the existence of piecewise continuous mild solutions in a Banach space X to the problem where d dt u(t) + Au(t) = g(t, u(t), u[v 1(t, u(t))]), t I = [, T ], t t k, u t=tk = I k (u(t k )), k = 1, 2,..., m, u() = u, v 1 (t, u(t)) = f 1 (t, u(f 2 (t,..., u(f m (t, u(t)))... ))) (1.2) and A is the infinitesimal generator of an analytic semigroup of bounded linear operators. The functions g and f i satisfy appropriate conditions. I k (k = 1, 2,..., m) are bounded functions for = t < t 1 <,..., < t m < t m+1 = T, and u t=tk = u(t + k ) u(t k ), u(t k ) and u(t+ k ) represent the left and right hand limits of u(t) at t = t k, respectively. For more details, we refer the reader to [19]. With strong motivation from [19, 27, 28, 29], we establish the sufficient condition for the existence as well as uniqueness of a solution for a quasi-linear functional differential equation with iterated deviating argument in a Banach space. We organize this article as follows. The preliminaries and assumptions on the functions f, h j and the operator A A(, u ) are provided in Section 2. The local existence and uniqueness of a solution to Problem (1.1) have established in Section 3. The application of the main results are illustrated by an example at the end. 2. Preliminaries and assumptions In this section, we recall some basic facts, lemmas and theorems that are useful in the remaining sections. We make assumptions on the functions f, h j and the operator A A(, u ) that are required for the proof of the main results. The material covered in this section can be found, in more detail, in Friedman [5] and Tanabe [31]. Let L(X) denote the Banach algebra of all bounded linear operators on X. For T [, ), let {A(t) : t T } be a family of closed linear operators on the Banach space X such that (B1) the domain D(A) of A(t) is dense in X and independent of t;

EJDE-214/249 EXISTENCE OF SOLUTIONS 3 (B2) the resolvent R(λ; A(t)) exists for all Re λ, for each t [, T ], and R(λ; A(t)) C, Re λ, t [, T ] λ + 1 for some constant C > (independent of t and λ); (B3) there are constants C > and ρ (, 1] with [A(t) A(τ)]A 1 (s) C t τ ρ, for t, s, τ [, T ], where C and ρ are independent of t, τ and s. Note taht assumption (B2) implies that for each s [, T ], A(s) generates a strongly continuous analytic semigroup {e ta(s) : t } in L(X). Also the assumption (B3) implies that there exists a constant C > such that A(t)A 1 (s) C, (2.1) for all s, t T. Hence, for each t, the functional y A(t)y defines an equivalent norm on D(A) = D(A()) and the mapping t A(t) from [, T ] into L(X 1, X) is uniformly Hölder continuous. The following theorem will give the existence of a solution to the homogeneous Cauchy problem du dt + A(t)u = ; u(t ) = u, t > t. (2.2) Theorem 2.1 ([5, Lemma II. 6.1]). Let the assumptions (B1) (B3) hold. Then there exists a unique fundamental solution {U(t, s) : s t T } to (2.2). Let C β ([t, T ]; X) denote the space of all X-valued functions h(t), that are uniformly Hölder continuous on [t, T ] with exponent β, where < β 1. Then the solution to the problem is given by the following theorem. du dt + A(t)u = h(t), u(t ) = u, t > t (2.3) Theorem 2.2 ([5, Theorem II. 3.1 ]). Let the assumptions (B1) (B3) hold. If h C β ([t, T ]; X) and u X, then there exists a unique solution of (2.3) and the solution u(t) = U(t, t )u + U(t, s)h(s)ds, t t T t is continuously differentiable on (t, T ]. We need the following assumption to establish the existence of a local solution to (1.1): (H1) The operator A = A(, u ) is a closed linear with domain D dense in X and the resolvent R(λ; A ) exists for all Re λ such that (λi A ) 1 C 1 + λ for some positive constant C( independent of λ). for all λ with Re λ (2.4)

4 R. HALOI EJDE-214/249 Then the negative fractional powers of the operator A is well defined. For α >, we define A α = 1 e ta t α 1 dt. Γ(α) Then A α is a bijective and bounded linear operator on X. We define the positive fractional powers of A by A α [A α ] 1. Then A α is closed linear operator and domain D(A α ) is dense in X. If ς > υ, then D(A ς ) D(Aυ ). For < α 1, we denote X α = D(A α ). Then (X α, ) is a Banach space equipped with the graph norm x α = A α x. If < α 1, the embeddings X α X are dense and continuous. For each α >, we define X α = (X α ), the dual space of X α, endowed with the natural norm x α = A α x. Then (X α, α ) is a Banach space. The following assumptions are necessary for proving the main result. Let R, R 1 > and B α = {x X α : x α < R}, B α 1 = {y X α 1 : y α 1 < R 1 }. (H2) The operator A(t, x) is well defined on D for all t [, T ], for some α [, 1) and for any x B α. Furthermore, A(t, x) satisfies [A(t, x) A(s, y)]a 1 (s, y) C(R)[ t s θ + x y γ α] (2.5) for some constants θ, γ (, 1] and C(R) >, for any t, s [, T ] and x, y B α. (H3) For α (, 1), there exist constants C f C f (t, R, R 1 ) > and < θ, γ, ρ 1 such that the non-linear map f : [, T ] B α B α 1 X satisfies f(t, x, x 1 ) f(s, y, y 1 ) C f ( t s θ + x y γ α + x 1 y 1 ρ α 1 ) (2.6) for every t, s [, T ], x, y B α, x 1, y 1 B α 1. (H4) For some α [, 1), there exist constants C hj C h (t, R) > and < θ i 1, such that h j : B α 1 [, T ] [, T ] satisfies h j (, ) =, h j (t, x) h j (s, y) C hj ( t s θi + x y γ α 1 ), (2.7) for all x, y B α, for all s, t [, T ] and j = 1, 2,..., m. (H5) Let u X β for some β > α, and u α < R. (2.8) (H6) The operator A 1 is completely continuous operator. Remark 2.3. We note that the assumptions (H1) and (H6) imply that A ν is completely continuous for any < ν 1. Indeed, we define A ν,ϑ = 1 Γ(α) We write A ν,ϑ = A 1 (A A ν,ϑ ). As A A ν,ϑ so A ν,ϑ is completely continuous. Also A ν completely continuous. ϑ e ta t ν 1 dt. is a bounded operator for each ϑ >, as ϑ. Thus A ν is,ϑ A ν

EJDE-214/249 EXISTENCE OF SOLUTIONS 5 Let us state the following Lemmas that will be used in the subsequent sections. Let C β ([t, T ]; X) denote the space of all Hölder continuous function from [t, T ] into X. Lemma 2.4 ([6, Lemma 1.1]). If g C β ([t, T ]; X), then H : C β ([t, T ]; X) C([t, T ]; X 1 ) by Hg(t) = U(t, s)g(s)ds, t t T t is a bounded mapping and Hg C([t,T ];X 1) C g Cβ ([t,t ];X), for some C >. As a consequence of Lemma 2.4, we obtain Corollary 2.5. For v X 1, define R(v; g) = U(t, )v + U(t, s)g(s)ds, t T. Then R is a bounded linear mapping from X 1 C β ([, T ]; X) into C([, T ]; X 1 ). 3. Main results We establish the existence and uniqueness of a local solution to Problem (1.1). Let I denote the interval [, δ] for some positive number δ to be specified later. For α 1, let C α = {u u : I X α is continuous} Then (C α, ) is a Banach space, where is defined as Let u = sup u(t) α for u C(I; X α ). t I Y α C Lα (I; X α 1 ) = { y C α : y(t) y(s) α 1 L α t s for all t, s I } for some positive constant L α to be specified later. Then Y α is a Banach space endowed with the supremum norm of C α. Definition 3.1. A function u : I X is said to be a solution to Problem (1.1) if u satisfies the following: (i) u( ) C Lα (I; X α 1 ) C 1 ((, δ); X) C(I; X); (ii) u(t) X 1 for all t (, δ); (iii) du dt + A(t, u(t))u(t) = f(t, u(t), u(w 1(u(t), t))) for all t (, δ); (iv) u() = u. We choose R > small enough such that the assumptions (H2) (H5) hold. Let K > and < η < β α be fixed constants, where < α < β 1. Define W(δ, K, η) = { x C α Y α : x() = u, x(t) x(s) α K t s η for all t, s I }. Then W is a non-empty, closed, bounded and convex subset of C α. We prove the following theorem for the local existence and uniqueness of a solution to Problem (1.1). The proof is based on the ideas of Haloi et al [8] and Sobolevskiĭ [26]. Theorem 3.2. Let u X β, where < α < β 1. Let the assumptions (H1) (H6) hold. Then there exist a solution u(t) to Problem (1.1) in I = [, δ] for some positive number δ δ(α, u ) such that u W C 1 ((, δ); X).

6 R. HALOI EJDE-214/249 Proof. Let x W. It follows from (H5) that x(t) α < R for t I (3.1) for sufficiently small δ >. By assumption (H2), the operator A x (t) = A(t, x(t)) is well defined for each t I. Also it follows from assumption (H2) and inequality (2.1) that [A x (t) A x (s)]a 1 C t s µ for µ = min{θ, γη}, (3.2) for C > is a constant independent of δ and x W. We note that A x () = A. Assumption (H1) and inequality (3.2) imply that (λi A x (t)) 1 C 1 + λ for λ with Re λ, t I, (3.3) sufficiently small δ >. Again from assumption (H3), it follows that [A x (t) A x (s)]a 1 x (τ) C t s µ for t, τ, s I. (3.4) It follows from the assumption (H2), (3.3) and (3.4) that the operator A x (t) satisfies the assumptions (B1) (B3). Thus there exists a unique fundamental solution U x (t, s) for s, t I, corresponding to the operator A x (t) (see Theorem 2.1). For x W, we put f x (t) = f(t, x(t), x(w 1 (t, x(t)))). Then the assumptions (H3) and (H4) imply that f x is Hölder continuous on I of exponent γ = min{θ, γη, θ 1 ρ, θ 2 ηγρ, θ 3 η 2 γ 2 ρ,...., θ m η m 1 γ m 1 ρ}. By Theorem 2.1, there exists a unique solution ψ x to the problem and given by dψ x (t) dt ψ x (t) = U x (t, )u + + A x (t)ψ x (t) = f x (t), t I; ψ() = u (3.5) Now for each x W, we define a map Q by Qx(t) = U x (t, )u + U x (t, s)f x (s)ds, t I. U x (t, s)f x (s)ds for each t I. By Lemma 2.4 and the assumption (H5), the map Q is well defined and Q : W C α. We show that Q maps from W into W for sufficiently small δ >. Indeed, if t 1, t 2 I with t 2 > t 1, then we have Qx(t 2 ) Qx(t 1 ) α 1 [U x (t 2, ) U x (t 1, )]u α 1 + 2 U x (t 2, s)f x (s)ds 1 U x (t 1, s)f x (s)ds α 1. (3.6) Using the bounded inclusion X X α 1, we estimate the first term on the right hand side of (3.6) as (cf. [5, Lemma II. 14.1]), ( U x (t 2, ) U x (t 1, ) ) u α 1 C 1 u α (t 2 t 1 ), (3.7)

EJDE-214/249 EXISTENCE OF SOLUTIONS 7 where C 1 is some positive constant. Using [5, Lemma 14.4], we obtain the estimate for the second term on the right hand side of (3.6) as 2 U x (t 2, s)f x (s)ds 1 C 2 M f (t 2 t 1 )( log(t 2 t 1 ) + 1), U x (t 1, s)f x (s)ds α 1 where M f = sup s [,T ] f x (s) and C 2 is some positive constant. Thus from (3.7) and (3.8), we obtain (3.8) Qx(t 2 ) Qx(t 1 ) α 1 L α t 2 t 1, (3.9) where L α = max{c 1 u α, C 2 M f ( log(t 2 t 1 ) +1)} that depends on C 1, C 2, M f, δ. Finally, we show that Qx(t + t) Qx(t) α K 1 ( t) η for some constants K 1 >, < η < 1 and for t [, δ]. For α < β 1, t t + t δ, we have Qx(t + t) Qx(t) α [ ] U x (t + t, ) U x (t, ) u α + + t U x (t + t, s)f x (s)ds U x (t, s)f x (s)ds α. (3.1) Using [5, Lemma II. 14.1] and [5, Lemma II. 14.4], we obtain the following two estimates [U x (t + t, ) U x (t, )]u α C(α, u )( t) β α ; (3.11) + t U x (t + t, s)f x (s)ds C(α)M f ( t) 1 α (1 + log t ). Using (3.11) and (3.12) in (3.1), we obtain U x (t, s)f x (s)ds α Qx(t + t) Qx(t) α ( t) η[ ] C(α, u )δ β α η + C(α)M f δ ν ( t) 1 α η ν ( log t + 1) for any ν >, ν < 1 α η. Thus for sufficiently small δ >, we have Qx(t + t) Qx(t) α K 1 ( t) η for all t [, δ], (3.12) some positive constant K 1. Thus Q maps W into W. We show that Q is continuous in W. Let t [, δ]. Let x 1, x 2 W. We put φ 1 (t) = ψ x1 (t) and φ 2 (t) = ψ x2 (t). Then for j = 1, 2, we have dφ j (t) dt Then from (3.13), we have d(φ 1 φ 2 )(t) dt + A xj (t)φ j (t) = f xj (t), t (, δ]; φ j () = u. (3.13) + A x1 (t)(x 1 x 2 )(t) = [A x2 (t) A x1 (t)]φ 2 (t) + [f x1 (t) f x2 (t)]

8 R. HALOI EJDE-214/249 for t (, δ]. We note that A x 2 (t) is uniformly Hölder continuous for τ t δ and for τ > which is followed form [5, Lemma II. 14.3] and [5, Lemma II.14.5]. Again Lemma 2.4 implies that A U x2 (t, s)f x2 (s)ds C 3 for some positive constant C 3. Thus we have the bound A φ 2 (t) C 4 t β 1 (3.14) for some positive constant C 4 and t (, δ]. Further, in view of (2.1) and (3.4), the operator [A x2 (t) A x1 (t)]a 1 is uniformly Hölder continuous for τ t δ and τ >. Hence, [A x2 (t) A x1 (t)]φ 2 (t) is uniformly Hölder continuous for τ t δ and for τ >. By Theorem 2.1, we obtain that for any τ t δ and τ >, φ 1 (t) φ 2 (t) = U x1 (t, τ)[φ 1 (τ) φ 2 (τ)] { } + U x1 (t, s) [A x2 (s) A x1 (s)]φ 2 (s) + [f x1 (s) f x2 (s)] ds. τ (3.15) Using the bound (3.14), we take the limit as τ in (3.15), and passing to the limit, we obtain { } φ 1 (t) φ 2 (t) = U x1 (t, s) [A x2 (s) A x1 (s)]φ 2 (t) + [f x1 (s) f x2 (s)] ds. Using (2.5), (2.6), (2.7) and [5, inequlity II. 14.12], we obtain Qx 1 (t) Qx 2 (t) α CC(R) (t s) α x 1 (s) x 2 (s) γ αs β 1 ds + CC f (t s) α{ x 1 (s) x 2 (s) γ α } + x 1 (w 1 (x 1 (s), s)) x 2 (w 1 (x 2 (s), s)) ρ α 1 ds, (3.16) where C is some positive constant. Now using the bounded inclusion X α X α 1, inequalities (2.6) and (2.7), we obtain x 1 (w 1 (x 1 (s), s)) x 2 (w 1 (x 2 (s), s)) ρ α 1 = x 1 (h 1 (t, x 1 (h 2 (t,..., x 1 (h m (t, x 1 (t)))... )))) x 2 (h 1 (t, x 2 (h 2 (t,..., x 2 (h m (t, x 2 (t)))... )))) ρ α 1 x 1 (h 1 (t, x 1 (h 2 (t,..., x 1 (h m (t, x 1 (t)))... )))) x 1 (h 1 (t, x 2 (h 2 (t,..., x 2 (h m (t, x 2 (t)))... )))) ρ α 1 + x 1 (h 1 (t, x 2 (h 2 (t,..., x 2 (h m (t, x 2 (t)))... )))) x 2 (h 1 (t, x 2 (h 2 (t,..., x 2 (h m (t, x 2 (t)))... )))) ρ α 1 L ρ α h 1 (t, x 1 (h 2 (t,..., x 1 (h m (t, x 1 (t)))... ))) h 1 (t, x 2 (h 2 (t,..., x 2 (h m (t, x 2 (t)))... ))) ρ + x 1 x 2 ρ α L ρ αc ρ h 1 x 1 (h 2 (t,..., x 1 (h m (t, x 1 (t)))... )) x 2 (h 2 (t,..., x 2 (h m (t, x 2 (t)))... )) γρ α 1 ] + x 1 x 2 ρ α

EJDE-214/249 EXISTENCE OF SOLUTIONS 9 L ρ αc ρ h 1 [L ρ α h 2 (t,..., x 1 (h m (t, x 1 (t)))... ) h 2 (t,..., x 2 (h m (t, x 2 (t)))... ) γρ + x 1 x 2 γρ α ] + x 1 x 2 ρ α... ] [1 + L ρ αc ρh1 + (L ρ α) 2 C ρh1 C ρh2 + + (L ρ α) m C ρh1... C ρhm x 1 x 2 κ α = C x 1 x 2 κ α, (3.17) where κ = min{ρ, γρ, γ 2 ρ,..., γ m 1 ρ} and C = 1 + L ρ αc ρ h 1 + (L ρ α) 2 C ρ h 1 C ρ h 2 + + (L ρ α) m C ρ h 1... C ρ h m. Using (3.17) in (3.16), we obtain Qx 1 (t) Qx 2 (t) α CC(R) (t s) α x 1 (s) x 2 (s) γ αs β 1 ds + (1 + C)CC δ 1 α f 1 α sup x 1 (t) x 2 (t) µ α t [,δ] Kδ β α sup where µ = min{γ, κ} and K = max sup Qx 1 (t) Qx 2 (t) α t [,δ] t [,δ] { CC(R) 1 α Kδ β α sup x 1 (t) x 2 (t) µ α,, (1+ e C)CC f 1 α t [,δ] }. Thus x 1 (t) x 2 (t) µ α. (3.18) This shows that the operator Q is continuous in W(δ, K, η). Again it follows from inequality (3.1) that the functions x(t) in W(δ, K, η) is uniformly bounded and is equicontinuous (by the definition of W(δ, K, η)). If we can show that the set {ψ x (t) : x W (δ, K, η)} for each t [, δ], is contained in a compact subset of C α, then the image of W(δ, K, η) under Q is contained in a compact subset of Y α which follows from the Ascoli-Arzela theorem. For each t [, δ], we have ψ x (t) = A ν Aν ψ x (t), for < ν < β α. As {A ν ψ x (t) : x W(δ, K, η)} is a bounded set and A ν is completely continuous, so {ψ x (t) : x W (δ, K, η)} for each t [, δ], is contained in a compact subset of C α. Thus by the Schauder fixed point theorem, Q has a fixed point x in W(δ, K, η); that is, x(t) = U x (t, )u + U x (t, s)f x (s)ds for each t I. It is clear from Theorem 2.2 that x C 1 ((, δ); X). Thus x is a solution to problem (1.1) on I. The solution to Problem (1.1) is unique with stronger assumptions. We outline the proof of the following theorem that gives the uniqueness of the solution. For more details, we refer to Haloi et al [8].

1 R. HALOI EJDE-214/249 Theorem 3.3. Let u X β, where < α < β 1. Let the assumptions (H1) (H5) hold with ρ = 1 and γ = 1. Then there exist a positive number δ δ(α, u ) and a unique solution u(t) to Problem (1.1) in [, δ] such that u W C 1 ((, δ); X). Proof. We define { } W(δ, K, η) = y C α Y α : y() = u, y(t) y(s) α K t s η for all t, s [, δ]. For v W and [, δ], we set w v (t) = Qv(t), where w v (t) is the solution to the problem dw v (t) + A v (t)w v (t) = f v (t), t [, δ]; dt (3.19) w() = u. That is, Qv(t) is given by Qv(t) = w v (t) = U v (t, )u + U v (t, s)f v (s)ds, t [, δ]. (3.2) We choose δ > such that Kδ β α < 1/2, where { CC(R) K = max 1 α, (1 + C)CC } f 1 α for some positive constant C. Then it follows from (3.18) that the map Q defined by (3.2) is contraction on W. Thus by the Banach fixed point theorem Q has unique fixed point in W. Remark 3.4. The value of δ in Theorem 3.2 and Theorem 3.3 depends on the constants C in (2.4), R, u β and R u α for < α < β 1. So, any solution u(t) on [, δ] is global solution to Problem (1.1), it is sufficient to show [A(t, u(t))] satisfies the a priori bound [A(t, u(t))] β u(t) D for any t [, T ] and for some positive constant D independent of t. 4. Application Let X = L 2 (Ω), where Ω is a bounded domain with smooth boundary in R n. For T [, ), we define Ω T = { (t, x, y, z) : x Ω, < t < T, y, z X }. We consider the following quasi-linear initial value problem in X [5, 8], w(t, x) + a β (t, x, w, Dw)D β w(t, x) t where β 2m = f(t, x, w(t, x), w(h 1 (w(t, x), t)), x), t >, x Ω, D β w(t, x) =, β m, t T, x Ω, w(, x) = w (x), x Ω, f(t, x, w(t, x), w(h 1 (w(t, x), t)), x) ( = b(y, x)w y, φ 1 (t) ( u x, φ2 (t) u(x,... φ m (t) u(t, x) ) ) ) dy (t, x) Ω T, Ω (4.1)

EJDE-214/249 EXISTENCE OF SOLUTIONS 11 φ j : R + R +, j = 1, 2, 3,..., m are locally Hölder continuous with φ() =, and b C 1 (Ω Ω; R). Here we assume the following two conditions [5]: (i) a β (,,, ) is a continuously differentiable real valued function in all variables for β 2m; (ii) there exists constant c > such that { ( 1) m Re a β (t, x, w, Dw)ζ β} c ζ 2m (4.2) β =2m for all (t, x) Ω T and ζ R n. We take X 1 H 2m (Ω) H m (Ω), X 1/2 = H m (Ω), X 1/2 = H 1 (Ω) and define A(t, u)u = a β (t, x, u, Du)D β u, A u = a β (, u, Du )D β u, β 2m β 2m where u D(A ) and D β β u u = x β1 1 xβ2 2... xβn n is the distributional derivative of u and β is a multi-index with β = (β 1, β 2,..., β n ), β i integers. It is clear from (4.2) that A(t) generates a strongly continuous analytic semi-group of bounded operators on L 2 (Ω) and the assumptions (H1), (H2) are satisfied [5]. We define u(t) = w(t, ). Then (4.1) can be written as du dt + A(t, u(t))u(t) = f(t, u(t), u(w 1(t, u(t)))), t > ; u() = u, (4.3) where w 1 (t, u(t)) = h 1 (t, u(h 2 (t,..., u(h m (t, u(t)))... ))). Let α = 1/2 and 2m > n. By Minkowski s integral inequality and imbedding theorem H m (Ω) C(Ω), we obtain f(x, ψ 1 (x, )) f(x, ψ 2 (x, )) 2 L 2 (Ω) b 2 (ψ 1 ψ 2 )(y, ) 2 dxdy Ω Ω b 2 (ψ 1 ψ 2 )(y, ) 2 dy c b 2 ψ 1 ψ 2 2 H m(ω) for a constant c >, for all ψ 1, ψ 2 H m (Ω). This shows that f satisfies (2.6). We show that the functions h i : [, T ] H m (Ω) [, T ] defined by h i (t, φ) = g i (t) φ(x, ) for each i = 1, 2,..., m, satisfies the assumption (2.7). Let t [, T ]. Then using the embedding H m (Ω) C(Ω), we obtain Ω h i (t, χ) = φ i (t) χ(x, ) φ i χ L (,1) C χ H m where C is a constant depending on bounds of φ i. Let t 1, t 2 [, T ] and χ 1, χ 2 H m (Ω). Using the Hölder continuity of φ and the imbedding theorem H m (Ω) C(Ω), we have h i (t, χ 1 ) h i (t, χ 2 ) φ i (t) ( χ 1 (x, ) χ 2 (x, ) ) + (φ i (t) φ i (s)) χ 2 (x, ) (Ω), φ i χ 1 χ 2 L (,1) + L φi t s θ χ 2 L (,1)

12 R. HALOI EJDE-214/249 C φ i χ 1 χ 2 H m (Ω) + L φi t s θ χ 2 H m (Ω) max{c φ i, L φi χ 2 }( χ 1 χ 2 H m (Ω) + t s θ ). Thus (2.7) is satisfied. We have the following theorem. Theorem 4.1. Let β > 1/2. If u X β, then Problem (4.3) has a unique solution in L 2 (Ω). References [1] H. Amann; Quasi-linear evolution equations and parabolic systems, Trans. Amer. Math. Soc. 293 (1986), No. 1, pp. 191-227. [2] H. Amann; Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser Boston, Inc., Boston, MA, 1995. [3] E. H. Anderson, M. J. Anderson, W. T. England; Nonhomogeneous quasilinear evolution equations, J. Integral Equations 3 (1981), no. 2, pp. 175-184. [4] L. E. El sgol ts, S. B. Norkin; Introduction to the theory of differential equations with deviating arguments, Academic Press, 1973. [5] A. Friedman; Partial Differential Equations, Dover Publications, 1997. [6] A. Friedman, M. Shinbrot; Volterra integral equations in Banach space, Trans. Amer. Math. Soc. 126 (1967), pp. 131-179. [7] C. G. Gal; Nonlinear abstract differential equations with deviated argument, J. Math. Anal. Appl. 333 (27), no. 2, pp. 971-983. [8] R. Haloi, D. Bahuguna, D. N. Pandey; Existence and uniqueness of solutions for quasilinear differential equations with deviating arguments, Electron. J. Differential Equations, 212 (212), No. 13, 1-1. [9] R. Haloi, D. N. Pandey, D. Bahuguna; Existence, uniqueness and asymptotic stability of solutions to a non-autonomous semi-linear differential equation with deviated argument, Nonlinear Dyn. Syst. Theory, 12 (2) (212), 179 191. [1] R. Haloi, D. N. Pandey, D. Bahuguna; Existence and Uniqueness of a Solution for a Non- Autonomous Semilinear Integro-Differential Equation with Deviated Argument, Differ. Equ. Dyn. Syst, 2(1),(212),1-16. [11] R. Haloi, D. N. Pandey, D. Bahuguna; Existence of solutions to a non-autonomous abstract neutral differential equation with deviated argument, J. Nonl. Evol. Equ. Appl.5 (211) p. 75-9. [12] M. L. Heard; An abstract parabolic Volterra integro-differential equation, SIAM J. Math. Anal. Vol. 13. No. 1 (1982), pp. 81-15. [13] T. Jankowski, M. Kwapisz; On the existence and uniqueness of solutions of systems of differential equations with a deviated argument, Ann. Polon. Math. 26 (1972), pp. 253 277. [14] T. Jankowski; Advanced differential equations with non-linear boundary conditions, J. Math. Anal. Appl. 34 (25) pp. 49-53. [15] A. Jeffrey; Applied partial differential equations, Academic Press, 23, USA. [16] T. Kato; Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jrgens), pp. 25-7. Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975. [17] T. Kato; Abstract evolution equations, linear and quasilinear, revisited. Functional analysis and related topics, 1991 (Kyoto), 13-125, Lecture Notes in Math., 154, Springer, Berlin, 1993. [18] K. Kobayasi, N. Sanekata; A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces, Hiroshima Math. J. 19 (1989), no. 3, 521-54. [19] P. Kumar, D. N. Pandey, D. Bahuguna; Existence of piecewise continuous mild solutions for impulsive functional differential equations with iterated deviating arguments, Electron. J. Differential Equations, Vol. 213 (213), No. 241, pp. 1 15. [2] M. Kwapisz; On certain differential equations with deviated argument, Prace Mat. 12 (1968) pp. 23-29. [21] A. Lunardi; Global solutions of abstract quasilinear parabolic equations, J. Differential Equations 58 (1985), no. 2, 228-242.

EJDE-214/249 EXISTENCE OF SOLUTIONS 13 [22] M. G. Murphy; Quasilinear evolution equations in Banach spaces, Trans. Amer. Math. Soc. 259 (198), no. 2, 547-557. [23] R. J. Oberg; On the local existence of solutions of certain functional-differential equations, Proc. Amer. Math. Soc. 2 (1969) pp. 295-32. [24] N. Sanekata; Convergence of approximate solutions to quasilinear evolution equations in Banach spaces, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 7, 245-249. [25] N. Sanekata; Abstract quasi-linear equations of evolution in nonreflexive Banach spaces, Hiroshima Math. J. 19 (1989), no. 1, 19-139. [26] P. L. Sobolevskiĭ; Equations of parabolic type in a Banach space, Amer. Math. Soc. Translations (2), 49 (1961) pp. 1-62. [27] S. Stević; Solutions converging to zero of some systems of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput. 219 (212), no. 8, 431 435. [28] S. Stević; Globally bounded solutions of a system of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput. 219 (212), no. 4, 218 2185. [29] S. Stević; Bounded solutions of some systems of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput. 218 (212), no. 21, 1429 1434. [3] S. Stević; Asymptotically convergent solutions of a system of nonlinear functional differential equations of neutral type with iterated deviating arguments, Appl. Math. Comput. 219 (213), no. 11, 6197 623. [31] H. Tanabe; On the equations of evolution in a Banach space, Osaka Math. J. Vol. 12 (196) pp. 363 376. Rajib Haloi Department of Mathematical Sciences, Tezpur University, Napaam, Tezpur 78428, India Phone +91-3712-275511-259753, Fax +91-3712-2676 E-mail address: rajib.haloi@gmail.com