Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

Abstract

This paper is concerned with the existence of mild and strong solutions on the interval <svg style="vertical-align:-2.22495pt;width:38.037498px;" id="M1" height="14.85" version="1.1" viewBox="0 0 38.037498 14.85" width="38.037498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,5.927,12.012)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.086,12.012)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,20.784,12.012)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g><g transform="matrix(.017,-0,0,-.017,32.105,12.012)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g> </svg> for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carath&#xeb;odory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval <svg style="vertical-align:-2.22495pt;width:38.674999px;" id="M2" height="15" version="1.1" viewBox="0 0 38.674999 15" width="38.674999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,5.927,12.162)"><path id="x1D6FF" d="M494 514l-10 -15q-56 76 -106.5 117t-110.5 41q-45 0 -45 -30q0 -28 109 -152q59 -68 86 -118.5t27 -107.5q0 -62 -29.5 -119.5t-86.5 -96.5q-64 -45 -138 -45q-76 0 -121.5 53t-45.5 136q0 74 40.5 135t98.5 95q43 26 87 42q-40 59 -57.5 92t-17.5 62q0 46 39 77.5&#xA;t90 31.5q54 0 97.5 -35.5t65 -78.5t28.5 -84zM359 234q0 84 -90 191q-47 -17 -74 -47q-81 -89 -81 -200q0 -71 32.5 -109.5t72.5 -38.5q47 0 80 33t46.5 77.5t13.5 93.5z" /></g><g transform="matrix(.017,-0,0,-.017,14.715,12.162)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,21.413,12.162)"><use xlink:href="#x1D447"/></g><g transform="matrix(.017,-0,0,-.017,32.734,12.162)"><use xlink:href="#x5D"/></g> </svg> for <svg style="vertical-align:-0.1638pt;width:36.5px;" id="M3" height="12.4375" version="1.1" viewBox="0 0 36.5 12.4375" width="36.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FF"/></g><g transform="matrix(.017,-0,0,-.017,13.56,12.162)"><path id="x3E" d="M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z" /></g><g transform="matrix(.017,-0,0,-.017,28.263,12.162)"><use xlink:href="#x30"/></g> </svg> that is small enough, and, then, by letting <svg style="vertical-align:-0.1638pt;width:48.400002px;" id="M4" height="12.4375" version="1.1" viewBox="0 0 48.400002 12.4375" width="48.400002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FF"/></g><g transform="matrix(.017,-0,0,-.017,16.398,12.162)"><path id="x2192" d="M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z" /></g><g transform="matrix(.017,-0,0,-.017,40.163,12.162)"><use xlink:href="#x30"/></g> </svg> and using a diagonal argument, we have the existence results on the interval <svg style="vertical-align:-2.22495pt;width:38.037498px;" id="M5" height="14.85" version="1.1" viewBox="0 0 38.037498 14.85" width="38.037498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,5.927,12.012)"><use xlink:href="#x30"/></g><g transform="matrix(.017,-0,0,-.017,14.086,12.012)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,20.784,12.012)"><use xlink:href="#x1D447"/></g><g transform="matrix(.017,-0,0,-.017,32.105,12.012)"><use xlink:href="#x5D"/></g> </svg>. This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.