ON THE PICARD PROBLEM FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN BANACH SPACES

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1 Discussiones Mathematicae Differential Inclusions, Control and Optimization ) ON THE PICARD PROBLEM FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN BANACH SPACES Antoni Sadowski Institute of Mathematics and Phsics Technical Universit of Warsaw, Branch P lock ul. Lukasiewcza 17, 9 4 P lock, Poland Abstract B. Rzepecki in [5] examined the Darboux problem for the hperbolic equation z x = fx,, z, z x ) on the quarter-plane x, via a fixed point theorem of B.N. Sadovskii [6]. The aim of this paper is to stud the Picard problem for the hperbolic equation z x = fx,, z, z x, z x ) using a method developed b A. Ambrosetti [1], K. Goebel and W. Rzmowski [2] and B. Rzepecki [5]. Kewords: boundar value problem, fixed point theorem, functionalintegral equation, hperbolic equation, measure of noncompactness. 2 Mathematics Subject Classification: 35L7. 1. Notations and formulations B E, ) we shall denote a real Banach space. The smbol R k, ) is reserved for n-dimensional Euclidean space. We introduce the notion and R + =<, ), Q = R + R + R 2 Ω = Q E E E. Let B be the famil of bounded sets of E. Then α : B R +, defined b αb) = inf{d > : B admits a finite cover b sets of diameter d},

2 32 A. Sadowski B B, is called Kuratowski s measure of noncompactness. Let σ, τ : R + E be two functions such that τ) = σβ)), where β : R + R + is a given function. We shall consider the following problem z x x, ) = fx,, zx, ), z x x, ), z x x, )) P.P ) zx, ) = σx) zβ), ) = τ) where f : Ω E is a given function. The above P.P ) problem is usuall called the Picard problem for hperbolic equations. 2. The main result The aim of this paper is to prove the following theorem Theorem 2.1. Assume that σ, τ : R + E are C 1 -mappings such that τ) = σβ)), where β : R + R + is a function of class C 1 satisfing the following condition β : [, M n ] [, M n ], where M n ), n N is an increasing and unbounded sequence. Assume further that f : Ω E is uniforml continuous on bounded subsets of Ω and 2.1) fx,, u, v, w) Gx,, u, v, w ) for x,, u, v, w) Ω. Suppose that for each bounded subset P of Q there exist nonnegative constants kp ) and LP ) < 1 2 such that 2.2) αfx,, U, V, w)) kp )αu) + αv )) and 2.3) fx,, u, v, w 1 ) fx,, u, v, w 2 ) LP ) w 1 w 2 for all x, ) P, u, v, w 1, w 2 E. For an nonempt bounded subsets U, V of E, let α denote Kuratowski s measure of noncompactness in E.

3 On the Picard problem for hperbolic Assume in addition that the function x,, r, s, t) Gx,, r, s, t) is nondecreasing for each x, ) Q i.e. r 1 r 2, s 1 s 2 and t 1 t 2 implies Gx,, r 1, s 1, t 1 ) Gx,, r 2, s 2, t 2 )) and the scalar inequalit 2.4) G x,, gs, t)dsdt, gx, t)dt, gx, ) gx, ) β) has a locall bounded solution g on Q. Under these assumptions, P.P ) has at least one solution on Q. For the proof we need the following two lemmas. Lemma 2.1. Let M, d) be a metric space and let A 1, A 2 be transformations mapping bounded sets of M into bounded sets of M. Assume that is a mapping such that F : A 1 M) A 2 M) M M df A 1 x, A 2, z 1 ), F A 1 x, A 2, z 2 )) Ldz 1, z 2 ) for x,, z 1, z 2 M, L ) and αf A 1 X A 2 X {z})) ψ 1 αa 1 X)) + ψ 2 αa 2 X)) for z M, X being a bounded subset of M, and ψ i : R + R + ; i = 1, 2. Then αf A 1 X A 2 X X)) 2LαX) + ψ 1 αa 1 X)) + ψ 2 αa 2 X)) for an bounded subset X of M. The proof of this Lemma is similar to that in [4]. Lemma 2.2. If W is a bounded equicontinuous subset of a Banach space of continuous E-valued functions defined on a compact subset P = [a 1, a 2 ] [b 1, b 2 ] of Q, then a2 b2 ) a2 b2 α W s, t)dsdt αw s, t))dsdt. a 1 b 1 a 1 b 1

4 34 A. Sadowski Lemma 2.2 is an adaptation of the corresponding result of Goebel and Rzmowski [2]. P roof of T heorem 2.1. Without loss of generalit, we ma assume that σ = and τ = see [3]). Problem P.P ) is equivalent to the functionalintegral equation 2.5) wx, ) = f x,, β) ws, t)dsdt, wx, t)dt, wx, ) Denote b CQ, E) the space of all continuous functions from Q to E CQ, E) is a Frechet space whose topolog is introduced b seminorms of uniform convergence on compact subsets of Q), and b X the set of all w CQ, E) with 2.6) wx, g x, ) x, ) Q. Let P be a bounded subset of Q. From the uniform continuit of f on bounded subsets of Ω there follows the existence of a function δ P :, ), ) such that x fx,, ws, t)dsdt, wx, t)dt, wx, )) 2.7) β ) x < ε fx,, ws, t)dsdt, wx, t)dt, wx, )) β ) w X; x, ) and x, ) P satisf the relations x x < δ P ε) and < δ P ε). Consider the set X X possessing the following propert: for each bounded subset P Q, ε > and x x < δ P ε), < δ P ε), x, ), x, ) P, the inequalit 2.8) holds for ever w X. wx, ) wx, ) 1 LP )) 1 ε

5 On the Picard problem for hperbolic The set X is a closed, convex and almost equicontinuous subset of CQ, E). To appl the fixed point theorem of B.N. Sadovskii [6] we define the continuous mapping T : CQ, E) CQ, E) b the formula 2.9) T w)x, ) = f Let w X. Then T w)x, ) 2.1) G x,, G x,, β) β) x,, β) ws, t)dsdt, wx, t)dt, wx, ). ws, t) dsdt, wx, t) dt, wx, ) g s, t)dsdt, g x, t)dt, g x, ) g x, ). Furthemore, for ε > and x, ), x, ) P such that x x < δ P ε), < δ P ε) we have see 2.3),2.7) and 2.8)) T w)x, ) T w)x, ) 2.11) + fx,, wx, t)dt, wx, )+ wx, t)dt, wx, )) x fx,, ws, t)dsdt, wx, t)dt, wx, ))+ β ) x fx,, ws, t)dsdt, wx, t)dt, wx, )) β ) x fx,, ws, t)dsdt, β ) β ) ws, t)dsdt, LP ) wx, ) wx, ) + ε 1 LP ) 1 ε.

6 36 A. Sadowski Thus, the inclusion TX ) X holds. Let n be a positive integer and let W be a nonempt subset of X. Put = [, M n ] [, M n ], k n = k ) and L n = L ). Now we shall show the basic inequalit see [5]): 2.12) sup exp p n )αt W x, ))) p 1 ) n k n M n + 1) + 2L n sup exp p n )αw x, )) where p n > n = 1, 2,...). B Lemma 2.2 see [5]), we obtain for a fixed x, ) the following inequalit α W s, t)dsdt 2.13) β) β) Mn Analogousl, we have α 2.14) exp p n t) expp n t)αw s, t))dsdt exp p n t) expp n t)αw s, t))dsdt sup exp p n t)αw s, t))) Mn expp n t)dsdt p 1 n M n expp n )supexp p n t)αw s, t))). = ) W x, t)dt αw x, t))dt exp p n t) expp n t)αw x, t))dt sup exp p n t)αw s, t))) expp n t)dt p 1 n expp n ) sup exp p n t)αw s, t)).

7 On the Picard problem for hperbolic The inequalit 2.12) is a simple consequence of 2.13), 2.14) and Lemma 2.1. Let p n > 1 2L n ) 1 k n M n + 1) n = 1, 2,...). Define ΦW ) = sup exp p 1 )αw x, )), sup exp p 2 )αw x, )),...) P 1 P 2 for an nonempt subset W of X. B Ascoli s theorem, the properties of α and inequalit 2.12) it follows that all assumptions of B.N. Sadovskii s fixed point theorem are satisfied. Consequentl, the mapping T has a fixed point in X. The proof of the Theorem is complete. References [1] A. Ambrosetti, Un teorema di essistenza per le equazioni differenziali nagli spazi di Banach, Rend. Sem. Mat. Univ. Padova ), [2] K. Goebel, W. Rzmowski, An existence theorem for the equations x = ft, x) in Banach space, Bull. Acad. Polon. Sci., Sér. Sci. Math ), [3] P. Negrini, Sul problema di Darboux negli spazi di Banach, Bolletino U.M.I. 5) 17-A 198), [4] B. Rzepecki, Measure of Non-Compactness and Krasnoselskii s Fixed Point Theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math ), [5] B. Rzepecki, On the existence of solution of the Darboux problem for the hperbolic partial differential equations in Banach Spaces, Rend. Sem. Mat. Univ. Padova ). [6] B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surves ), Received 25 April 23