Discussiones Mathematicae Differential Inclusions, Control and Optimization 23 23 ) 31 37 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},
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.
On the Picard problem for hperbolic... 33 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
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
On the Picard problem for hperbolic... 35 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.
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)).
On the Picard problem for hperbolic... 37 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 39 1967), 349 36. [2] K. Goebel, W. Rzmowski, An existence theorem for the equations x = ft, x) in Banach space, Bull. Acad. Polon. Sci., Sér. Sci. Math. 18 197), 367 37. [3] P. Negrini, Sul problema di Darboux negli spazi di Banach, Bolletino U.M.I. 5) 17-A 198), 156 16. [4] B. Rzepecki, Measure of Non-Compactness and Krasnoselskii s Fixed Point Theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math. 24 1976), 861 866. [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 76 1986). [6] B.N. Sadovskii, Limit compact and condensing operators, Russian Math. Surves 27 1972), 86 144. Received 25 April 23